Quantum Mechanics: A Discussion

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Paul Ballard

On the basis of these arguments from the theory of relativity, de Broglie showed that the relations E = hν and p = E/c were partially successful in describing the particle properties of light. (Indeed they can be interchanged to describe the wave properties of matter.) The second equation can be rewritten in terms of the light’s wavelength λ, giving p = h/λ.

When evaluated, the wavelength λ was found to be of the order of the atomic dimensions and, at the atomic level, the wave assocoiated with the electron would produce important new effects. Among these were the appearance of discrete frequencies of vibration resulting from confinement of the waves within the electron. De Broglie was able to calculate both frequencies and corresponding energies of the discrete modes of vibration of these waves. The Bohr energy levels could be explained in terms of an wave, with the assumption that this wave is related to its frequency.

Later experiments by Clinton Davisson and Lester Germer on the scattering of electrons from metallic crystals revealed a pattern of strong and weak scattering similar to the fringes obtained by passing a beam of strong light quanta through a set of slits. The regular array of atoms in the crystal were playing a role equivalent to the light slits in Davisson’s and Germer’s experiments; when the length of the assumed wave was calculated on the basis of the observed pattern of strong and weak scattering, it was a brilliant confirmation of de Broglie’s theory. More recent experiments showed that other particles – protons, neutrons, electrons etc. – had similar wave-like properties.

At this point new and somewhat paradoxical limitations on the wave theory were discovered. Erwin Schrodinger believed that de Broglie’s relativistic four-dimensional model had yielded no practical results and proposed a multi-dimensional model. To quote Schrodinger:

‘ΨΨ is a sort of weight function in the configuration of the system. The wave mechanical configuration of the system is a superposition of many, strictly speaking all kinematically possible point mechanical configurations. Thereby every point mechanical configuration contributes to the true wave mechanical configuration with a certain weight which is given precisely by ΨΨ.

‘If one likes paradoxes, one can say that the system is found simultaneously in all kinematic locations, but not in all of them in equal strength...’

Schrodinger originally proposed that the electron should be thought of as a continuous distribution of time-dependent waves; the density of this wave is denoted by Q = ½Ψ, which is half the angle of the wave function divided by its length. Unfortunately this interpretation was tenable only as long as the Schrodinger wave remained confined to the atom, but outside the atom the electron is always found in a comparatively small region of space so in general its wave density does not agree with Schrodinger’s value.

To deal with this problem Bohr proposed that the wave intensity does not represent the actual charge density of the electron but rather its probability density, conceived as a small localised particle. It was only possible to verify Bohr’s hypothesis indirectly. Thus in a transition between allowed energy levels, the change of the Schrodinger wave from one mode of vibration to another was now interpreted in terms of a continuously changing probability that the electron had one energy or another, which could be verified by experiment. The probability of the transition between the allowed energy levels was dependent on the amplitude of the wave vibrations.

As early as 1912 it had been noted that when a beam of x-rays of well-defined wavelength is scattered by material of low atomic weight, some of the scattered x-rays have a longer wavelength than before. Several explanations were proposed, only to be disproved. In 1922 Arthur Compton gave an explanation which fitted with observations by applying the laws of conservation of energy and momentum to the collision of an x-ray photon with a single electron.

The ‘Compton Effect’ brought particle theory and wave theory of light into confrontation. Uninhibited by factions in European physics, John Slater asked why radiation cannot be both wave and particle. Slater took Bohr’s theory (described above) of harmonic vibration and proposed that atoms in ‘excited states’ emitted electromagnetic waves corresponding to transitions to lower states of ‘excitement’ than allowed for by Bohr’s theory, but that these electromagnetic waves are a peculiar kind which do not carry energy. Bohr himself adopted the term “virtual oscillation.”

A strange combination of the determinate and statistical components of systems was identified in an apparently contradictory form, which had never been encountered before. In 1924 Satyendra Bose derived Planck’s distribution law, which applied to the kinetic movement of molecules, affected by black (infra-red) radiation, without reference to electromagnetic theory. He applied appropriate statistics to the photons within a container in which photons were considered to be massless particles capable of existing in two states of polarisation given by p = hc/ν.

Einstein saw that the Bose method of counting can also be applied to statistics of indistinguishable molecules. The difference between molecules and photons is that for the former, the number of particles is held constant in calculating the maximum entropy in the system, i.e. the equilibrium state. Einstein believed that the radiation was more than a mere analogy and that molecules and photons must have wave and particle characteristics.

Once we admit that the entire conceptual framework of the existing quantum theory is inadequate for detailing all of the properties of individual systems, then an unlimited number of possibilities open up. At the sub-atomic level it is suggested that we have to deal with some kinds of system that have non-linear oscillations, but of a type of non-linearity which leads to discrete frequencies satisfying the Planck relation E = hν and the de Broglie relation p = h/λ. Once this is achieved, we would then be able to explain the appearance of discrete energy levels in matter and electromagnetic energy in quanta. Moreover the transitions between discrete energy levels would be accounted for.

To overcome Heisenberg’s famous indeterminacy principle, we may ask ourselves: ‘Suppose there is some underlying irregular, but defined motion of the electron, which could explain the probabilities defined by the Schrodinger wave function Ψ. Would it be possible by actually observing the motions of electrons to ascertain their character?’

In the case of observing an atomic particle, such as an electron going round a nucleus, with a microscope, the light used in the microscope is always in the form of discrete packets or quanta, and we cannot help disturbing the electron when we look at it: we must use at least one quanta of light when we look at it. When the quanta collides with the electron there will be a minimal disturbance in the latter’s motion, which comes from the light we use in the process of observation. To reduce this disturbance we might use electromagnetic waves of a lower frequency, so we obtain a smaller quantum. But there is another difficulty – light acts like both a particle and a wave. The wavelength is proportional to the frequency ν – if the frequency is low, the wavelength will be big, and the image in the microscope will be so poor that we will not know where the electron is.

The particle character of light cannot avoid disturbing the particle’s momentum, creating an unpredictable and uncontrollable disturbance, denoted by λp. Alternatively, treating light as a wave, we cannot avoid an uncertainty λν in the position of the electron. Calculation leads to Heisenberg’s indeterminacy relation, λpλν=h. In other words, uncertainty in momentum multiplied by uncertainty in position is equal to Planck’s constant.

This result demonstrates the reciprocal relationship between the precision of calculation of the momentum and that of the position. If the uncertainty in position is very small then the uncertainty in momentum cannot be, whereas the more accurately the position is determined the less accurately the momentum is determined.

Quantum theory implies that every process of measurement will be subject to the same ultimate limitations on its precision, which led Heisenberg to regard the indeterminacy relationships as being manifestations of a very fundamental and pervasive principle operating throughout the whole of the natural law. Most of the proponents of this point of view see future developments in physics making the behaviour of things even less definable in terms of current quantum theory: they see the indeterminacy principle as an absolute and final limitation on our ability to establish the state of things by means of measurement.

This assumption has far reaching consequences, for even if a sub-quantum mechanical level containing hidden variables exists at the level of quarks, or conceivably at sub-quark level, these variables would never play any real part in the prediction of experimental results. Six forms of quarks have now been detected – the last being the top quark, whose existence was confirmed at Fermilab in 1995 by accelerating particles to collide at very near the speed of light, but the quarks cannot be directly observed. Outside of particle accelerators quarks always operate in groups and are confined to protons and neutrons, which suggests that they are a fundamental particle.

Quantum theory was originally conceived as being applicable only at the atomic level, but it has now been found to apply to some sub-atomic particles, including the nucleus.

John Neumann, a supporter of Schrodinger, used abstruse mathematics to prove that hidden variables cannot be added to quantum mechanics without destroying its verified results. However David Bohm, Louis de Broglie and others produced hidden variable theories which were capable of yielding at least some of the results of quantum mechanics.

The usual interpretation of quantum mechanics has led to the renunciation of the concept of motion within the same domain. Consider the experimental methods of observing the approximation of positions and velocity of an individual electron. If a free electron of high energy passes through a photographic plate, it leaves a record of its track in the form of small grains of silver, which appear in the microscope. These grains of silver are deposited as a result of the interaction of the electron with the atoms nearby – the interaction must take place in the form of quanta, hence the indeterminacy principle will apply. We know that an indeterminate amount of momentum is transmitted to the electron in each interaction, but we cannot predict exactly where the electron will go after leaving the photographic plate.

According to the former view, the track of grains of silver indicates that a real electron has moved continuously through space somewhere near the grains, and by interaction has caused the formation of the grains. However in the now orthodox interpretation of quantum theory it is incorrect to assume that this really happened: all we can say is that certain grains appeared, but we must not suppose that the grains were produced by a real object moving through space. Thus the notion of the continuously moving electron is purely a metaphysical one, which cannot be experimentally verified.

There were a few physicists, such as Einstein and Planck, who continued to believe that a more complete theory explaining individual quanta should be sought. However, most physicists believe that Einstein lost the debate with Bohr and others (popularly known as the Copenhagen Agreement) held around 1926. At that time Bohr’s principle of complementarity best summed up the mainstream view. This stated that we are restricted to complementary pairs of imprecisely defined concepts, such as position and momentum, wave and particle etc. and this has remained the orthodox theory up to the present.

An alternative interpretation of quantum theory was suggested by David Bohm in Quantum Theory (Dover Publications, 1989). Although the wave and particle concept are each, by themselves, incapable of dealing with the full richness of the properties of matter, only two possibilities were considered by Bohr and other mainstream physicists: pure waves or pure particles. The possibility that both concepts may be combined was not considered.

Bohm suggested that the wave and particle could be brought together in some kind of interconnection. He postulated that associated with each of the fundamental particles is a body existing in a small area of space, smaller than the size of an atom. It is assumed that there is a wave associated, and always found with the body. The wave is assumed to be an oscillation in a new kind of field, which is represented mathematically by the Ψfield of Schrodinger, which ceases to be a function of (x,y,z,t). We assume that Schrodinger’s Ψfield and the body are interconnected in the sense that the Ψ field exerts a new kind of quantum force on the body – a force which manifests itself strongly at the atomic level, but not on a large scale. The body may exert a reciprocal force on the Ψ field which is small enough to be neglected at the quantum mechanical level but not at the sub-quantum level. If the above tendency were present the body would eventually find itself in the place where the Ψ field had its highest intensity. It is assumed that this tendency is resisted by random motions undergone by the body, which are analogous to Brownian movements. These may originate from variations in the strength of the field or by interaction with new kinds of entities at a lower level. The net result will be to produce a mean distribution which favours the regions where the Ψ field is most intense.

In the case of the system containing the two slits A and B, every electron is assumed to have the same momentum, and therefore the same wave function, which takes the form of a plane wave perpendicular to the slit system. These waves will diffract through the slit system producing a pattern of high and low intensity. The small body connected with the electron undergoes random motion, and follows an irregular path from a point. Each electron then arrives at the screen at a particular point. After a large number of electrons have passed through the slits we obtain a pattern of points in which the density of the electrons is proportional to the field intensity h/2. They tend to arrive at the point of highest field intensity due to the effects of quantum force. If slit B is closed the wave pattern ceases to have strong and weak fringes, and a different pattern of electrons is obtained at the screen.

David Bohm’s theory that light is both a particle and wave provides an explanation for the light interference patterns produced by the two slits; his theory is more complete than Bohr’s metaphysical version and meets Planck’s stipulation that a theory must be capable of explaining individual quanta. It also accords with Einstein’s famous maxim “God does not play dice” as the ‘random’ nature of quantum mechanics is, according to Bohm’s theory, more apparent than real.

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