Probability in quantum physics
9.1. Emergence of the wave theory of probability
This section critically reviews the history of quantum physics. The development of its basic concepts involved a good number of years of concerted efforts
on the part of many theoretical and experimental physics in many countries. The basic ideas of the theory were essentially established during a thirty year period (1900-1930), yet at the end of the 20th century there was still no agreement about the interpretation
of its foundations. The subsequent sections discuss the mathematical framework, which was basically established in the years 1925-1930 by physicists such as Louis de Broglie, Erwin Schrödinger, Werner Heisenberg, Max Born, Wolfgang Pauli, Pascual Jordan,
and Paul Dirac, and by mathematicians such as John von Neumann. The so-called
Hilbert-space representation is not the only one, but suffices for the purpose
to show the probabilistic character of quantum physics, and to point out how it differs from classical probability theory (chapter 8).
general theory of quantum physics addresses five related problems:
(a) To find the spectrum of possible properties of the system under
(b) To give an objective description of the (initial) state of the system, to the extent that it is specified, and to
the extent that it is at random (9.4).
(c) To determine the relative statistical weights associated with the possible properties of
the system relative to its state. This implies the discussion of the external (modal) symmetries (9.5, 9.6) as well as the internal structure, partly expressed by internal symmetries (9.7).
(d) To determine the temporal development of the state during the time from one interaction to the next, and to treat the problem of interference (9.8).
(e) To explain the actualization of one of the possible properties via an interaction, which implies the distinction between possessed properties and latent propensities (9.8).
It is most remarkable that at least the first four problems can be treated within the context of a single concept, that of a complex Hilbert space, with
its associated hermitean operators. This concept is an abstract one, and has many realizations. All Hilbert spaces with the same number of dimensions are isomorphic to each other. The basic hypothesis of quantum physics says that the set of possible states
of a system is isomorphic to all Hilbert spaces of a certain dimensionality, which depends on the typical structure of the system.
Any property of the system is related to a coordinate system (a set of basis functions) in Hilbert space, such that the property’s spectrum
is related to the dimension of that space. Properties with a number of possible values less than the dimension of the Hilbert space are called degenerate for that system. Degeneracy is always connected to some kind of symmetry.
The probability associated with a certain value of some property is determined jointly by the spectrum of that property and the state at the moment the interaction
revealing (probing) that property takes place. Thus, while the concept of a Hilbert space provides the description of probabilities in an isolated system, at the same time it anticipates interaction. Its use breaks down as soon as we want to investigate the
interaction itself. So the fifth problem mentioned above is at best partly solved.
Operators in Hilbert space
A Hilbert space is the dense and complete set of all linear combinations of a number of basis functions with complex coefficients (2.7). In this space for any pair of functions
f1 and f2 a linear functional (f1,f2) exists and is called the scalar product. Now the concept of a linear operator is introduced as a mapping of the Hilbert space onto itself,
more or less similar to a rotation in an Euclidean space of two or three dimensions.
If f and g are arbitrary functions in the Hilbert space H, then A is a linear operator if it transforms the function f into Af such that Af is a function in H. The identity operator I
transforms each function into itself, and the zero operator reduces each function to zero. All linear operators in a Hilbert space form a group with respect to addition, with the zero operator as identity element, and with –A=(-1)A
as the inverse of A. In general, multiplication of operators is not commutative. A and B are said to commute if AB=BA.
So the fifth problem mentioned above is at best partly solved.
Quantum physics is especially interested in hermitean operators
(for which A=A+, the adjoint operator)and in unitary
operators (defined by UU+=I, the identity operator).
Hermitean operators, eigenvectors and eigenvalues
An operator is called hermitean
or self-adjoint if for any pair of functions f and g in H, (f,Ag)=(Af,g). Each hermitean operator generates a basis in the Hilbert space. This means, for any hermitean operator A there exist vectors ni such that Ani=aini,
where ai is a real number. If normalized, the so-called eigenvectors or eigenfunctions ni of A have the properties of basis functions in H: (ni,ni)=1, (ni,nj)=0, for any i and j, i≠j. Moreover,
the set of ni’s is complete, which means that any function in the Hilbert space can be written as a linear combination of those eigenvectors.
The real eigenvalues ai can serve to distinguish the eigenvectors. Therefore, if two mutually orthogonal eigenvectors have the same eigenvalue, all vectors in the two-dimensional space consisting of the linear combinations of these
two eigenvectors are also eigenvectors. Hence an eigenvalue determines a subspace in Hilbert space, whether one-dimensional (non-degenerate eigenvalue) or multi-dimensional (degenerate eigenvalue).
If two hermitean operators commute, they have the same set of eigenvectors, but with different eigenvalues, and different degeneracy. To every unit vector ni of a basis in Hilbert space is connected
a hermitean operator Pi, which transforms any function into its projection onto that unit vector. Thus, because f=Σ(f,ni) ni, we have Pif=(f,ni)ni. For the basis vectors themselves, Pini=1
and Pinj=0 if i≠j. Hence the eigenvalues of Pi are either one or zero (the latter is highly degenerate), and the projection operators can be used to describe yes-no experiments.
A unitary operator U is defined by the property UU+=I, the identity operator.
Thus unitary operators can form a multiplication group with I as the identity element, and U+ as the inverse of U. The application of a unitary operator to a basis leads to a new basis having the same the orthogonality
and normalization properties. With this change of basis, a hermitean operator A is transformed into U+AU. If A and U commute, A is not changed (U+AU=U+UA=A). Therefore,
unitary operators are very useful in describing symmetry operations, in which transformations of the state of the system are made without changing its properties.
Unitary operators turn out to be particularly useful for the description of the spatial and temporal homogeneity for isolated systems (9.5). For spatial isotropy one finds that the degeneracy of eigenvalues is not complete,
so that the corresponding unitary operator is a two- or more-dimensional matrix (9.6).
9.2. Problems concerning wave packets
Probability is a measure over an ensemble, a set of possibilities (8.3). If the set of possibilities is continuous this is a field over a space or a region
in a space. Stating this once again shows the static character of classical probability theory, and points, at the same time, to the way to open it up in a kinematic sense. For the kinematics of a field leads to the theory of waves, in particular the concept
of a wave packet as an aggregate of waves (chapter 7). This theory is not only applicable to physical fields in physical space, but to any field in any space, including probability. The actualisation of any possibility requiring a physical interaction finishes
the development of probability.
Many sounds are signals. A signal being a pattern of oscillations moves as an aggregate of waves from the source
to the detector. This motion has a physical aspect as well, for the transfer of a signal requires energy. But the message is written in the oscillation pattern, being a signal if a person or an animal receives and recognizes it.
A signal composed from a set of periodic waves is called a wave packet. Although a wave packet is a kinetic subject, it achieves its foremost meaning if its physical interaction
is taken into account. The wave-particle duality has turned out to be equally fundamental and controversial. Neither experiments nor theories leave room for doubt about the existence of the wave-particle duality. However, it seems to contradict common sense,
and its interpretation has been the object of hot debates.
Common sense dictated waves and particles to exclude each other, meaning that light
is either one or the other. When the wave theory turned out to explain more phenomena than the particle model, the battle appeared to be over (T&E, 6.3, 6.4).
Light is wave motion, as was confirmed by Maxwell’s theory of electromagnetism. Nobody realized that this conclusion was a non sequitur. At most, it could be said that light has wave properties, as follows from the interference experiments of Young and
Fresnel, and that Newton’s particle theory of light was refuted.
A dualistic world view
19th-century physics discovered and investigated many other rays. Some looked like light, such as infrared and ultraviolet radiation (about 1800), radio
waves (1887), X-rays and gamma rays (1895-96). These turned out to be electromagnetic waves. Other rays consist of particles. Electrons were discovered in cathode rays (1897), in the photoelectric effect and in beta-radioactivity. Canal rays consist of ions
and alpha rays of helium nuclei (T&E, 6.5).
At the end of the 19th century, this gave rise to a rather neat and rationally satisfactory world view. Nature consists partly of particles, for the other part of waves, or
of fields in which waves are moving. This dualistic world view assumes that something is either a particle or a wave, but never both, tertium non datur.
It makes sense to distinguish a dualism, a partition of the world into two compartments, from a duality, a two-sidedness. The dualism of waves and particles rested on common sense, one could not imagine an alternative. However,
20th-century physics had to abandon this dualism perforce and to replace it by the wave-particle duality. All elementary things have both a wave and a particle character (7.8).
Almost in passing, another phenomenon, called quantization, made its appearance. It turned out that some magnitudes are not continuously variable. The mass of an atom can only have a certain well-defined value. Atoms emit light
at sharply defined frequencies. Electric charge is an integral multiple of the elementary charge. In 1905 Albert Einstein suggested that light consists of quanta with energy E = hf. In Niels Bohr’s atomic theory (1913), the angular
momentum of an electron in its atomic orbit is an integer times Max Planck’s reduced constant.
Until Erwin Schrödinger and Werner Heisenberg in 1926 introduced modern quantum mechanics, repeatedly atomic scientists found new quantum numbers with corresponding rules.
Louis de Broglie
In 1923, Louis de Broglie published a mathematical paper about the wave-particle character of light. 
Applying the theory of relativity, he predicted that electrons too would have a wave character. The motion of a particle or energy quantum does not correspond to a single monochromatic wave but to a group of waves, a wave packet. The speed of a particle cannot
be related to the wave velocity (l/T=ƒ/s), being larger than the speed of light for a material particle. Instead, the particle speed corresponds to the speed of the wave packet, the group velocity. This is the derivative of
frequency with respect to wave number (df/ds) rather than their quotient. Because of the relations of Planck and Einstein, this is the derivative of energy with respect to momentum as well (dE/dp). At most, the group velocity equals
the speed of light.
In order to test these suggestions, physicists had to find out whether electrons show interference phenomena. Experiments by Clinton Davisson and Lester Germer in America and by George P. Thomson in England
(1927) proved convincingly the wave character of electrons, thirty years after Thomson’s father Joseph J. Thomson established the particle character of electrons. As predicted by De Broglie, the linear momentum turned out to be proportional to the wave
number. Afterwards the wave character of atoms and nucleons was demonstrated experimentally.
It took quite a long time before physicists accepted
the particle character of light. Likewise, the wave character of electrons was not accepted immediately, but about 1930 no doubt was left among pre-eminent physicists.
This meant the end of the wave-particle (or matter-field) dualism, implying all phenomena to have either a wave character or a particle character, and the beginning of wave-particle duality being a universal property of
matter (7.8). In 1927, Niels Bohr called the wave and particle properties complementary.
Bohr also asserted that measurements can only be analyzed in classical mechanical terms, using arguments derived from Immanuel Kant.
The dual character of
An interesting aspect of a wave is that it concerns a movement in motion, a propagating oscillation. Classical mechanics
restricted itself to the motion of unchangeable pieces of matter. For macroscopic bodies like billiard balls, bullets, cars and planets, this is a fair approximation, but for microscopic particles it is not.
The experimentally established fact of photons, electrons, and other microsystems having both wave and particle properties does not fit the still popular mechanistic world view. However, the theory of characters (10.2) accounts for this fact as follows.
The character of an electron consists of an interlacement of two characters, a generic kinetic wave character and an accompanying specific
particle character that is physically qualified (7.8). The specific character (different for different physical kinds of particles), determines primarily how the particles concerned interact with other physical subjects, and secondarily which magnitudes play
a role in interaction. These characteristics distinguish electrons from other particles like muons, from spatially founded systems like protons and atoms, and from photons and similar particles having a kinetic foundation.
Interlaced with the specific character is a pattern of motion having the kinetic character of a wave packet. Electrons share this generic character with all other particles.
In experiments demonstrating the wave character, there is little difference between electrons, protons, neutrons, or photons. The generic wave character has primarily a kinetic qualification and secondarily a spatial foundation. The specific physical
character determines the boundary conditions and the actual shape of the wave packet. Its wavelength is proportional to its linear momentum, its frequency to its energy. A free electron’s wave packet looks different from that of an electron bound in
a hydrogen atom. The wave character representing the electron’s motion has a tertiary characteristic as well, anticipating physical interaction. The wave function describing the composition of the wave packet determines the probability of the electron’s
performance as a particle in any kind of interaction.
Properties of wave packets
A purely periodic wave is infinitely extended in both space and time. It
is unfit to give an adequate description of a moving particle, being localized in space and time. A packet of waves having various amplitudes, frequencies, wavelengths, and phases delivers a pattern that is more or less localized. The waves are superposed
such that the net amplitude is zero almost everywhere in space and time. Only in a relatively small interval (to be indicated by Δ) the net amplitude differs from zero.
Rectilinear motion of a wave packet at constant speed is described by four magnitudes. These are the position (x) of the packet at a certain instant of time (t), the wave number (s) and the frequency (f).
The packet is an aggregate of waves with frequencies varying within an interval Δf and wave numbers varying within an interval Δs.
Generally, the wave packet in the direction of motion has a minimum dimension Δx such that Δx.Δs>1. In order to pass a certain point, the packet needs a time Δt, for which Δt.Δf>1.
If the packet is compressed (Δx and Δt small), the packet consists of a wide spectrum of waves (Δs and Δf large). Conversely, a packet with a well defined frequency (Δs and Δf small) is
extended in time and space (Δx and Δt large). It is impossible to produce a wave packet whose frequency (or wave number) has a precise value, and whose dimension is simultaneously point-like. If the variation Δs is small,
the length of the wave packet Δx is large, whereas if the packet is localized, the wave number needs to show a large variation.
a wave packet is longer than one might believe. A photon emitted by an atom has a dimension of Δx=cΔt, Δt being equal to the mean duration of the atom’s metastable state before the emission. Because
Δt is of the order of 10-8 sec and c=3*108 m/sec, the photon’s ‘coherence length’ in the direction of motion is several metres. This is confirmed by interference experiments, in which the photon
is split into two parts, to be reunited after the parts have transversed different paths. If the path difference is less than a few metres, interference will occur, but this is not the case if the path difference is much longer. The coherence length of photons
in a laser ray is many kilometres long, because in a laser, Δt has been made artificially long.
An oscillating system emits
or absorbs a wave packet as a whole.During its motion, the coherence of the composing waves is not always spatial. A wave packet can split itself without losing its kinetic coherence. This coherence is expressed by phase relations, as can
be demonstrated in interference experiments as described above. In general, two different wave packets do not interfere in this way, because their phases are not correlated. This means that a wave packet maintains its kinetic identity during
its motion. The physical unity of the particle comes to the fore when it is involved in some kind of interaction, for instance if it is absorbed by an atom causing a black spot on a photographic plate or a pulse in a Geiger-Müller counter. Emission and
absorption are physically qualified events, in which an electron or a photon acts as an indivisible whole.
The identification of a particle
with a wave packet seems to be problematic for various reasons. The first problem, the possible splitting and absorption of a wave packet, is mentioned above.
Second, the wave packet of a freely moving particle always expands, because the composing waves having different speeds.
Even if the wave packet is initially well localized, gradually it is smeared out over an increasing part of space and time. However, the assumption that the wave function satisfies a linear wave equation is a simplification of reality. Wave motion can be non-linearly
represented by a ‘soliton’ that does not expand. Unfortunately, a non-linear wave equation is mathematically more difficult to treat than a linear one.
Third, in 1926 Werner Heisenberg observed that the wave packet is subject to a law known as indeterminacy relation, uncertainty relation or Heisenberg relation (7.6). As a matter of fact, there is as little agreement about its definition
as about its name.
Combining the relations Δx.Δs>1 and Δt.Δf >1 with those of Planck
(E=hf) and Einstein (p=hs) leads to Heisenberg’s relations for a wave packet: Δx.Δp>h and Δt.ΔE>h.
The meaning of Δx etc. is given above. In particular, Δt is the time the wave packet needs to pass a certain point.
This interpretation is the oldest one, for the indeterminacy relations – without Planck’s constant - were applied in communication theory long before the birth of quantum mechanics.
It is interesting to observe that the indeterminacy relations are not characteristic for quantum mechanics, but for wave motion. The relations are an unavoidable consequence of the wave character of particles and of signals. I prefer this interpretation because
it remains closest to experimental facts, but I shall discuss some alternative, more theoretical, interpretations, in particular paying attention to the Heisenberg relation between energy and time.
Quantum mechanics connects any variable magnitude with a Hermitean operator having eigenfunctions and eigenvalues (2.4, 9.1). The eigenvalues are the
possible values for the magnitude in the system concerned. In a measurement, the scalar product of the system’s state function with an eigenfunction of the operator is the square of the probability that the corresponding eigenvalue will be realized.
If two operators act successively on a function, the result may depend on their order. The Heisenberg relation Δx.Δp>h
can be derived as a property of the non-commuting operators for position and linear momentum. In fact, each pair of non-commuting operators gives rise to a similar relation. This applies, e.g., to each pair out of the three components of angular momentum. Consequently, only one component of an electron’s magnetic moment (usually
along a magnetic field) can be measured. The other two components are undetermined, as if the electron exerts a precessional motion about the direction of the magnetic field.
Remarkably, there is no operator for kinetic time. Therefore, some people deny the existence of a Heisenberg relation for time and energy.
On the other hand, the operator for energy, called Hamilton-operator or Hamiltonian, is very important. Its eigenvalues are the energy levels characteristic for e.g. an atom or a molecule. Each operator commuting with the Hamiltonian represents a constant
of the motion subject to a conservation law (5.3).
Mean standard deviation
From the wave function, the probability to find a particle in a certain state can be calculated. Now the indeterminacy is a measure of the mean standard deviation, the statistical
inaccuracy of a probability calculation. The indeterminacy of time can be interpreted as the mean lifetime of a metastable state. If the lifetime is large (and the state is relatively stable), the energy of the state is well defined. The rest energy of a short
living particle is only determined within the margin given by the Heisenberg relation for time and energy.
This interpretation is needed to
understand why an atom is able to absorb a light quantum emitted by another atom in similar circumstances. Because the photon carries linear momentum, both atoms get momentum and kinetic energy. The photon’s energy would fall short to excite the second
atom. Usually this shortage is smaller than the uncertainty in the energy levels concerned. However, this is not always the case for atomic nuclei. Unless the two nuclei are moving towards each other, the process of emission followed by absorption would be
impossible. Rudolf Mössbauer discovered this consequence of the Heisenberg relations in 1958. Since then, the Mössbauer effect became an effective instrument for investigating nuclear energy levels.
The position of a wave packet is measurable within a margin of Δx and its linear momentum within a margin of Δp. Both are as small as experimental
circumstances permit, but their product has a minimum value determined by Heisenberg’s relation. The accuracy of the measurement of position restricts that of momentum.
Initially the indeterminacy was interpreted as an effect of the measurement disturbing the system. The measurement of one magnitude disturbs the system such that another magnitude cannot be measured with an unlimited
accuracy. Heisenberg explained this by imagining a microscope exploiting light to determine the position and the momentum of an electron.
Later, this has appeared to be an unfortunate view. It seems better to consider the Heisenberg relations to be the cause of the limited accuracy of measurement, rather than to be its effect.
The Heisenberg relation for energy and time has a comparable consequence for the measurement of energy. If a measurement has duration Δt, its accuracy cannot be better than ΔE>h/Δt.
The so-called measurement problem constitutes the nucleus of what is usually called the interpretation of quantum mechanics.
It is foremost a philosophical problem, not a physical one, which is remarkable, because measurement is part of experimental physics, and the starting point of theoretical physics. After the development of quantum physics, both experimental and theoretical
physicists have investigated the relevance of symmetry, and the structure of atoms and molecules, solids and stars, and subatomic structures like nuclei and elementary particles (chapter 11). Apparently, this has escaped the attention of many philosophers,
who are still discussing the consequences of Heisenberg’s indeterminacy relations.
The law of conservation of energy
In quantum mechanics,
the law of conservation of energy achieves a slightly different form. According to the classical formulation, the energy of a closed system is constant. In this statement, time does not occur explicitly. The system is assumed to be isolated for an indefinite
time, and that is questionable. Heisenberg’s relation suggests a new formulation. For a system isolated during a time interval Δt, the energy is constant within a margin of ΔE≈h/Δt. Within this
margin, the system shows spontaneous energy fluctuations, only relevant if Δt is very small.
According to quantum field theory, a physical vacuum is not an empty space. Spontaneous fluctuations may occur. A fluctuation leads to the creation and
annihilation of a virtual photon or a virtual pair consisting of a particle and an antiparticle, having an energy of ΔE, within the interval Δt<h/ΔE. Meanwhile the virtual particle or pair is able to
exert an interaction, e.g. a collision between two real particles. Virtual
particles are not directly observable but play a part in several real processes.
The probability interpretation of the wave packet
The amplitude of waves in water,
sound, and light corresponds to a measurable physical real magnitude. In water this is the height of its surface, in sound the pressure of air, in light the electromagnetic field strength. The energy of the wave is proportional to the square of the
amplitude. This interpretation is not applicable to the waves for material particles like electrons. In this case the wave has a less concrete character, it has no direct physical meaning. The wave is not even expressed in real numbers, for the wave function
has a complex value.
In 1926, Max Born offered a new interpretation, since then commonly accepted.
He stated that a wave function (real or complex) is a probability function. In a footnote added in proof, Born observed that the probability is proportional to the square of the wave function.
A wave function is prepared at an earlier interaction, for instance, the emission of the particle. It changes during its motion, and one of its possibilities
is realized at the next interaction, like the particle’s absorption. The wave function expresses the transition probability between the initial and the final state.
This probability may concern any measurable property that is variable. Hence, it does not concern natural constants like the speed of light or the charge
of the electron. According to Born, the probability interpretation bridges the apparently incompatible wave and particle aspects.
Wave properties determine the probability of position, momentum, etc., traditionally considered properties of particles.
used statistics as a mathematical means, assuming that the particles behave deterministic in principle (8.1). In 1926, Born’s probability interpretation put a definitive end to mechanist determinism, having lost its credibility before because of radioactivity.
Waves and wave motion are still determined, e.g. by Schrödinger’s equation, even if no experimental method exists to determine the phase of a wave. However, the wave function determines only the probability of future interactions.
In quantum mechanics, the particles themselves behave stochastically.
An even more strange feature of quantum statistics is that chance is
subject to interference. In the traditional probability calculus (8.3) probabilities can be added or multiplied. Nobody ever imagined that probabilities could interfere. Interference of waves may result in an increase of probability, but to a decrease as well,
even to the extinction of probability. Hence, besides a probability interpretation of waves, there is a wave interpretation of probability.
Outside quantum mechanics, this is still unheard of, not only in daily life and the humanities, but in sciences like biology and ethology as well. The
reason is that interference of probabilities only occurs as long as there is no physical interaction by which a chance realizes itself.
The absence of physical interaction is an exceptional situation. It only occurs if the system concerned has no internal interactions (or if these are frozen), as long as it moves freely. In macroscopic bodies, interactions occur continuously and interference
of probabilities does not occur. Therefore, the phenomenon of interference of chances is unknown outside quantum physics.
Reduction of the wave
The mathematical concept of probability or chance anticipates the physical relation frame, because only by means of a
physical interaction a chance can be realized. An open-minded spectator observes an asymmetry in time. Probability always concerns future events. It draws a boundary line between a possibility in the present and a realization in the future. For this realization,
a physical interaction is needed. The wave equation and the wave function describe probabilities, not their realization. The wave packet anticipates a physical interaction leading to the realization of a chance, but is itself a kinetic subject, not a physical
subject. If the particle realizes one of its possibilities, it simultaneously destroys all alternative possibilities. In that respect, there is no difference between quantum mechanics and classical theories of probability.
As long as the position of an electron is not determined, its wave packet is extended in space and time. As soon as an atom absorbs the electron at a certain position, the
probability to be elsewhere collapses to zero. This so-called reduction
of the wave packet requires a velocity far exceeding the speed of light. However, this reduction concerns the wave character, not the physical character of the particle. It does not counter the physical law that no material particle can move faster than light.
Likewise, Schrödinger’s equation describes the states of an atom or molecule and the transition probabilities between states. It does not account
for the actual transition from a state to an eigenstate, when the system experiences a measurement or another kind of interaction.
Is the problem of the reduction of the wave packet relevant for macroscopic bodies as well? Historically, this question is concentrated in the problem
of Schrödinger’s cat, locked up alive in a non-transparent case. A mechanism releases a mortal poison at an unpredictable instant, for instance controlled by a radioactive process. As long as the case is not opened, one may wonder whether the cat
is still alive. If quantum mechanics is applied consequently, the state of the cat is a mixture, a superposition of two eigenstates, dead and alive, respectively.
The principle of decoherence, discovered at the end of the 20th century, provides a satisfactory answer. For a macroscopic body, a state being a combination of eigenstates will spontaneously change very fast into an eigenstate, because of
the many interactions taking place in the macroscopic system itself. This solves the problem of Schrödinger’s cat, for each superposition of dead and alive transforms itself almost immediately into a state of dead or alive. The principle of decoherence is part of a realistic interpretation of quantum physics.
It does not idealize the ‘collapse’ or ‘reduction’ of the wave packet to a projection in an abstract state space. It takes into account the character of the macroscopic system in which a possible state is realized by means of a physical
As soon as it was established, both experimentally and theoretically, that all particles move like wave packets,
Born recognized these waves as expressing probability. It should be emphasized that as long as only kinematics is concerned, this interpretation is not needed. Therefore the wave theory was discussed separately (chapter 7), because it is of a purely
modal character – in contrast to probability theory. The relation of wave theory and probability concerns the kinematic aspect of the latter, and manifests itself if one wishes to account for the individuality of the physically qualified particles.
Both theories have an anticipatory character. For the wave motion of physically qualified individual particles this implies that the probability interpretation becomes necessary as soon as one wishes to consider the
individual interaction of the particles with some other system (for example, but not necessarily, a measuring apparatus). A purely kinematic theory breaks down for this interaction.
The theory of waves as applicable to the kinematics of probability is not restricted to sinusoidal or spherical waves. It is the phase which shows the modal kinematic character of wave theory. The phase is not directly
relevant for the probability, which is only determined by the amplitude. The most peculiar consequence of the phase formalism is that it leads to interference. Because probability is no longer static, but can be propagated, the addition of probabilities has
to be considered.
In classical theory, addition of probabilities can only lead to an increase, because probability is essentially a positive
measure. But the phase relations between interfering waves make it possible that waves may annihilate each other. In order to save the positive-definiteness of probability as a measure, it is defined as the square of the absolute value of the amplitude of
the wave. (It is now also clear why wave packets must be normalized to unity). Especially this interference of probability, completely unknown in classical static probability theory, shows the relevance of the kinetic aspect of probability. It is one important
new feature of quantum statistics. A second innovation concerns the description
of the transition from an initial state to a final state (9.8), which is also intimately related to wave theory.
9.3. Static quantum probability theory
In order to discuss the physical interpretation of this formal, mathematical scheme, one has to state in which way this formalism
is an objective representation of physical states of affairs. Such a statement has the character of a hypothesis, which can never be proved analytically, but must be corroborated by its consistency with other theories and experimental results.
Section 3.2 defined a magnitude as a property of comparable subjects, which allows a quasi-serial ordering of those subjects. Now one asserts
that any physical magnitude for a system is represented by a hermitean operator A in a Hilbert space. The eigenvalues of A (whether discrete, denumerable, or continuous) compose the spectrum of possible real values for some property of that
particular system – i.e., the spectrum is in part determined by the typical structure of the system. This interpretation rests on the fact that the basis functions of any orthogonal basis in a linear function space can always be quasi-serially ordered.
The state of an isolated system corresponds to a normalized function or vector in the Hilbert space.
Non-isolated systems can be included, if the action from the outside is static (e.g., an electric or magnetic field), and if there is no reaction of the system to the outside world.
If f represents the state of the system, the statistical distribution over the spectrum of a physical magnitude is determined by the
scalar product of f and the eigenvectors of the corresponding operator A. Therefore, for an eigenvector ni of A, whose eigenvalue is ai, (ni,f)(f,ni)
is the probability that the subject will adopt the value ai for A by virtue of an interaction in which that magnitude is displayed. This applies in particular, but not exclusively, to a measurement of this magnitude.
The state functions f are restricted to normalized vectors in Hilbert space because the probability must be normalized. This can be seen because
Σ(nj,f)(f,nj)=(f,f) includes all possibilities, and must therefore be one. This means that if f itself happens to be an eigenvector of A with eigenvalue af, then the
probability of finding af for the magnitude is unity, and the probability of finding any other value is zero. (This ‘certainty’ in finding a particular value for a property is only possible for the idealized pure states). The
mean value for the measured property with respect to a given initial state f is simply (f,Af).
From classical properties to quantum propensities
far the static theory already shows some remarkable differences from the classical theory discussed in section 8.3. In the first place, the spectrum of possible values for a property of a system reflects more directly its spatial character, being represented
by a coordinate system, a basis in Hilbert space. It still refers back
to the numerical modal aspect, as is fitting for a magnitude. The spectrum of possibilities, being determined, as it is, by the internal typical structure of the system, is shifted therefore to the law side. Hence, the spectrum, rather than being primarily
a subject-subject relation, determines the latter. Consequently, a magnitude is not a property, but a propensity or disposition. No longer does it belong to the subject, but rather to the typical law to which the subject responds.
This is an important point to keep in mind in our forthcoming discussion.
Relations between states
Also the concept of probability itself, as used here, differs
from the classical one, in which probability is a function over the possibilities. In quantum physics, probability is a functional, a scalar product between the state function f and the eigenvector corresponding to a particular possible
case. Thus in quantum physics probability is a relation between the state of a system and the state of some reference system. It only has meaning if interpreted as some propensity, which can manifest itself as a property only in an interaction. Actually,
in classical theory as well, the probability is determined by both the spectrum of possibilities and the initial state, but the latter is assumed to be completely at random. In this respect the new theory is much richer than the old one, because it allows
specification of the initial state. Probability is a joint quantitative measure of the state of the system and the reference system.
the discovery that spatial position and kinematic motion are subject-subject relations, relative to spatial or kinematic reference systems, one finds now that the physical state of a system can only be understood relative to a physical reference system.
Such a system is objectively represented by a basis in Hilbert space, but is in any actual interaction a second physical system, with which the former system may interact.
This second system may be a measuring instrument, in the same way spatial coordinate systems and kinematic reference systems may be constructed from concrete metre sticks and clocks. But in an abstract analysis of physical interaction
it is not necessary to restrict oneself to measuring instruments, neither in geometry and kinematics, nor in physics. On the contrary, in each of these three cases the relation between a subject and a reference system or measuring instrument is merely one
of many concretizations of an abstract relation, the modal subject-subject relation. Indeed, for the spatial, kinetic, and physical modal aspects the modal subject-subject relation turns out to be, respectively, relative position, relative motion, and relative
interaction. Therefore it is justified to conclude that the theory has a relational character, and that the modal aspects are truly relation frames.
9.4. State preparation, randomness, and complementarity
The meaning of a state as represented by the state
function f is given in its relation to a reference system representing a possible second system, with which it may interact. The state itself is the result of a previous interaction, and is intermediary between the two interactions. In experimental
physics one speaks of a state selector as a device which e.g. singles out a certain state from an incoming beam of particles, excluding all other states. Hence it is a yes-no experiment, which can be described with a projection operator. Alternatively, a state
selector is characterized by an eigenvector of an operator representing the property according to which the incoming particles are discriminated. The state of the particles emerging from a pure-state selector is independent of the state of the incoming particles,
whereas the effect of a mixed-state selector depends partly on this preceding state.
Restricting oneself to a pure non-degenerate state selector (which gives the maximum obtainable determination of a state) one finds that this state represents
randomness in two ways. The first is connected with the phase of the state as follows. The scalar product (f,ni) determines the probability that a system in the state f will adopt the eigenvalue ai as a result
of an interaction characterized by the corresponding operator A. In particular, these probabilities are invariant under multiplication of the state vector by a complex number exp.iq, where q is a real number called the phase of the
exponential function. Therefore, in state selection, the phase associated with the state f is completely undetermined, and must be assumed to be a random parameter.
Next recall that the state f produced by a state selector is an eigenvector of the operator A corresponding with some property according
to which the systems are selected. This means that for a later interaction characterized by an operator B, f is not an eigenstate of B unless A and B commute. Thus, e.g., if A is the momentum operator,
and B the position operator, (which operators do not commute), then the particles emerging from the state selector may be said to have a precisely determined momentum, but not a precise position at any time before the next interaction takes place.
However, this statement is liable to be misunderstood, as the history of quantum physics has shown. The propensities of physically qualified subjects
as represented by hermitean operators have the character of a law, and therefore never belong to the subject. They display themselves only if the system interacts with some other subject. As long as we talk about an isolated system in the
state f, we have to assume that any propensity has a potential character, even if f happens to be a pure eigenstate of some operator. This latter case must be understood as a limiting case, and as such it is comparable with the state of rest
in kinematics and a one-dimensional space (a point) in geometry. As long as the system does not interact, the state is more or less autonomous with respect to any property. Indeed, the state has a potential
or latent character.
In this respect the quantum physical concept of a state is profoundly different from the classical molecular state, or the micro- or macrostates of macroscopic systems (8.5).
The classical state is a mere enumeration of actual properties of the system, whereas the state in quantum physics has a potential character for a single system. For an ensemble of similar systems in the same state, it determines the properties in the mean. It must be emphasized that this potential character of the state strictly
pertains to isolated systems only and is therefore somewhat abstract. Any concrete system always interacts with other systems in various ways, and therefore its state is always actualized in some sense or another.
The incompatibility of properties represented by non-commuting operators has given rise to the Copenhagen school, mainly represented by Niels Bohr, and strongly bears the influence of some
widely differing philosophies, in particular Kantian mechanism. As a result,
every proponent of complementarity has his own interpretation of it. One difficulty in understanding this concept is that it was introduced from the very beginning of the development of quantum physics. Therefore, its meaning has evolved along with the theory
For some people complementarity is the same as the wave-particle duality. For others it refers to the relation between the microsystems
governed by quantum physics and the macroscopic measuring instruments which are supposed to be describable in classical terms.
Sometimes it even denotes the psychic subject-object relation between the observer and the observed system. Shortly before the final establishment of Schrödinger’s wave theory and Heisenberg’s matrix mechanics in 1925, Bohr, Kramers and others
paid much attention to the complementary relation between a causal and a spatiotemporal description of physical processes. In its simplest form the principle of complementarity merely expresses the fact, mentioned above, that an eigenvector f of an
operator A cannot be an eigenvector of a different operator B, unless A and B commute.
The fact that the position and momentum operators do not commute has particularly been the cause of much discussion. In the first place, it refutes the
classical mechanist maxim that the state of any physical system must be describable in modal kinematic terms, i.e., by its actual position and momentum at a given time. In the Copenhagen school, in an effort to focus on the classical maxim, attempts
have sometimes been made to reduce all types of complementarity to this one of position and momentum (considered as ‘primitive’). But other kinds of incompatible magnitudes cannot be reduced in this way (9.6), and this idea had to be abandoned.
The need for a kinematic description disappears as soon as one recognizes the mutual irreducibility of the physical and kinetic modal aspects, which implies
that the state of a physically qualified system must be referred to a physical coordinate system and not to a kinetic one. The situation is quite analogous to Einstein’s criticism of the classical theory of motion, where in discussing kinetic
temporal relations, kinetic frames of reference have to be used, in which, e.g., static simultaneity is relativized. Similarly, for the study of the relations between physically interacting systems, one has to refer to physical frames of reference. This principle
regarding physical frames of reference has consequences even for temporal relations which are not original to the physical relation frame, such as position and momentum.
Thus the potential character of the state must be discussed taking into account all three basic distinctions of the philosophical framework introduced
in chapter 1. Distinguishing law side and subject side, leads to the discovery that the properties of a system have the character of a law, and cannot be possessed by the system, apart from its interaction with other systems. The distinction of modality and
typical individuality allows for the contingency of the state concurrently with the fact that the main properties in which one is interested (position, momentum, energy) posses a modal, universal nature. Differently structured systems can interact just because
they have such universal properties in common. Finally, the distinction of the relation frames helps to understand the relativization of the pre-physical modal relations in physically qualified relations.
9.5. Modal symmetry: energy and momentum
Both in classical and quantum
probability, symmetry considerations may help to solve the problem at hand. Symmetry can be divided into two types, modal or external, and typical or internal. In both cases we can translate the problem of symmetry into spatial terms. Then one inquires as
to which transformations in Hilbert space leave both the statistical distribution over some hermitean operator and its eigenvalue spectrum unchanged. All transformations of this kind are represented by unitary operators (which commute with the hermitean operator
being considered). Further, with any kind of symmetry, there corresponds a group of unitary operators. Thus the theory of groups is of great use for the solution of symmetry problems.
The case of modal symmetries involves transformations which depend on the mutual irreducibility of the first four modal aspects, whereas the case of typical symmetry depends on the typical structure of the system
concerned, and is therefore of a less general character (9.7). The modal symmetries depend on the isotropy and homogeneity of time and space, and the Galilean or Lorentz invariance of uniformly moving physical systems. Being concerned with isolated systems,
I shall postpone the discussion of the internal time evolution until section 9.8, and start with stationary states. This leads to the conservation laws of energy and momentum (the present section) and of angular momentum (9.6). These laws were discovered prior
to the quantum era, yet are far easier to derive in the quantum theory than in classical physics.
A unitary operator can be described by a
matrix which can be broken down into a number of matrices of smaller dimensionality. The simplest case is that of a matrix of dimension 1 whose element is a single complex number of absolute value 1. This one-dimensional matrix is nothing but the phase-factor,
the exponential function exp.iq, the phase q being some real number.
If we multiply all functions in Hilbert space with the
same phase-factor, then all scalar products (f,ni), as well as all mean values (f,Af) for any linear hermitean operator A, are invariant under such a transformation. This phase-factor invariance allows us to study the
conservation laws of energy and momentum for stationary states.
The temporal development of an isolated system
Consider an isolated system whose state
at a certain time t is denoted by the function f(t). We suppose that the state f(t1) develops continuously from the state f(to). Taylor’s theorem yields
which formally can be written as
= [exp i(t1–to)(1/i) ∂/∂t] f(to)=[exp.i(t1–to)H]f(to)
where we have introduced the hermitean Hamiltonian operator H=(1/i)∂/∂t as the generator of the unitary operators
describing the temporal homogeneity.
Now we demand that the states f(t1) and f(to) differ only by a phase factor. Because of the previous result, this phase factor can
only have the form of exp.iω(t1 – to), where ω is some real number:
where g(ω) is a Hilbert space vector independent of time, and characterized by the real number ω. In terms of g(ω), the general state vector is
If we apply the Hamiltonian H to f, we have
Thus the real number ω is an eigenvalue of H. Since ω can be any real number, the spectrum of H is continuous, and consists
of all positive and negative real numbers. The value of ω depends on the reference system, and may therefore be determined by a state selector. If the latter is incompatible with H or if it only produces mixed states, the state of the system
corresponds with a statistical distribution over the spectrum of H. In the pure-state case, ω is the same for all vectors in the Hilbert space representing the internal structure of the system. This space is therefore an invariant sub-space
of a much larger Hilbert space (of higher dimensionality) which includes all possible values for ω.
In a completely analogous way, the
system at different positions r (or relative to different spatial coordinate systems) can be considered:
where P=(1/i)V = (1/i)(∂/∂x,∂/∂y,∂/∂z).
If g(k) is a vector independent of position, characterized by the three-dimensional vector k, then the general state vector can be
represented by f(r,k)=(exp.ik.r)g(k)
is again a hermitean operator with eigenvalues k, whose spectrum covers triplets of all possible positive and negative real numbers. It is the generator of the unitary operators exp.ir.P, which form a multiplication
group isomorphic to the three-dimensional addition group of vectors k.
So far we have only studied the temporal and spatial
homogeneity of the systems, referred to different temporal and spatial frames. In order to include uniform motion, we first connect the two cases to obtain
This result is just the plane wave discussed in chapter 7. Thus in a very natural way one arrives at the plane wave representation of temporal and spatial homogeneity, which in classical physics can only be found
in a very laborious way. In chapter 7 we saw that the number ω has the character of a frequency and is proportional to energy, and that the vector k is the wave vector proportional to the linear momentum. That is to
say that ω and k are constants of the motion. By considering Galilean transformations between inertial reference systems moving relative to each other, one finds as in classical physics
ω=k2/2m, where k is the absolute value of k, and m is a frame-independent magnitude, called the mass of the system. Therefore, the system satisfies Schrödinger’s equation:
If the Lorentz instead of the Galilean transformations are considered, the relation between energy E and momentum p becomes E2=m2c4+p2c2.
This means that for a certain value of the momentum (or wave vector) there are two possible values of the energy (or frequency), one positive and one negative. This was first pointed out by Paul Dirac, and because this relation is universally valid, each particle
has a corresponding antiparticle, whose energy, or alternatively, charge is of opposite sign to that of its counterpart (11.6). Thus the positively charged positron corresponds to the negatively charged electron, and the negatively charged antiproton corresponds
to the positively charged proton. Additionally, as regards some properties of nuclear reactions, a particle is the antipode of its antiparticle, and vice versa. But essentially they have the same rest mass.
In general, a moving system will not be represented by a single vector, but by a mixture or wave packet
where the amplitudes A(k) and the phases φ are determined by the state selector.
Hence the absolute value of h(r,t)drdt is the probability that the system will be in the spatial interval between r and r+dr during the temporal interval
between t and t+dt. Thus the amplitudes A(k) are subjected to the normalization condition
In a similar way, the probability expresses that the energy and the linear momentum have values in a certain interval. The general theory of the Hilbert space formalism now leads to the same Heisenberg relations between the minimal
spreads in the statistical distributions of energy/momentum and time/position, as can be found from simple wave theory (chapter 7).
A somewhat more complicated problem is concerned with the rotational invariance of isotropic space. In classical physics it had
already been shown that this symmetry leads to the conservation of angular momentum. A group-theoretical analysis rules out a description of this symmetry in terms of one-dimensional matrices, which precludes the phase factor formalism.
At some stage of the development of quantum physics it was assumed that all relevant operators could be derived from those of energy, position and momentum, just as in classical
physics. One problem in the application of this so-called correspondence
principle is that the momentum and position operators do not commute with
each other, which has no analogy in classical physics. But even if this
difficulty is circumvented, one arrives at incorrect or incomplete results, as in the case of angular momentum.
In the ordinary representation
of classical physics for simple systems (consisting of mass points), the angular momentum is L=rxp, where r is the position vector, p the linear momentum, and rxp
is a vector product. Since the position operators involve simple multiplication by x, y and z, the correspondence principle yields the operator Lz=(1/i)(x∂/∂y-y∂/∂x)
for the z-component of the vector L. The corresponding eigenvalue equation can be solved. The eigenvalues are just the integers, m=+1, +2,
etc. This solution is wrong, in so far it gives only integral values for m, whereas experiments show that m can also have half-integral values, m=+(1/2), +(3/2), etc. It is impossible to find these half-integral values by merely looking for an analogy in classical mechanics.
arrives at the correct solution in two ways. First the angular momentum components, lx, ly, lz as defined above satisfy the commutation relations
lxly-lylx=ilz (cyclic in x,
y and z)
If this relation is not taken as a result, but as a starting point, one finds both integral and half-integral eigenvalues.
Alternatively, and without reference to classical physics, group theory shows that the unitary operators describing rotational invariance are reducible
to matrices with one or more dimensions. Matrices of even dimensionality correspond to half-integral values of the spin, and those of odd dimensionality to integral values. If the dimensionality is n, the eigenvalues are: -½(n–1),
…, +½(n–1), at unit intervals, such that the total number of different eigenvalues is just equal to n. The number n depends on the structure of the system, and is an invariant. The number ½(n–1),
the highest eigenvalue for each spin component, is also an eigenvalue of the total angular momentum operator J, which must be sharply distinguished from the three components lx, ly, lz.
Only those eigenstates of the system can be superposed which have the same eigenvalue of J. This is a so-called superselection rule,
to be discussed later. On the other hand, for example, the superposition of eigenstates with different eigenvalues for the operator lz is always possible. Thus for a spin-zero particle (e.g., a pion), n=1, J=0, and m=0.
For a spin one-half particle (e.g., an electron or a proton), n=2, J=½, and m can have two values: +½; and for a spin-one particle (e.g., a photon), n=3, J=1;
and m=+1 or 0. More complicated systems like nuclei and atoms may have states with different eigenvalues for J.
Although spin has a discrete spectrum, it is an external parameter just like energy and momentum. That is, it depends partly on the external reference system with respect to which the system is orientated. Now
in a physical sense orientation can only be given by a field, which in every spatial point has a certain direction (the z-direction, say). With respect to this direction, the lz component may be assumed to have a certain value,
whereas the fact that lz does not commute with lx and ly implies that in this case the state of the system (now an eigenstate of lz) cannot be an eigenstate of lx
or ly. The system cannot have spin components perpendicular to the field.
For some systems (like electrons) the spin is
associated with a magnetic moment, such that a magnetic field can orient the spin of the system. For other systems (such as light quanta), the direction of propagation is the determining factor. Light quanta can be transversely polarized in two mutually orthogonal
directions, corresponding with m=+1 (the third eigenvalue, m=0, can only be realized in longitudinal, virtual photons, as a consequence of the typical structure of electromagnetic interaction).
Indeterminacy of the spin
As observed, with every possible direction in space there corresponds a spin-component operator lz, which
does not commute with the other operators, lx and ly. Thus if a system is oriented with respect to a certain direction, it cannot at the same time be oriented with respect to another direction. Consider a beam of electrons
moving along the x-direction in the presence of a magnetic field pointing in the z-direction. This beam will split up into two parts, because the spin operator lz has two eigenvalues, +½. If one beam passes through another magnet parallel to the former, no further splitting will occur, which shows that after passing the first magnet all electrons in one beam are in an eigenstate of lz,
and remain so before and during their passage through the second magnet. But if the second magnet has its field in the y-direction, the beam will split again in two parts. If one of them passes a third magnet, again in the z-direction, a
beam splitting will occur, showing that the orientation with respect to the y-axis (in the second magnet) destroyed the earlier orientation with respect to the z-axis.
Maybe Ballentine overlooked this in his thought experiment in which he tries to show the possibility of determining the spin of a particle along two different directions, y and z.
His experiment proceeds as follows. Suppose two particles with J=½ emerge from a decaying particle of spin zero. This means (because of angular momentum conservation) that the two particles must have opposite spins. If the decaying particle
was at rest, the two emerging particles will also have opposite momentums. Now suppose one measures the spin of one particle with a magnet in the y-direction, and that of the other with a magnet in the z-direction. Then, according to Ballentine
we have determined both the y- and the z-components of the spin of each particle.
However, it appears that Ballentine is
wrong assuming that there is no magnetic field at the place where the original particle decays. Each emerging particle experiences the field of the other. Therefore, each particle is already oriented before it arrives at the measuring magnet. And with this
magnet, not only the original orientation is destroyed, as we saw above, but also the connection with the other particle.
the two particles as a single system, which means that their combined state must be described by a single vector in a Hilbert space. But the fact that their spin is oriented relative to each other as described above implies that this Hilbert space for the
two-particle system can be separated into two one-particle Hilbert spaces, which are mutually orthogonal, such that a measurement on one system has no bearing on the other. Of course, if the original orientation immediately after the decay can be fixed, there
is a statistical correlation between the measurements of lz for both systems, if the measuring magnets are both oriented in the z-direction.
Ballentine’s experiment was proposed as an alternative to the famous Einstein-Podolsky-Rosen paradox,
in which the momentum of one system is determined by the momentum of the another, with which the former has interacted shortly before the measurement. In this case one makes use of the conservation of linear momentum. But then again, account is not taken of
the destruction of the previous state of the measured system due to the measurement of its momentum. This means that nothing can be inferred about the state of the unmeasured system, whose state is orthogonal to that of the measured system.
The objection may arise here that in Compton’s experiment it is shown that the measurement of the momentum of one colliding particle gives the value
of the other one because of the law of conservation of momentum, which therefore is also valid for microsystems. But in this case the value of the momentum is much larger than its statistical spread, so that the latter can be neglected. For low-energy experiments
as discussed by Einstein, Podolsky and Rosen, the spread in momentum becomes proportionally more significant (7.6).
For a clear understanding
of these paradoxes, one has to keep in mind that the Hilbert space formalism is only applicable to isolated systems. The paradox results due to the jump from one isolated system to another. First one considers the two colliding or emerging particles as one
system. If we consider the two particles separately, we must then consider them either as interacting with each other (which means they are not isolated), or as having a state prepared by the preceding interaction. But, in the latter case, this state need
not be an eigenstate with respect to the measuring instrument. The ‘projection postulate’ (according to which the state of the system changes in a measurement into an eigenstate determined by the measuring instrument) applies to individual systems.
Therefore it must not be applied to the state of the two particles together, if the measurement is only done on one of them. It is only the state of the latter particle which is subjected to the projection postulate.
9.7. Typical symmetry
The unitary operators describing
the symmetry of a system always form a group. The group of external temporal or spatial translations and the Lorentz or Galilean groups of kinetic motion are continuous. On the other hand, those describing rotational symmetry are discrete, and this is also
often the case for internal symmetries.
With full knowledge of the internal structure of a system the symmetry properties would not be needed,
because they are inherent in the structure. But more often than not, the symmetry relations are better known than the detailed structure, and therefore form a great help in designing and interpreting experiments. This is also the case where the system is in
principle known, but mathematically difficult to treat, as in molecules and solids.
Classically one speaks of an internal symmetry whenever
it can be determined a priori that for two or more cases a certain probability must be the same, regardless of its precise value. For example, Laplace’s equally favourable cases are determined by symmetry. But only if all possible cases are equally favourable
(as in dice throwing) is it sufficient to find the probability value itself. This is not usually the case in physically qualified systems.
idea that symmetry leads to equal probability does not rely on a complete lack of knowledge or lack of sufficient reason as was sometimes assumed in order to reconcile statistical methods with a deterministic philosophy. This idea can even lead to absurdities
if taken literally. Take, for example, the case of throwing two dice simultaneously. Beginning with the argument of a complete lack of knowledge one should argue that all possible results, 2, 3, …, 12, are equally probable, which is patently false.
Symmetry is relational with respect to the environment of the subject concerned. That is, it anticipates possible interaction. It is not an epistemological
lack of knowledge, which we confront, but an ontological indifference on the part of one subject (the environment) which interacts with another subject, whose symmetry is being considered.
Now in quantum physics the probability of finding a certain state depends jointly on the initial state, and the operator describing the interaction, namely the reference system. But when speaking of symmetry,
one abstracts from the initial state. One wishes to discuss the symmetry of the law for the system, with respect to some reference system. Therefore, contrary to what was stated in the preceding paragraph, the probability cannot be immediately discussed.
Instead the eigenstates and eigenvalues of the operator must be considered in relation to the reference system.
A hermitean operator determines
the spectrum of possible values which can be assumed by a certain physical magnitude. Thus an eigenstate of the operator can be characterized by its corresponding eigenvalue, which is the value of the physical magnitude for the system in that state. For example,
a measurement does not directly determine the eigenstates, but the corresponding eigenvalues, and this is also the case in a state selector. Contrary to the eigenstates, the eigenvalues can be manipulated. This implies that if two eigenstates have the same
eigenvalue, the two eigenstates cannot be distinguished, neither in a selector, nor in a measurement.
The occurrence of two different eigenstates
having the same eigenvalue is sometimes accidental. However, this accidental degeneracy does not interest us. More important is the degeneracy due to some kind of symmetry. For external symmetries, the temporal homogeneity for example implies that an infinity
of eigenstates (differing by a phase factor) have the same energy.
The internal symmetry of a system can be described by a set of unitary operators.
If one has just two mutually orthogonal eigenvectors with the same eigenvalue for some hermitean operator A, then there is a unitary operator U which transforms one eigenstate into the other, and its inverse U+ which performs
the reverse transformation. Together with the identity operator, I=UU+, they form a group. If A has more than two orthogonal eigenvectors which are degenerate, then the corresponding group of unitary operators has more than two
members. These unitary operators commute with the hermitean operator A, which is therefore left invariant under the symmetry operations. This shows again the extreme importance of group theory for the analysis of typical structures. Even if group
theory cannot give the eigenvalues of A, it can tell which of them are degenerate.
A complete set of commuting operators
Some operators commuting
with A (and therefore having the same set of eigenvectors), may break the symmetry leading to degenerate eigenvalues for A. Now it is an important hypothesis that a complete set of compatible magnitudes exists for any typical structure. That
is, there exists a complete set of mutually commuting hermitean operators, such that each eigenstate can be uniquely characterized by a set of eigenvalues or quantum numbers.
Group theory can show which eigenvalues of some operator of the complete set are degenerate. This set of operators ultimately determines both the full set of possible states and their relative weights (jointly with the initial state of the system).
Such a complete set is not unique, however. There are always several mutually incompatible sets of compatible magnitudes. Thus, any component of the spin
can belong to such a set, but different spin components are mutually incompatible, and cannot simultaneously belong to the same set.
is no criterion known for the completeness of such a set. The possibility always exists of finding a new kind of symmetry applicable to the system. For example, for a long time it was thought that a complete set of magnitudes for a free electron consists of
momentum (or energy as an alternative), mass, spin, and one of the spin components. Later it was realized that the electron could exist in two charge-eigenstates, one negative for the common electron, and one positive for the positron. Thus charge must be
added to the set. Finally yet another attribute, the lepton number L, emerged, compatible with all the members of the set (11.2).
Superposition and superselection
If two eigenstates of some operator (such as the Hamiltonian in connection with the energy) can be superposed, then the system is said to be in the state
objectified by the linear superposition of those two eigenstates. In particular, if two eigenstates, for some operator, are degenerate, then any linear superposition of the two states is also an eigenstate of that operator. One consequence of the complete
set of compatible operators is that no two eigenstates can be degenerate with respect to all operators of the set.
However, not all eigenstates
can be superposed. For instance, of the properties mentioned above for a free electron, only the eigenstates of the momentum and of the components of the spin can be superposed. The mass, charge, total spin, and the lepton number are subject to so-called superselection
rules, stipulating that their eigenstates cannot be superposed. Even when considering the electron and the positron being different states of the same system, these states are not superposable. There are no states with partly positive and partly negative charge.
This distinction between superposable and non-superposable properties allows us to distinguish between two basically different kinds of typical structures (10.5).
In conclusion, we find the somewhat surprising result that in quantum physics the symmetry of the system does not immediately lead to a prediction concerning
the probability of a state, but rather to one concerning a possible value. If, according to symmetry, different eigenstates have the same eigenvalue, these states cannot be distinguished by a measurement that discriminates on the basis of the eigenvalues.
If the initial state is fixed (e.g., by a state selector), then the probabilities associated with the different eigenstates possessing the same eigenvalues may be quite different.
However, if, for the sake of argument, one assumes the initial state to be arbitrary, then all eigenstates have the same probability, and thus the relative weights of the eigenvalues are proportional
to their degree of degeneracy. And this is once again the same as in classical physics, which usually assumes a completely random initial state.
9.8. The dynamic development of a physical system
Section 9.5 discussed the conservation laws of energy and momentum with respect to an
isolated system, finding that its external motion is just the plane wave motion dealt with in chapter 7. This is the case e.g. for electrons. For complicated systems, like atoms, the energy operator, the Hamiltonian H, not only has relevance to external
motion (kinetic energy), but also to internal motion. This consists at least of potential energy as well as kinetic energy due to the relative motions of the electrons and the nucleus in the atom.
The ‘central dynamical postulate’ says that for any isolated system the temporal evolution of its initial state is determined by the Hamiltonian. In particular, if the initial state happens to be
an eigenstate of the Hamiltonian (i.e., if it is prepared by a pure-state energy selector), the state will remain the same, if it is a stationary state, with a definite eigenvalue, the energy of the system. Strictly speaking, only the ground state of an atom
is stationary, and any other state will sooner or later decay. Implicitly in such a decay process, however, is the occurrence of a reaction to the environment, which means the system is not strictly isolated.
The Hamiltonian can tell us something about the probability of decay, i.e., the mean decay time. An unstable state can always be conceived as the superposition of several stationary states,
and the Hamiltonian determines the probability that the system will be found in a state consisting of the ground state of the original system plus a free photon. The only requirement is that the energy of the system in its ground state plus the energy of the
emitted photon be equal to the energy of the initial unstable state. This is only an approximation, however, and a more satisfactory theory is found in quantum electrodynamics, which starts from the interaction between the system and an electromagnetic field.
If the initial state is not an eigenstate of the Hamiltonian, then its change will be determined by its relation to the electromagnetic field.
the initial state is a mixture, it can be shown that it becomes more and more mixed. In fact, for mixed states a magnitude can be defined having the same properties as the classical entropy. For a pure state, this magnitude is zero whereas for a mixed state
it is positive. For an isolated system it increases during its temporal evolution, which is therefore subjected to the physical time order of irreversibility.
Besides the above mentioned distinctions between pure and mixed states, another one is found in interference, the most characteristic kinetic property of waves. Interference is only possible if (and as long as) the interfering waves are coherent,
i.e., if their phase relations are not at random, but determined by some previous interaction, namely, the preparation of the wave packet. Thus one speaks of the ‘coherence length’ of a wave packet. Now a pure state has an infinite coherence, and
a mixed state has a finite one. The possibility of interference is therefore limited for a mixed state (and, of course, every actual state of any concrete system is more or less mixed). Thus any distinction between pure and mixed states has to do with the
irreversibility as the physical time order, and is related to some interaction. It turns out that the pure state is an idealized boundary case, just as a state of rest is a boundary case of motion.
Most authors on the foundations of quantum physics take for granted that an isolated system should be described with a complex Hilbert space. Tolman calls the phase factor randomness (9.4) the basic
postulate of quantum mechanics, but Jauch states that as yet no one has
shown that a real Hilbert space fails to do the job. I hold that complex
numbers are most suited to a description of interference phenomena, which is one of the characteristic differences between classical and modern disclosed kinematics.
The Hilbert space concept is mostly confined to isolated systems. It implies the possibility of describing the internal structure of systems like nuclei, atoms, molecules and solids, and also simple cases of collisions (which are describable
as isolated events). The theory allows one to incorporate internal interactions. The problem of the actualization of possibilities is then circumvented, because one restricts oneself to the calculation of stationary states and the mean values of properties.
As to its external relations, the theory can do little more than describe the particle’s motion. With respect to external interactions, it can only give probabilities, in an anticipatory way.
Thus we find that the Hilbert space formalism has a very wide scope since it can solve four problems mentioned in section 9.1. But is has its limitations too. It cannot solve the fifth problem, concerning the
actualization of one of the possible states. This is often called the measurement
problem, although it pertains to any external interaction. This is not always recognized, because in many cases of external interaction, it is possible to solve the problem by including the two interacting subjects into one single system, in which case
the Hilbert space formalism is applicable. This method is very successful in describing the interaction of, e.g., electrons and nuclei, and somewhat less successful in the description of the interaction between an atom and an electromagnetic field. The latter
does not lend itself for treatment as an isolated system.
In order to see the relevance of interaction to the understanding of our problem,
consider the following example. It is a well-known statement that an atom has discrete energy levels, each corresponding with an eigenstate of the Hamiltonian operator. For the sake of argument, let us only consider two such states, the ground state and the
first excited state. The Hilbert space formalism says that the atom may also be in a mixed state, described by a Hilbert space vector which is a linear combination of the two eigenstates. To this mixed state there corresponds no fixed energy value. How should
this be interpreted?
As long as we consider the atom as isolated, we are at a loss, because every isolated system has a fixed energy. However,
no system is really isolated. This problem can be studied with the help of quantum electrodynamics, observing that each atom interacts with an electromagnetic field, such that the mixed state can be considered as the ground state together with a virtual photon
– in which case the energy of the atom is not fixed, but neither is the atom isolated.
There is also a probabilistic interpretation.
In fact, one will never have a single atom, but rather a dilute gas. The interactions between the atoms may be so small, that for the calculations the atoms may be assumed to be isolated. At the same time, the interaction may be large enough (e.g., via collisions:
the temperature of the gas must be well above zero) in order to have atoms both in the ground state and in the first excited state, in a ratio determined by the Boltzmann factor (8.5), which therefore gives the probability that a certain atom is in one of
these states. This ratio is immediately related to the mixed-state vector, for which an interpretation was sought. Thus by introducing a weak interaction, small enough not to disturb the calculation results, a very natural interpretation of these mixed
states can be given.
Also in these cases, however, no actualization is described, and the success of the theory lies in the predictability
of observational probabilities, stationary states, etc. It has been suggested
that the main reason why the Hilbert space formalism fails is based on the superposition principle, by which the linear combination of two possible states is again a possible state. Hence, if a state is a linear combination of two eigenstates for some operator,
the system can be said to have simultaneously two different values for the magnitude represented by that operator. This principle is only applicable to potential states, not to actual ones. Whereas the possible energies of a system can be superposed,
as can be verified in interference experiments, actual energies cannot.
Above superposable and non-superposable properties of isolated systems
were distinguished. Only for the latter it can be said that the system possesses the property. Thus an electron has a negative charge, a positron has a positive charge, but neither possesses a certain energy value. Only if the electron
has actualized one of its possibilities in some external interaction, may it be said to have a certain energy value. This is not only the case in a measurement or observation, but also e.g. in a collision between two electrons, if after the collision
takes place, there is a correlation between their states. Evidently a theory of external interaction must do away with linearity and superposability. This was recognized some time ago by several physicists, among them Werner Heisenberg, Louis De Broglie, Jean-Pierre
Vigier, and David Bohm. However, the mathematical difficulties involved
in a non-linear theory are enormous, and the theory is progressing only very slowly, if at all.
So the conceptual difficulties involved in
quantum theory can be understood if keeping in mind that the theory only applies to isolated systems and potential interactions. This poses the question as to what extent the concept of an isolated system is fruitful. Strictly speaking, isolated
systems do not exist, and the concept itself is even objectionable, since physical systems are characterized by their interaction as the basic physical subject-subject relation. It is a subject-subject relation, an interaction between two physically qualified
systems which is at stake. It is not a psychically qualified subject-object relation between the observer and the observed system, as was believed by Niels Bohr and Werner Heisenberg, e.g.
I do not say that the latter relation is of no interest. However, if one wishes to study the latter, one must first of all be aware of the former subject-subject relation and its implications, precisely because any observer (a psychic subject) is a physical
subject as well.
The answer to our question must be found in the study of the individuality structures which are physically qualified. Both
the driving motive and the success of quantum theory must be sought in this field of research.
First, it is just the (limited) individuality
of physical systems which makes it both necessary and fruitful to consider them in isolation, in an intermediary state between two interactions. Thus the weakness of quantum theory is its strength all the same. Secondly, the distinction of potentiality and
actualization will be related to the distinction between the thing-like and the event-like character of physically qualified individual subjects. Both will be considered in part II of Laws for dynamic development.
For the sake of argument it will be assumed that the number of dimensions of Hilbert space is denumerable.
This means that an operator in the Hilbert space manifests itself as a ‘matrix’ by which the components of a function in the space are transformed.
 To each linear operator A corresponds
an adjoint operator A+, such that (f,Ag) = (A+f,g); this implies that I+=I.
 Cp. Jauch 1968, 73: … every measurement
on a physical system can be reduced, at least in principle, to the measurements with a certain number of yes-no experiments.’ It should be realized that a yes-no experiment is a theoretical idealization. In principle, no actual physical measuring instrument
can be reduced to a perfect yes-no experiment because of noise. Just because every physical instrument is a quantum mechanical system itself, this noise cannot be made as small as one likes. This refutes the possibility of accepting yes-no experiments as an
empirical basis for quantum physics, although they may be used as a theoretical starting point.
 Achinstein 1991, 24. Decisive was Foucault’s experimental confirmation in 1854
of the wave-theoretical prediction that light has a lower speed in water than in air. Newton’s particle theory predicted the converse.
 See Hanson 1963, 13; Jammer 1966, 31.
Cathode rays, canal rays and X-rays are generated in a cathode tube, a forerunner of the television tube, fluorescent lamp and computer screen.
 Pais 1991, 150. Planck’s reduced constant is
h/2π. In Bohr’s theory the angular momentum L=nh/2π, n being the orbit’s number. For the hydrogen atom, the corresponding energy is En=E1/n2,
with E1=-13.6 eV, the energy of the first orbit.
The group velocity df/ds=dE/dp equals approximately Df/Ds. E/p>c and dE/dp<c follow from the relativistic relation between energy and momentum, E=(Eo2+c2p2)1/2,
where Eo is the particle’s rest energy. Only if Eo=0, E/p=dE/dp=c. Observe that the word ‘group’ for a wave packet has a different meaning than in the mathematical
theory of groups.
Bohr 1934, chapter 2; Bohr 1949; Meyer-Abich 1965; Jammer 1966, chapter 7; 1974, chapter 4; Pais 1991, 309-316, 425-436. Bohr’s principle of complementarity presupposes that quantum phenomena only occur at an atomic level, which is refuted in solid state
physics. According to Bohr, a measuring system is an indivisible whole, subject to the laws of classical physics, showing either particle or wave phenomena. In different measurement systems, these phenomena would give incompatible results. This view is out
of date. [Sometimes, non-commuting operators and the corresponding variables (like position and momentum) are called ‘complementary’ as well, at least if their commutator is a number.]
Kastner 2013, 31-33.
Even in classical physics, the idea of a point-like particle is not well-defined. Both its mass density and charge density are infinite, and its intrinsic angular momentum cannot be defined.
 Light in vacuum is
The values of ‘1’ respectively ‘h’ in de mentioned relations indicate an order of magnitude. Sometimes other values are given, e.g. h/4p instead of h, see Messiah 1961, 133.
If Δx.Δs=Δt.Δf=1, the wave packet’s speed v=Δx/Δt=Δf/Δs is approximately the group velocity df/ds, according
to De Broglie.
In communication technology, Δf is the bandwidth, see Bunge 1967a, 265. Bunge denies that wave-particle duality exists in quantum mechanics, see ibid. 266, 291. In his formulation, the single concept of a quanton replaces the concepts of wave
and particle. However, this masques the fact that in the quanton a physical and a kinetic character are interlaced.
 See e.g. Margenau 1950, chapter 18; Messiah 1961, 129-149; Jammer 1966, chapter
7; 1974, chapter 3; Omnès 1994, chapter 2; Torretti 1999, chapter 6.
From the commutation properties of the operators referring to the components of angular momentum for an electron (having rotational symmetry), one derives the integral eigenvalues for the orbital angular momentum as well as the half-integral eigenvalues for
the intrinsic angular momentum or spin, see Messiah 1961, 523-536.
Bunge 1967a, 248, 267.
I leave here aside the important distinction between a time dependent and a time independent Hamiltonian, the former describing transition processes, the latter stationary states.
 Heisenberg 1930, 21-23.
Kastner 2013, 202: ‘The interpretive challenge of quantum theory is often presented in terms of the measurement problem: i.e., that the formalism itself does not specify that only one outcome happens, nor does it explain why or how that particular outcome
happens. This is the context in which it is often asserted that the theory is incomplete and is therefore in need of alteration in some way.’
 In fact, the value of ΔE is less significant
than the relative indeterminacy ΔE/E. For a macroscopic system the energy E is so much larger than ΔE that the energy fluctuations can be neglected, and the law of conservation of energy remains valid.
Such virtual processes are depicted in the so-called Feynman-diagrams.
The probability to find a particle in the volume element between r and r+dr is y(r)y*(r)dr, hence the scalar product y(r)y*(r)
is a probability density.
Cartwright 1983, 179. Of course, the probability is not given by a single wave function, but by a wave packet. If this consists of a set of orthogonal eigenvectors, a matrix represents the transition probability.
‘The true philosophical import of the statistical interpretation consists in the recognition that the wave-picture and the corpuscle-picture are not mutually exclusive, but are two complementary ways of considering the same process’, M. Born, Atomic
physics, 1944, quoted by Bastin (ed.) 1971, 5.
The fact that quantum physics is a stochastic theory has evoked widely differing reactions. Einstein considered the theory incomplete. Born stressed that at least waves behave deterministically, only its interpretation having a statistical character. Bohr
accepted a fundamental stochastic element in his world-view.
Heisenberg 1958, 25.
Observe that an interference-experiment aims at demonstrating interference. This is only possible if the interference of waves is followed by an interaction of the particles concerned with, e.g., a screen.
For the relevance of interactions for the interpretation of quantum physics, see Healey 1989.
Theoretically, this means the projection of the state vector corresponding to the state before the measurment onto one of the eigenvectors of Hilbert space, representing the state of the system after the measurement. Omnès 1994, 509:
‘No other permanent or transient principle of physics has ever given rise to so many comments, criticisms, pleadings, deep remarks, and plain nonsense as the wave function collapse.’ In particular, the assumptions that probability is an expression
of our limited knowledge of a system and that the observer causes the reduction of the wave packet, have led to a number of subjectivist and solipsist interpretations of quantum physics and related problems, of which I shall only briefly discuss that of Schrödinger’s
Omnès 1994, 84: ‘This transition therefore does not belong to elementary quantum dynamics. But it is meant to express a physical interaction between the measured object and the measuring apparatus, which one would expect to be a direct consequence
of dynamics’ Cartwright 1983, 195: ‘Von Neumann claimed that the reduction of the wave packet occurs when a measurement is made. But it also occurs when a quantum system is prepared in an eigenstate, when one particle scatters from another, when
a radioactive nucleus disintegrates, and in a large number of other transition processes as well … There is nothing peculiar about measurement, and there is no special role for consciousness in quantum mechanics.’ But contrary to Cartwright (ibid.,
198) stating: ‘… there are not two different kinds of evolution in quantum mechanics. There are evolutions that are correctly described by the Schrödinger equation, and there are evolutions that are correctly described by something like van
Neumann’s projection postulate. But these are not different kinds in any physically relevant sense’, I believe that there is a difference. The first concerns a reversible motion, the second an irreversible physical process, cp. Cartwright 1983,
179: ‘Indeterministically and irreversibly, without the intervention of any external observer, a system can change its state … When such a situation occurs, the probabilities for these transitions can be computed; it is these probabilities that
serve to interpret quantum mechanics.’
The principle of decoherence is in some cases provable, but is not proved generally, see Omnès 1994, chapter 7, 484-488; Torretti 1999, 364-367. Decoherence even occurs in quite small molecules, see Omnès 1994, 299-302. There are exceptions too,
in systems without much internal energy dissipation, e.g. electromagnetic radiation in a transparent medium and superconductors, see Omnès 1994, 269. The decoherence approach has been criticized, see e.g. Kastner 2013, 14-16.
Heisenberg’s relations can now be interpreted as determining the minimum statistical spread or standard deviation in measurements.
 Popper 1967 underestimates this difference between classical
and quantum statistics in his critique of the ‘great quantum muddle’.
This is an idealized statement, because usually the state of a system must be represented by a statistical mixture, to be described with the help of a so-called density operator (9.8).
It is no longer a Boolean algebra; cf. Weizsäcker 1971, 237.
Houtappel et al. 1965, 611.
Tolman 1938, 349 ff.
Heisenberg 1958, 38.
Margenau 1950, 175, 335.
On the historical development of the concept of complementarity, see Jammer 1966, 345ff, 1974; on the views of Bohr, see Bohr, 1934; 1949; 1958; 1963; Meyer-Abich 1965, chapter 3; Bunge 1959b, 173-209; Feyerabend 1962; Hanson 1959; Holton 1973, 115-161; Hooker
1972; Pais 1991.
Bohr 1949, 209, 210.
From a historical point of view, this interpretation rests on a misunderstanding (first by Pauli) of Bohr’s ideas; see Meyer-Abich 1965, 152.
 Jauch 1968, 69, 112ff.
This procedure was invented long before the rise of quantum physics, by Lagrange; cf. Jammer 1966, 224.
Jauch 1968, 198; Kaempffer 1965, Chapter 9.
A pure state cannot be normalized, and is therefore unsuited for the representation of a physical system.
This view is not justified, not even for classical physics: the formulation as given in the text is a simplification.
 See for the correspondence principle, Jammer 1966, 109-118; Meyer-Abich 1965; Hanson
1958, 1959, 1963, 60-67; Messiah 1958, 29-31; Heisenberg 1958, Chapter 3; Petersen 1968.
Margenau, Cohen 1967, 85, 86. For this section, see e.g. Messiah 1958, Chapter 3 and 13, p. 195ff, 508ff; Jauch 1968, Chapter 14; Kaempffer 1965, Chapter 1, 12.
 Ballentine 1970; see also Frisch (in:
Bastin (ed.) 1971, 20) on a similar thought experiment; see also Jammer 1974, 235, 302ff, 309.
Einstein, Podolsky, Rosen 1935; see also Bohr 1949, 231ff; Margenau 1950, 261ff; Feyerabend 1962; Jammer 1974; Hooker 1972. In 1982 Alain Aspect performed experiments which seem to confirm the view expressed in this section.
According to the Heisenberg relations, the states of the two particles after the collision cannot be pure momentum (or energy) states, because of the finite interval (or duration, respectively), in which the interaction took place.
In more complicated systems like atoms there are also ‘selection rules’ which sometimes refer to the same kind of operators (total angular momentum, e.g.). However, the transitions which are ‘forbidden’ by these rules are often simply
less probable than other transitions because the symmetry underlying these rules is ‘broken’ by some perturbing interaction; see any textbook on quantum physics.
Jauch 1968, 121; see also Weizsäcker 1971, 252.
Popper 1967, 25, 26.
Bohm, in: Bastin (ed.) 1971, 102ff.
See e.g., Heisenberg 1930, 2; Heitler 1949, 191.