Quantum Mechanics: New Approaches to Selected Topics

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POLARIZED PHOTONS FOR PEDESTRIANS

Photon polarization was used by Dirac as an introduction to quantum mechanics. Chapter 1 presents an expanded treatment of this simple case which introduces many basic concepts, phenomena and paradoxes of quantum mechanics at a level which requires only the simple mathematics of 2x2 matrices. This chapter is easily read by a student familiar with elementary matrix algebra and the Dirac notation for vector spaces and matrix elements.

All the mathematical formalism necessary for the quantum-mechanical description of photon polarization is introduced naturally at the classical level: the description states of polarization by complex vectors in a two-dimensional space, of apparatus which change the polarization by matrices, of polarization measurements by expectation values of Hermitean operators which also describe changes of polarization produced by measuring apparatus and whose eigenvectors are the particular states which pass through the measuring apparatus unaffected (e.g. those polarized in the direction of the principal axes of a nicol prism) and the use of unitary transformations between different bases.

Unpolarized and partially polarized light introduce the quantum nature of light already at the classical level, since they cannot be described as continuous classical waves but only as series of discrete short pulses or quanta with no correlation between the polarizations of successive pulses. The density matrix formalism is introduced to describe partially polarized light in the two-dimensional vector space.

Sections 1.5–1.9 show how this classical description must be modified in view of the quantum nature of electromagnetic radiation; i.e. that it is composed of discrete quanta. The basic features of the quantum description are obtained by considering how a single photon must behave in the various experimental situations discussed previously for the classical case, with the aid of the correspondence principle which requires that experiments performed with very large numbers of photons should give the classical results. The results of certain experiments are shown to be unpredictable even if all the available information about the initial conditions of the experiment are known. This leads to a discussion of the statistical description, the interaction of the observer with the system, the incompatibility of different measurements and wave-particle duality.

CHAPTER 1

POLARIZED PHOTONS AND QUANTUM THEORY

1.1 Polarization measurement for a classical beam

Consider a polarization measurement on a classical beam of polarized light. The measuring apparatus is a Nicol prism which splits the beam into two components, one horizontally polarized and one vertically polarized as shown in fig. 1.1.

Fig. 1.1

The relative magnitudes of the horizontally and vertically polarized components depend upon the properties of the incident beam. Measuring the intensities of these two components gives information about the incident beam, but does not generally supply sufficient information to determine its polarization. For example, if the intensities of the horizontal and vertical components are found to be equal, the incident beam might be circularly polarized left-handed or right-handed, plane polarized at an angle of ± 45° with respect to the horizontal, or elliptically polarized in any one of a continuum of states. To specify the polarization of the incident beam completely, we need to measure not only the relative magnitudes of the horizontal and vertical components, but also the relative phase.

More information about the incident beam is obtained from two successive measurements of the type shown in fig. 1.1, with the Nicol prism in different orientations. The second measurement could be, for example, with the Nicol prism rotated by an angle of 45°, thus separating the beam into two plane polarized components with the planes of polarization at angles of +45° and ‒45° with respect to the horizontal. In the case considered above, where the horizontally and the vertically polarized components were found to be equal, the second measurement would enable a choice to be made between circular polarization, plane polarization at an angle of 45°, and elliptical polarization. However, even two measurements do not completely specify the polarization of the incident beam. There remains one ambiguity. Suppose, for example, that the intensities of the two 45° components were also found to be equal, as well as the horizontal and vertical components. This would definitely establish the beam to be circularly polarized, but would not distinguish between right-circular and left-circular polarization.

The number of measurements required to specify the initial polarization is seen by the following analysis. The beam is completely specified by giving the values of the magnitudes and the phases of its two components; i.e., by the values of four parameters. If we are not interested in the absolute intensity of the beam or its absolute phase, but only in the state of polarization, we require only two parameters: the ratio of the amplitudes and the relative phase. Two measurements give us two equations from which we can determine these two parameters. However, measurements of intensities which are quadratic functions of the amplitudes give quadratic equations which have two solutions. These two solutions produce the ambiguity which in the above example fails to distinguish between the right and left circular polarization.

Thus more than a single measurement of the type shown in fig. 1.1 is generally needed to determine the state of polarization of the incident beam. If we were limited to a single measurement, we would not in general be able to specify the polarization completely. Such limitations might occur in practice, for example, if the beam consisted of a short pulse of radiation too short for two successive experiments. Such a difficulty might be overcome by splitting the beam into two components, e.g. with a half-silvered mirror, and performing two independent experiments on each component. However, the incident light pulse might be of such low intensity that our detecting apparatus could not measure it accurately if the intensity were further reduced by splitting the beam. In classical physics, these difficulties are only practical ones which can always be surmounted in principle by building better apparatus. It is always possible in classical physics to measure the state of polarization of an incident beam. We shall see that in quantum physics we can be limited to performing only one experiment on a particular incident beam. In such a case, we shall not generally be able to determine its state of polarization completely.

The measuring apparatus shown in fig. 1.1 also modifies the incident beam and determines the state of polarization of each emergent beam. The two beams are always horizontally polarized and vertically polarized respectively, independent of the state of polarization of the incident beam. The incident beam merely determines the relative magnitudes of these two components; their states of polarization are determined by the measuring apparatus; i.e., by the orientation of the Nicol prism. If the Nicol prism is rotated by an angle of 45°, the two beams emerging from the prism are plane polarized at angles of + and ‒45° with respect to the horizontal, independent of the polarization state of the incident beam. Again, the incident beam determines only the relative magnitudes of these components, the apparatus determines their polarization. This is an interesting example of the interaction between the measuring apparatus and the particular system being measured. The system being measured is changed or transformed by the apparatus in a manner which is characteristic of the apparatus.

In classical physics this interaction and transformation is completely reversible and has no special significance. One could, for example, reconstruct the original incident beam from the two components produced in fig. 1.1 by feeding them through an appropriate optical system into a Nicol prism which combines the two components again as shown in fig. 1.2. If no measurements are made in the space between the two Nicol prisms, the system is simply a complicated black box in which the incident beam enters and exactly the same beam comes out. The intensities of the individual horizontal and vertical components can be measured while they are separated in the space between the two nicols. Thus in classical physics ail the information about the incident beam can be measured in an apparatus from which the incident beam emerges unchanged.

Fig. 1.2

In practice the measurement of the individual components between the nicols in fig. 1.2 requires taking a small amount of energy from each of these components in order to produce a detectable signal in the measuring apparatus. If the incident beam were very weak, this removal of energy might change the relative magnitude and phase of the two components to make the beam emerging from the second nicol quite different from the incident beam. It would therefore not be possible to measure the intensities of the horizontal and vertical components without changing the incident beam. In classical physics such limitations are considered to be due only to the deficiencies of the apparatus. It is always possible in principle to find a better and more sensitive apparatus which can measure the horizontal and vertical components without appreciably affecting the beam. In quantum physics, there are limits to the sensitivities of these measurements. These make it impossible to measure the intensities of both the horizontal and vertical components without perturbing the beam to an extent which prevents reconstructing the incident beam.

There are certain special cases for which much of the above discussion does not apply. In these cases a single measurement suffices to determine the state of polarization of the incident beam completely and the polarization of the incident beam is not changed in passing through the apparatus. For the system shown in fig. 1.1, this occurs when the incident beam is either fully horizontally polarized or fully vertically polarized. The full intensity is observed in one of the two channels and zero intensity in the other. The exact state of polarization of the incident beam is then known from a single measurement and the emerging beam is in exactly the same polarization state as the incident beam. If the nicol were rotated by an angle of 45°, this situation would no longer be true for incident beams which were plane-polarized horizontally or vertically. Instead, plane-polarized beams fully polarized at angles of ± 45° with respect to the horizontal would now pass through the nicol unchanged, give full intensity in one channel and zero in the other, and allow a complete specification of the polarization of the incident beam to be made on the basis of this single experiment. For any orientation of the nicol, there are always two such states of polarization for the incident beam which satisfy the above conditions. These two states can be conveniently described as the ‘eigenstates’ of the operation produced by the nicol, as shown above.

1.2 A matrix representation of light polarization

The electric vector for a beam of polarized light propagating in the z-direction can be written in the form

where ex and ey are unit vectors in the x and y-directions respectively. A convenient and compact way to represent this is by a vector or column matrix with complex components

where E1 and E2 . The x-direction is chosen as horizontal; the j-direction as vertical. In this notation a beam of amplitude E, polarized in the vertical (i.e., y) direction, would be represented by the vector

There are two directions of 45° polarization. When the vectors Ex and Ey are equal and in phase we call the angle +45°; when the vectors Ex and Ey are equal and opposite we call the angle ‒45°.

The experiment shown in fig. 1.1 is easily described in terms of these matrices. Let ξ as defined in eq. (1.2) represent the state of the incident beam. Then the two beams emerging at points A and Β in fig. 1.1 are represented by the vectors

These vectors are related to the vector representing the incident beam by the matrix equations

where A and Β are the matrices

The intensities of the beams observed at A and Β are given by the norms of the vectors ξΑ and ξΒ. From eq. (1.4),

as expected. The intensities can also be expressed in terms of the vector ξ denoting the incident beam and the operators A and Β characteristic of the apparatus. The operators A and Β are both Hermitean and are both projection operators; i.e.,

Thus

The difference between the intensities observed at A and at Β is given by

where the matrix Μ is defined as

The results of these intensity measurements are thus expressed simply in terms of ‘diagonal matrix elements’ or ‘expectation values’ of the Hermitean operators A, B, and Μ with the initial vector ξ. We also see that the intensities as given by eq. (1.7) depend only on the magnitudes of the components E1 and E2 and therefore give no information about the relative phase.

In the preceding section we saw that an incident beam which is fully polarized horizontally or vertically is unchanged in passing through the nicol and one intensity measurement gives complete information. This is described very simply in our matrix notation. If the incident beam is fully polarized horizontally, E2 = 0, ξΑ = ξ and ξΒ = 0. The incident vector ξ is an eigenvector of the matrices A, B, and Μ which represent the transformation of the incident beam by the apparatus. When the vector representing the incident beam is an eigenvector, these matrices do not change the vector but merely multiply it by a constant; i.e., they do not change the state of polarization but may change the intensity. Similarly, if the incident beam is fully polarized vertically, E1 = 0, ξΑ = 0, ξΒ = ξ and ξ is again an eigenvector of the matrices A, B, and M.

1.3 Matrix transformations

Let us now consider the matrix description of an experiment with the nicol prism of fig. 1.1 rotated by an angle of 45° to make the component polarized at an angle of +45° appear at A and the component polarized at ‒45° appear at B. Beams of amplitude E, polarized at angles of +45° and ‒45°, respectively, are represented by the vectors

If the incident beam is represented by the vector ξ, eq. (1.2), the beams observed at A and Β are represented by the vectors

where primes are used in order to avoid confusion with the preceding case eq. (1.4). The corresponding intensities are given by the norms of these vectors

Relations analogous to eqs. (1.5), (1.9) and (1.10) are obtained by defining the analogous matrices

The matrices Α’, B’ and Μ’ are all Hermitean as before, and A’ and B’ are both projection operators, although none of the three is diagonal like the corresponding matrices in the preceding simple case. The two vectors (1.12) representing plane-polarized beams at angles of +45° and ‒45° are the eigenvectors of the matrices A’, B’, and M’, This is in accord with the physical observations that beams polarized at angles of +45° and ‒45° pass through the apparatus without being changed and that a single measurement can determine the complete state of polarization of such a beam.

A simpler description of this second experiment is obtained if the base vectors are chosen to represent beams polarized at angles of +45° and ‒45° rather than horizontally and vertically. In this case, beams of amplitude Ε polarized at angles of +45° and ‒45°, respectively, are represented by the vectors

Let the incident beam be represented by

represents the component polarized at ‒45°. In this representation the beams observed at A and Β are represented by the vectors

and the corresponding intensities are given by

The matrices Α’, B’ and M’ in this representation now become

Eqs. (1.18), (1.20), (1.21) and (1.22) look quite different from the corresponding equations (1.12), (1.13), (1.14), and (1.15) which describe exactly the same experiment using a different set of base vectors. However, eqs. (1.19), (1.20), (1.21), and (1.22) look exactly like eqs. (1.2), (1.4), (1.7), (1.6), and (1.11), which describe the experiment with the horizontal-vertical nicol, using a different set of base vectors. That two different experiments have the same formal description is not as strange as it seems, because these two different experiments are not really so different. They differ only in the rotation of the apparatus by 45° relative to our x and y-coordinate axes. If we rotate our coordinate axes along with the apparatus, then the second experiment in the rotated coordinate system looks exactly like the first experiment in the unrotated coordinate system. Our second description of the second experiment using base vectors at angles of +45° and ‒45° corresponds to rotating the coordinate system along with the apparatus and therefore gives exactly the same formal description as that of the first experiment in the unrotated coordinate system.

The two seemingly different descriptions of the second experiment are related to one another by a rotation of the coordinate axes by 45°. This transformation from one set of base vectors to another should be expressible as a unitary transformation on the vectors and matrices

where Κ represents any one of the matrices A, B or M, and ξ is an arbitrary vector. The unitary matrix U required to describe the transformation can be easily found. It is just the orthogonal matrix representing a rotation by an angle of 45°

Substituting the expression (1.24) for the matrix U into eq. (1.23a) shows that the matrix (1.24) does indeed perform the transformation indicated on the matrices Α’, Β’, and Μ’. Substituting eq. (1.24) for U and the corresponding quantities E1 and E2

Eqs. (1.25) show that the expressions for the intensities observed at A and Β in the two descriptions, eqs. (1.14) and (1.21), are equal, although the two sets of equations appear to be quite different. This is characteristic of two descriptions of the same physical experiment in different bases or coordinate systems. The numbers describing the components of a vector or matrix in different coordinate systems (or their representatives in different bases) are, in general, quite different from one another. However, the numbers describing the result of a physical measurement must be the same in any coordinate system.

It is now interesting to examine the unitary matrix (1.24). Since it is unitary, it has eigenvectors and eigenvalues of modulus unity. The eigenvectors of the matrix U are just

which are indeed of modulus unity. The eigenvectors (1.26) represent those states of polarization which are not changed by the transformation (1.24) which is a rotation by an angle of 45°. Examination of the specific form of the eigenvectors (1.26) shows that these represent the two states of circular polarization. These circularly polarized states each remain circularly polarized in the same sense after a rotation of 45°. The only change produced by the transformation is the introduction of a phase factor of +45° or ‒45° depending on the sense of polarization. These phase factors eiπ/4 and e –iπ/4 are just the two eigenvalues of modulus unity.

1.4 Unpolarized and partially polarized light

What is unpolarized light? The light from the sun is unpolarized. If sunlight is passed through the apparatus of fig. 1.1, half will come out horizontally polarized and half vertically polarized. If the nicol is rotated by an angle of 45°, half will come out polarized at +45°, half at -45°. If the nicol is rotated by any angle, the beam will always be .split into two equal components. So far, the behavior of the beam is like that of a circularly polarized beam, which is also split by a nicol into two equal components, independently of the orientation of the nicol. But if the unpolarized beam is passed through a quarter-wave plate which would change circularly polarized light into plane-polarized light, the unpolarized beam remains unpolarized. The unpolarized beam is an equal mixture of horizontally and vertically polarized components, but there is no definite phase between the two components.

An unpolarized beam cannot be represented by an expression of the form (1.1) or by the matrix representation (1.2). Such expressions always have a definite phase between the horizontal and vertical components. The existence of unpolarized light already gives an indication of the quantum nature of light. Unpolarized light cannot be described as a single simple classical wave of the form (1.1) or as any linear combination of such waves. Unpolarized fight can be described classically as a series of very rapid short bursts or pulses of light, each having a different polarization, with no correlation between the polarization of different pulses. If these pulses and the interval between them are very short compared to the characteristic times of the measuring apparatus, they will be detected as a continuous beam, and any polarization measurement will give an average of the polarizations of the individual pulses.

There are also partially polarized beams. Suppose an unpolarized beam is passed through an apparatus like that of fig. 1.2, and half of the vertical component is absorbed in the space between the two nicols. The reconstructed beam will now have a vertical component which is half the intensity of its horizontal component, but with no definite phase relation between the two components. It can also be considered as a mixture of a horizontally polarized beam and an unpolarized beam. Such a beam is called a partially polarized beam.

Let us now examine the matrix description of unpolarized and partially polarized beams. For example, consider a beam of short pulses of which one third are horizontally polarized, one third polarized at +45° and one third polarized at ‒45°. If the total intensity of the beam is E, the three components are represented by vectors

However, these three components cannot be combined into a single vector because they are incoherent. They represent pulses occurring at different times which cannot interfere with one another.

The results of any experiment with this beam can be calculated with our matrix formulation by calculating the intensities for each of the three components and then adding the intensities. Thus, the difference between the intensities observed at A and Β in the apparatus of fig. 1.1 is given by eq. (1.10) as

This notation becomes very cumbersome if there are many components in the beam. A more convenient description of a partially polarized beam is obtained by use of a matrix instead of a vector.

Let us rewrite eq. (1.28) specifying the matrix indices explicitly

This expression can be simplified by defining the matrix

Then

ji is called the ‘density matrix’.

Eq. (1.30b) shows that the four elements of the density matrix provide sufficient information about the incident beam for the calculation of the result of the experiment of fig. 1.1. Similarly for any measurement whose result is expressed as the expectation value of a matrix, like eq. (1.10), the corresponding expression for a partially polarized beam is given by the corresponding expression of the form (1.30b); namely the trace of the product of the density matrix and the matrix describing the measurement.

Eq. (1.30b) shows that the density matrix contains all the necessary information about an unpolarized or partially polarized beam to describe all intensity measurements which sum over all the pulses in the beam and do not resolve the individual components. The density matrix is thus a generalization of the vector notation (1.2) which describes a beam that can be an ‘incoherent mixture’ of several components as well as a ‘pure state’ of definite polarization.

For an unpolarized beam, which can be considered as half horizontally polarized and half vertically polarized, the density matrix is proportional to the unit matrix,

An unpolarized beam can also be considered as half polarized at +45°, and half at ‒45°, or as a 50-50 mixture of any two components which are described by two orthogonal vectors. The definition (1.30a) shows that the same expression (1.31) is obtained in all these cases.

The total intensity of a beam is given by the trace of the density matrix,

1.5 Introduction of quanta

We know that electromagnetic radiation cannot be infinitely subdivided into beams of smaller and smaller intensity. The smallest possible unit of radiation is the fight quantum which has energy E=hv, where h is Planck’s constant and ν is the frequency of the radiation. These quanta, or photons, are indivisible and any device measuring beam intensity can only detect an integral number of quanta.

What happens when a single photon is introduced as the incident beam into the apparatus of fig. 1.1? This photon cannot be split with one part coming out of the nicol at A and the other part coming out at B. The photon is indivisible; if it comes out at all, it must come out either at A or at B. However, if a classical beam polarized at an angle of 45° is introduced into the apparatus half the intensity comes out at A as horizontally polarized light and half the intensity comes out at Β as vertically polarized fight. In order to obtain a unified description of light which includes classical beams (many photons) as well as single photons, we must reconcile the classical results with the indivisibility of photons on the quantum level.

The classical experiment uses a beam consisting of a very large number of photons so that the quantum structure is not observed. If this beam is fully polarized at an angle of 45°, all the photons in the beam have a 45° polarization. The classical result tells us that half of the incident photons come out at A as horizontally polarized photons, and the other half come out at Β as vertically polarized photons. We can obtain a consistent description both of the single-photon and the many-photon case by saying that a single photon may come out either at A or at Β with equal probability. We cannot predict which way a given single photon will go. However, when many photons are incident, on the average half of them come out at A and half come out at B.

This probability or statistical description represents a definite break with concepts of classical physics. If in classical physics we know all possible information about a given system at a given time, we can predict its future behavior exactly by use of the equations of motion. In this example we are given a system where we know everything about it at a given time. There is a single photon polarized at an angle of 45° with respect to the horizontal incident upon a nicol prism. However, we are completely unable to predict whether this photon will come out at A or at B.

One might compare the statistical nature of this description with a random process such as flipping a coin. If the coin is symmetrical and there is no bias in the flipping mechanism, one expects on the average that the number of’heads’ equals the number of’ tails’. On the other hand, one cannot predict the result of a single flip of the coin. If the coin is flipped a number of times, the number of ‘heads’ should be approximately equal to the number of ‘tails’ with a difference between them due to the statistical fluctuations which is of the order of the square root of the total number of times the coin is flipped. The percentage fluctuation therefore decreases as the number of flippings increases. If the coin is flipped a million times, the statistical fluctuations are of the order of 0.1 percent. The photon experiment appears to resemble the coin experiment. If many 45° polarized photons are incident upon the Nicol prism, the number observed at A is approximately equal to the number observed at Β with statistical fluctuations of the order of the square root of the total number of photons. However, there is a crucial difference between the coin flipping example and the photon experiment. A coin is a macroscopic object which we believe to be governed by the laws of classical physics. If we know the exact position and orientation of the coin, the forces acting upon it at the time it is flipped and the elastic and frictional constants of the coin and of the surface on which it falls, we should in principle be able to predict each time whether it will fall heads or tails. If we are unable to do so and can only give a statistical description, it is only because of our ignorance. We do not know everything that we could in principle know about the coin.

One can ask if the same description applies to the incident photon. It might be something much more complicated than we think, with a complex structure involving other degrees of freedom of which we are as yet unaware. If we knew all about this substructure and knew the values of all these ‘hidden variables’ we might be able to predict how each individual photon would pass through the nicol and whether it would come out at A or at B. The quantum theory asserts that this is not the case and that our inability to predict whether an individual photon will come out at A or at Β is a fundamental property of nature. So far there has been no experimental evidence for the existence of these hidden variables and it is generally believed that they do not exist*. On the other hand, the quantum theory which assumes this unpredictability has had great success in describing many phenomena.

The matrix representation developed in treating the classical problem is very useful in discussing the quantum-mechanical case. Let the vector ξ, eq. (1.2), represent the state of polarization of a photon.

Let us normalize the vector ξ for a single photon so that

The classical results for the intensities obtained at A and Β (eqs. (1.7), (1.9), and (1.10)) can now be interpreted as giving the average number of photons observed at A and Β and the average value of the difference. If there is only one photon, the average number of photons appearing at any point must be less than or equal to one and this ‘average number’ is simply the probability that the photon appears at this point.

Thus in the quantum case the diagonal matrix elements, or expectation values, of the Hermitean operators, A, B, and Μ with the initial vector ξ give the mean values of experimental measurements on a single photon, or the average of a series of measurements on many photons.

1.6 The interaction of the observer with the system

Consider the apparatus of fig. 1.1 as a measuring apparatus to determine the state of an incident photon. Suppose a single photon is observed at A. We know that the incident photon was not fully polarized vertically, because then it would have had to come out at B. However, there is little more that can be said about the state of the incident photon, since any state other than full vertical polarization would give some probability that the photon would be observed at point A. There is no possibility of finding out any more information about this one photon. Once it is observed at point A, it is already fully horizontally polarized, and it is no longer possible to make any measurements to determine what the polarization was before passing through the nicol. Thus the process of measurement has given us some information about the state of the photon but has also disturbed this photon in such a way as to destroy all record of its previous state. Suppose we now repeat this experiment many times with many photons all prepared in the same way, so that we know that they are all initially in the same state. We thus measure the probabilities for finding the photon at A and at B. If the incident photon is described by a normalized vector ξ (eq. (1.2)), this experiment repeated on many photons measures the absolute values of the complex numbers E1 and E2. The relative phase of these numbers is not measured.

Consider now a single photon incident upon the apparatus shown in fig. 1.2 in which it passes through two nicol prisms, the first of which splits it into horizontal and vertical components, and the second of which brings them together and reconstitutes the incident beam. If no measurements are made in the space between the two nicols, one has classically a complicated black box in which the incident beam enters and exactly the same beam comes out at the other end. Since the classical experiment means an incident beam of many photons all having the same polarization and all of which emerge from the black box having the same polarization as they had initially, each individual photon must come out of the box with the same polarization it had upon entering. Otherwise, there would be an inconsistency between the single photon and the classical result.

One can now ask the question, ‘if a single photon enters the apparatus, polarized at an angle of 45° with respect to the horizontal, and leaves the apparatus also polarized at an angle of 45°, which path did it take in going between the two nicols – did it go through point A or through point B?’ This question cannot be simply answered without immediately encountering contradictions. If the photon went through point A, it must have been horizontally polarized at that time and there was then no photon at point B. How then could the photon regain its 45° polarization in going from A through the second nicol? In the classical problem there is no paradox because the incident beam is a wave which splits into two components after passing through the first nicol and then recombines after passing through the second. In order to explain the observed experimental results with single photons, each individual photon, although it is an indivisible quantum, must still somehow split into two components in passing between the two nicols and these parts must come together again in the second nicol.

The photon thus has both particle and wave properties.

As long as the photon is undisturbed by measuring apparatus which attempts to determine exactly where it is, it propagates as a wave described by Maxwell’s equations. It is only when the photon interacts with matter that the particle-like properties manifest themselves in requiring that the total quantum of energy, E=hv, and of momentum, p=h/λ, must be absorbed in a single event. The relation between the wave and particle aspects is again provided by the statistical interpretation. The place where the photon will be absorbed cannot be predicted with certainty; rather the probability that the photon is absorbed in any one place is proportional to the intensity of the wave at that point. Once the photon is absorbed, it is gone. The amplitudes indicating the propagation of this photon in other places immediately drop to zero as soon as the photon has been absorbed.

Let us now return to the question of which way the photon went between the two nicols in fig. 1.2. There is no clear answer to this question, somehow it went both ways. That quantum theory cannot tell which way the photon went should not be considered a failure of the theory. Rather, quantum theory says that this question has no meaning. The quantum theory states that a photon is something more complicated than a scaled-down version of a macroscopic particle or of a macroscopic wave which we encounter in our everyday experience. There is really no reason why these sub-microscopic particles which we can never see directly need be scaled-down versions of things which we can see. The photon is thus an object which propagates through space like a wave, but can suddenly appear as a particle in a manner which can only be predicted statistically.

There is an important distinction between the two questions (1) ‘Did the photon go through A or through Β in fig. 1.2?’ and (2) ‘Which way will the incident photon come out of the nicol in fig. 1.1 - at A or at B?’ Both cannot be answered by the quantum theory. Question (1) is meaningless and based on the erroneous assumption that a photon is a particle like a scaled-down version of macroscopic particles. This question does not concern any quantity which can be measured and is therefore something like the medieval question of how many angels can dance on the point of a pin. Question (2) is meaningful and refers directly to physically measurable quantities; e.g., signals observed in photomultipliers at A and B. That this question cannot be answered is an important property of nature. It represents a definite break with classical physics which requires that all experimentally observable quantities should be predictable if information completely specifying the state of the system has been given previously.

Let us now return to the experiment of fig. 1.2 and check whether it is indeed impossible to measure whether the photon has gone by point A or point B. Suppose a very sensitive apparatus is placed at A so that the horizontally polarized photon passing through point A may scatter an electron into a photomultiplier. Because this requires a transfer of energy and momentum from the photon to the electron, the photon’s energy and momentum and therefore its frequency and wavelength are changed after the collision with the electron. This change produces a change in the phase of the horizontally polarized wave as it passes through the second nicol. Thus even if both the horizontal and vertical components propagate as classical waves, the two can no longer recombine in the nicol to reproduce the incident beam.

In a classical experiment one can make the perturbation of the horizontal component by the electron scattering negligibly small so that the incident beam is still reproduced. However, there are limits to the smallness of this perturbation because of the quantum nature of the electron: i.e., because there is also wave-particle duality in the electron. The initial position and momentum of the electron are uncertain in a manner described by quantum theory. This uncertainty produces an uncertainty in the position along the photon path where the momentum is transferred and in the amount of momentum transferred. Both of these uncertainties contribute to an uncertainty in the phase of the horizontal component as it passes through the nicol. No matter how one arranges the apparatus it turns out that this uncertainty in phase is so large that all coherence between the horizontal component and the vertical component is destroyed. Thus even if the incident beam is considered as a classical wave which splits into two components it has the phase of the horizontal component altered by its interaction with the electron and the resultant beam emerging from the second nicol is no longer a beam with the same polarization (e.g., 45°) as the incident beam. It is an incoherent superposition of horizontally and vertically polarized components; i.e. an unpolarized beam. In our wave-particle description, we can simply say that the incident photon entered the nicol and propagated like a wave through the two channels A and B. However, as soon as a scattered electron was observed at A we knew that the photon had passed through A and was horizontally polarized. It therefore passed through the second nicol without any vertically polarized companion and emerged as a horizontally polarized photon.

1.7 States which are not changed by measurement

When a single photon enters the apparatus of fig. 1.1 we are in general unable to predict whether it will come out at A or at B. On the other hand, if it does come out at A then it is fully horizontally polarized and has been irreversibly altered by passing through the nicol. A series of such measurements on a number of photons identically prepared generally does not give enough information to allow a determination of the initial state of the photon.

In the classical discussion we noted special cases in which a single measurement sufficed to determine the state of polarization of the incident beam completely, and where the state of polarization of the incident beam was not changed in passing through the apparatus. These special cases have the same properties for photons. In the experiment of fig. 1.1 a photon which enters the nicol fully horizontally polarized, will certainly be observed at point A and its polarization will be unchanged; i.e., it will remain horizontal. An analogous situation obtains for an incident vertically polarized photon which can be predicted with certainty to appear at point Β with an unchanged vertical polarization. In an experiment with a number of photons, all prepared in the same way, and horizontally polarized, one would observe that ail come out of the nicol at A and none at B. One would then conclude on the basis of this single measurement that these photons were all fully horizontally polarized. Again the situation is analogous for incident vertically polarized photons.

As in the classical case, this no longer occurs if the nicol is rotated by an angle of 45°. Then incident horizontally or vertically polarized photons emerge from the nicol, sometimes at A and sometimes at B, with their polarizations changed to +45° or ‒45°, respectively. On the other hand, the behavior of photons polarized at +45° can be predicted with certainty. They pass through the nicol without having their state of polarization changed. If a large number of identically prepared 45° polarized photons are measured, they appear either all at A or all at B, depending on whether the initial polarization is +45° or ‒45°. A complete specification of the state of the incident photons is possible on the basis of this one experiment. For any orientation of the nicol, there are similarly always two orthogonal states of polarization for incident photons which pass through the nicol unchanged and whose observation at A or Β can be predicted with certainty. The same is true for combinations of Nicol prisms with quarter-wave plates before and after the prism such that a classical incident beam is split into components at A and at Β which are circularly or elliptically polarized. For these cases, there are again two orthogonal photon states, of circular or elliptical polarization respectively, which pass through the apparatus unchanged and for which their observation at A or Β can be predicted with certainty.

The above discussion can be stated concisely in the matrix representation. For all orientations of the nicol, or of combinations of nicols and quarter-wave plates, the vectors representing the beam or the photon at A and Β can be expressed in terms of some linear operator or matrix operating on the vector denoting the incident beam. The states represented by the eigenvectors of these matrices are those states whose polarization remains unchanged in passing through the apparatus. Since these matrices are all 2x2, they have two linearly independent eigenvectors. There are therefore always just two states for the incident photon which have the special properties discussed above.

For each orientation of the nicol, or arrangement of nicol and quarter-wave plates, there are two different states of polarization for which the result of the measurement can be predicted with certainty. There is no single state of polarization for the photon such that results can be predicted with certainty for each of two different types of experiments.

Consider for example, the experiment with the nicol which separates the incident beam into horizontal and vertical components, and an experiment with quarter-wave plates and a nicol which separates the incident beam into left-circularly and right-circularly polarized components. If the incident photon is plane-polarized, horizontally or vertically, the result of the first experiment can be predicted with certainty, but the result of the second cannot. If the incident photon is right-circularly or left-circularly polarized, the result of the second experiment can be predicted with certainty, but that of the first cannot. It is impossible to prepare a photon in such a state of polarization that the results of both of these experiments could be predicted with certainty. This is not surprising since there is no paradox in the statement that a photon cannot be fully plane-polarized and fully circularly polarized at the same time. That these two types of measurements are incompatible is not startling, since full circular polarization and full plane polarization are known to be incompatible even for a classical beam of polarized light. We shall find, however, that this concept of incompatibility extends to other properties which are not incompatible in classical physics. For example, it is impossible to prepare a particle in a state where a measurement of its position and a measurement of its momentum can both be predicted with certainty.

Remember, however, that the inability to predict both polarization measurements with certainty is a purely quantum effect, arising from the indivisibility of the photon. Classically one could predict with certainty that a circularly polarized beam incident upon the apparatus of fig. 1.1 would emerge with half of its intensity at A and half at B.

1.8 Measurements with single photons

When a classical beam is passed through the apparatus of fig. 1.1 the fraction of the intensity appearing at A or at Β can have any value from 0 to 1, depending upon the incident beam. When a single photon is passed through the apparatus, the only possible values are 0 and 1, and the continuous spectrum of values in between are not allowed. These allowed values for a single photon measurement are very simply expressed in the matrix representation, they are just the eigenvalues of the matrix which describe the particular measurement, e.g., the matrices A, B and M, eqs. (1.6) and (1.11).

The use of matrices to describe measurements allows the following properties of measurements involving single photons