The complex state vector, observables and properties

In quantum mechanics, the system that is being studied is described by a wave or state function which is dependent on the whole experimental arrangement. This function includes all the

496 Laurikainen 1973, 180-184. Laurikainen believes that the concept of space does not have a clear meaning in the realm of elementary particles since general relativity, a non-linear theory, is not compatible with linear quantum mechanics.

497 Auyang 1995, 68. For example R. Penrose and J. Polkinghorne assume that physicists generally consider wave functions to be real.

possible states of the system, and these can be calculated to develop in a deterministic manner in a mathematical state-space. In a measurement situation, the quantum system is however

”projected” onto normal space and the result obtained is one of the many possible presentations of the system and a realisation of one of its possible values. For example, the system being investigated can, if required, be localised in a single position, but measurements naturally cannot provide knowledge about whether particles have positions other than those which result from specific interactions.

In classical physics, the state means a system’s space-time situation, its position and its velocity in space at a specific moment. Examination of the fundamental equations of quantum mechanics shows that quantum theory employs a definition of state that is quite unlike that employed by classical mechanics.⁴⁹⁹ The state function in quantum mechanics does not bestow any specific position or momentum on particles.⁵⁰⁰ It cannot be thought of as describing more a particle than a wave⁵⁰¹, and the state vector’s relationship to the classical concept of an object remains

obscure. Also, the thought that an object ”owns” even its primary properties becomes a problem when no clear and observable properties can be attached to the wave function itself, only the possibility of different observable realisations in different interactive situations. Dirac

characterised the new situation produced by quantum mechanics by saying that ”an observable introduces a set of basic states in which the characteristics of a state can be revealed.⁵⁰²

While all observables in quantum physics are represented by self-adjoint operators on the state space, not all adjoint operators represent observables. Among the infinitely many self-adjoint operators in a Hilbert space, physicists use no more than a handful.⁵⁰³ In quantum mechanics, if the eigenvalue of an operator can be measured, it corresponds to an observable.⁵⁰⁴ Since a state vector can be treated as a sum or organised source of different kind of observables, quantum mechanics is often presented as indicating that the different objects and phenomena which influence the macroscopic world are fundamentally indivisible and profoundly

interdependent.

499 Nagel 1961, 306. The state-description employed in quantum theory is extraordinarily abstract. The so-called Psi-function does not lend itself to a intuitively-satisfactory non-technical exposition.

500 Velocity does not have a clear role at the micro level and momentum cannot be considered to be classical, but the relations to energy are similar to those at classical level. Hodgson 1991, 258-260.

501 When introducing students to the handling of quantum physics, they are told: “We have abandoned the notion of a wave packet as representing a particle. This notion was helpful to us in making the Schrödinger equation plausible, but now it is Ψ(x,t) and its probabilistic interpretation that tell us where the particle is, without the particle being thought of as 'made up out of waves'." Gasiorovits 1974, 54

502 Quoted in Auyang 1995, 20.

503 Auyang 1995, 87.

Many of the problems of interpreting state vectors are connected with the fact that quantum characteristics are irreducibly complex. This is explicitly evident in the Schrödinger equation and other fundamental relationships such as commutation. If, as Sunny K. Auyang believes, state vectors depict real complex quantum objects, the complex nature of quantum-state space is neither an accident nor a mathematical convenience. The entire complex state vector is

significant. Its meaning is destroyed if we try to separate the real and imaginary parts. Auyang believes that quantum objects and quantum properties really exist at the quantum level, and argues on their relationship with observed observables and their eigenvalues as follows:

”In classical mechanics a particles position and momentum assume defined values in certain coordinate system. The same holds for quantum systems. But there are differences. The state space of quantum mechanics has a richer intrinsic structure.

The Hilbert space has a built-in metric structure embodied in the inner product which enables the quantum state to internalize coordinatization as a kind of relation among states. Thus quantum properties can assume definite values in bases or coordinate systems defined within the state space itself. The most interesting bases are associated with observables. The basis of an observable A constitutes a

representation of a state vector.”⁵⁰⁵

Because of their complexity, quantum observables should not be confused with their classical namesakes. For example, quantum momentum has a richer structure than classical momentum and some observables such as spin have no classical counterparts. The quantum predicates of amplitude also describe characteristics more complicated than can be handled by the classical predicates of eigenvalues. A specific eigenvalue ai , has the unique significance of being the one that can be found in experiments on a single system, but there are great objections to ascribing it as the property of a quantum system. The nature of eigenvalues conflicts with that of state vectors, which claim to be summaries of quantum properties; the one is real and the other is complex. A state vector also has its governing equation of motion, a specific eigenvalue that is determined by experiment does not. State vectors are not kickable within quantum mechanics:

we cannot manipulate a quantum system to obtain a specific eigenvalue in an experiment. Only in special cases analogous to the eigenstates can a quantum system be "aligned" in such a way that a classical predicate becomes a good abbreviation.⁵⁰⁶ Because of the difficulties, Auyang concludes that eigenvalues should not be considered properties of quantum objects. The explicit stipulation of some quantities that can be measured justifies the name “observable”, but

504 In principle for each possible observable can be constructed its own operator.

505 Auyang 1995, 68, 73.

506 Auyang 1995, 19, 80.

eigenvalues or spectral values are only parts of the structure of observables. Their relationship remains unexplained.

”Abstractly, the eigenvalues of an observable can be regarded as labels of the eigenstates. Physically the labels are realized in classical objects we can measure.

How they are realized no one knows. Eigenvalues are analogous to symptoms of a disease, they indicate something that does not show up. Unlike amplitudes, the occurrence of an eigenvalue needs the extra condition of classical realization.

Practically, an indicator is somehow triggered in measurements and experiments.

Unperformed experiments have no results, but this does not imply that the quantum systems on which the experiment might have been performed has no properties.”⁵⁰⁷

In quantum mechanics, phases are also important. To describe a state more definitely we have to use representations which reveal that state’s relations with other states. The relation is contained in complex relative phases which account for the peculiar phenomena associated with quantum mechanical interference. That is why we need complex numbers ci instead of their moduli ‌‌ /‌ci / when expanding the state vector. The set of moduli is truly measured, but the relative phases are somehow destroyed in experiments; exactly how this happens we do not know. Measurements always return to real numbers into which complex numbers cannot be homomorphically embedded. It is like trying to squeeze a three-dimensional something into a two-dimensional plane: some damage is unavoidable. The reduction of a multi-dimensional quantum description to nothing more than the checking of attributes (eigenvalues) is not due to our clumsiness, the cause is more basic. It may be due to the fundamental limitations of our form of observation.

Specific representations are necessary for us to acquire empirical knowledge of the objective world.⁵⁰⁸ Humans clearly cannot investigate quantum states other than by individual

coordinatization, which they themselves may influence:

”An observable coordinatizes the quantum world in a particular way with its eigenstates, and formally correlates the coordinate to classical indicators, the eigenvalues. An observable introduces a representation of the quantum state space by coordinatizing it. Within the representation a quantum state acquires a definite description in terms of amplitude. It realizes a general conceptual distinction of physical state and its specific representation. An amplitude is ascribed to the quantum system only with the choice of a representation, just as coordinates are assigned to a classical particle only in a coordinate system.”⁵⁰⁹

The indispensability of observables and the representations they introduce is apparent in

quantum mechanics. The representation-free form of a state space M of a physical system is too

507 Auyang 1995, 79.

508 Auyang 1995, 74.

abstract and by itself insufficient for physical theories, since the theories must predict the behaviour of particular objects so that they can be supported by experiments. M, being a total abstraction from all particularities and observational conditions, is newer observed: observations are always of particular representations of M. What we observe in experiments is characterised by initial and boundary conditions that are expressed in coordinates, but to say that observables are conventional does not imply that they are phantasmal. Once an observable is chosen, its eigenstates that realize the coordinate bases are as physical as any other state, and the classical quantities that realize its eigenvalues are concrete. Thus the conventionality of representations does not lead to relativism. Quantum mechanics prescribes rigid transformation rules among the various representations that leave the quantum state invariant.⁵¹⁰

An attempt can also be made to visualise the abstract relationship between quantum states and observables by employing the language used for wave functions. As has already been pointed out, in quantum theory the wave function of the system under investigation can always be presented as an expansion of the desired wave-types. In this way, different families of waves correspond to different physically measurable attributes.⁵¹¹ A fixed spatial position is associated with a momentum wave. Momentum is associated with spatial sine waves, and energy is

connected to temporal sine waves. Spin is connected to spherical-harmonic waveforms.

Waveforms and the connection with their attributes tells us why some attributes are quantised and others are not. Quantised attributes are connected to restricted waveforms such as spherical-harmonic waves whose vibrations are limited to the spherical surface. Clearly, each possible waveform corresponds to some dynamic attribute which can, in principle, be measured. The number of different waveforms is infinite.

This can lead to the conclusion that quantum theory does not directly describe independent objects in space-time any more than it describes their enduring attributes. With the help of quantum theory, a desired object can be connected to a wave function whose form incorporates information about all the possible observable attributes of that object. Observable attributes are not however manifested without interaction or measurement. During measurement, the wave function is cut to the wave expansion of the desired attribute in which each term has its own amplitude. The square of this amplitude gives the probability that the value in question will be

509 Auyang 1995, 85-86.

510 Auyang 1995, 87, 96.

511 Herbert 1985, 102. Quantum entities have two kinds of properties: static and dynamic. For example, mass, charge, and spin are static, while position, momentum and the direction of spin are dynamic.

reached if measurement is carried out. It is truly strange that real measurements reinforce the probabilities that quantum mechanics predicts.

Whether the abstract quantum-mechanical state function describes a real quantum object, a world of possibilities, knowledge of the observer or anything else, this mathematical construction connects concrete observations of real phenomena in many ways which are impossible to comprehend within the framework of reality provided by classical physics. These confusion-causing and difficult-to-interpret features of quantum mechanics are examined more closely in the following section.

4.2..2. Discontinuity and wave-particle dualism

In classical terms, the world was assumed to be made up of separate mechanically interacting objects which had different objective properties such as size, mass or velocity. In principle, these properties could be assigned any values, and changes in them from one state to another were believed to take place in a continuous manner through all the intermediate stages. The discovery that the interaction of matter and radiation was connected with a new universal constant,

Planck’s constant h, set a limit on the minimum size of any effect and at the same time made some atomic particle states discontinuous, i.e. quantised.

In quantum mechanics there is a certain probability that an electron in a hydrogen atom, for example, can be found in one position, and there is also a certain probability of it being in another position. Electrons no longer have orbits but physicists often speak of ”electron clouds”

that are of different sizes and shapes. These configurations are known as quantum states.⁵¹² Different states are associated with different energies. When an atom’s energy changes, the electron makes a transition between two different states. In doing this, the atom emits or absorbs a light quantum (or photon), whose energy corresponds to the difference between the energy states. The atom’s transference from one stationary state to another cannot however be visualised in space-time. The change is usually presented as taking place in a quantum jump, in which

512 Morris 1997, 93. Subatomic particles are in general considered to be in mixtures of states. One can speak of the probability of a subatomic particle being in this or that state. Not only is an electron in many different places at once, it can simultaneously occupy an infinite number of different energy states.

internmediate states are impossible.⁵¹³ In micro-physics, we also encounter new quantised properties such as spin, charm and strangeness. Although these are properties whose

conservation in particle reactions is confirmed, it is difficult to construct a clear understanding of their fundamental nature.

Arthur March, a German professor of physics, tried to conceptualise the ideas of the Copenhagen school on the foundations of quantum mechanics in a clear and careful manner. He reflected the new situation in his remark that any phenomenon that occurs in the micro-world consists of elementary processes or acts which, by virtue of natural law, cannot be analysed. Hence, the micro-world is by nature atomic not only in respect to matter but to events as well. We shall never know what happens in an atom during the process that leads to the production or

annihilation of a photon. The emission or absorption of light as well as the scattering of a photon by an electron are examples of elementary processes or acts which resist any attempt to analyse them. We cannot therefore apply the principle of causality to these processes. The atomicity of events appears to us as a discontinuity in the course of events, and only probability relations exist between present and future.⁵¹⁴

This situation can also be illustrated by saying that the wave properties associated with particles make the earlier deterministic space-time description impossible: particles are not only

mechanically interacting objects that can be idealised as mass points. At the same time, waves make possible probability forecasts concerning atomic events.⁵¹⁵ Categories of reality previously illustrated using separate wave and particle metaphors now appear to be in some way linked. An clear and visualizible representation of the new connection between particles and waves or matter and radiation cannot however be found in the classical ”billiard ball” world. Particles are located at specific points, but waves spread throughout the whole of space. Also, quantum field theories cannot claim credit for visualisable clarity, through which the connection between waves and particles has anyway become increasingly clear. In quantum field theories, all elementary particles are considered to be quanta in the fields that they are connected to. For example, a photon is understood as a quantum in a electromagnetic field which mediates

513 This strange situation can somehow be visualised by using the simple example of the waves on a string. There are always some positions where the string is at rest. We can see how the wave-like properties of particles lead directly to ’energy quantization’ without solving the Schrödinger equaton. Hey and Walters 1987, 42-44.

514 March 1951, 1-3 and March 1957, 47-50.

515 Laurikainen 1993, 155.

electromagnetic interactions between electrically charged particles.⁵¹⁶

The probabilistic structure of quantum mechanics incorporates the possibility that a particle can be found in locations that are absolutely forbidden to it in the classical world. In classical terms, there can be an energy barrier which separates one region of space from another so that particles below some energy threshold cannot move from one region to another. What is remarkable about quantum ’particles’ is that they do not behave like classical objects. There is a finite probability that they are able to ’tunnel through’ the forbidden region and appear on the other side. This

’barrier penetration’ or ’quantum tunnelling’ is responsible for nuclear fission and the burning of hydrogen in stars. In modern technology, it is a commonplace quantum phenomena that forms the basis for a number of electronic devices such as new type of microscopes.⁵¹⁷

The developers of quantum mechanics were confused by the fact that atomic objects were described in some experimental situations as being particles and in others as being waves. In mathematical representations of a studied system, for example, an electron could be associated with a certain individual waveform whose form allowed the calculation of its position

probabilities. When an individual electron is observed, it always appears as a localised particle.

Groups of electrons are, on the other hand, usually considered to be cooperating with each other in making waves. In Young’s double-slit experiment, where an electron beam is directed through two thin slits, the diffraction pattern formed in the shade is similar to that observed when

investigating the diffraction of light waves. The intensity distribution obtained can be explained by assuming that the waves passing through both slits interfere with each other. In the diffraction pattern formed in the shade, individual electrons strike in those places where waves diffracted through the neighbouring slits reinforce each other (are in phase), while places where the diffracted waves are out of phase remain dark.

The intensity distribution obtained from the double-slit experiment clearly differs from the patterns which result when the individual patterns obtained by opening one slit at a time are combined. When particles are allowed to pass through the two open slits one at a time, the diffraction pattern gradually built up in the shade is the double-slit interference pattern, not the pattern obtained by superimposing the patterns obtained by opening each slit in turn. In

516 Quantum field theories deal with two kinds of interacting fields. The quanta of matter fields are fermion-type particles and interactions or forces are transmitted by fields which consist of bosons. Particles and fields form one

516 Quantum field theories deal with two kinds of interacting fields. The quanta of matter fields are fermion-type particles and interactions or forces are transmitted by fields which consist of bosons. Particles and fields form one

In document Quantum Metaphysics : The Role of Human Beings within the Paradigms of Classical and Quantum Physics (sivua 193-200)