Constituents of Matter / 5 / 5.2 / 5.2.4

5.2.4 Quantum Superposition Of States

Because an object of explanation is not an object of experience, it does not possess any definite physical properties. When we conceive of the electron which explains Davisson-Germer as a giving together from unity of a particle and a wave, we must give up both the physical properties we usually assign to a particle as well as the physical properties we usually assign to a wave; that object of explanation has no location, no momentum, and no wavelength. However, if we conceived of that object as a synthesis of a particle and a wave, the properties we have assigned to each would have to somehow survive the putting together, which raises the question of how one might reconcile the contradictory nature of those properties.

To see how that reconciliation was accomplished by the founding fathers, let a linearly polarized beam of light fall upon a tourmaline crystal and measure how the fraction of it that passes through the crystal depends upon the angle between the crystal’s optical axis and the initial direction of polarization. This measurement is predicated upon having manufactured the incident beam of light, say by including a Nichols prism between a source of un-polarized light and the crystal, because the arrangement involving such a prism and a source of un-polarized light is well explained by assigning linear polarization to light. The results of the measurement are described in Dirac’s book on quantum mechanics as follows:

Let us assume that a light ray traverses a tourmaline crystal which has the property of letting pass only linearly polarized light in a direction orthogonal to its optical axis. If the incident ray is polarized in a line perpendicular to the optical axis, it will pass the crystal undisturbed; if parallel, it will not pass at all; while if it is polarized in a direction which forms an angle $\alpha$ with the said axis, only a fraction ${\mathrm{s}\mathrm{i}\mathrm{n}}^2\alpha$ of the incident ray will pass the crystal.

This outcome could be explained most easily by conceiving of the incident light as a mixture of two beams: one linearly polarized perpendicular to the optical axis of the crystal and the other linearly polarized parallel to the optical axis. Since the energies of the two components should add up to the total energy of the incident light, we assume that those energies must be to each other as ${\mathrm{s}\mathrm{i}\mathrm{n}}^2\alpha$ is to ${\mathrm{c}\mathrm{o}\mathrm{s}}^2\alpha$. Consequently, the polarization of the object incident upon the crystal must be conceived as a giving together of the linear polarization which explained the manufacture of the incident beam of light and the specific mixture of the two orthogonally polarized beams of light which explains the results of the measurement.

This syndosis, unlike the syndosis of a particle and a wave, is not ontological in nature because it is not a giving together in unity of two objects of explanation but rather of two properties we assign to one and the same object of explanation: light. But, like its ontological counterpart, this syndosis gets understood as a synthesis of two physical properties assigned to an object of experience; one property is the linear polarization we have assigned to the object in order to explain its manufacture, the other is the mixture of the two orthogonal polarizations we have assigned to it in order to explain the observation. These two terms must then be put together in such a way as to reflect the angular relationship between the components of the mixture. Putting together directional properties in this manner is unremarkable; we have been accustomed to doing so for centuries ever since Newton put together the directional properties of mechanical force by the parallelogram rule. Since the electromagnetic theory of light assigns it a polarization described by the direction of the electric component of the electromagnetic field, any linear polarization can be decomposed into two orthogonal ones by the same rule that allows us to decompose the electric force.

However, this unremarkable superposition rule was to be made quite remarkable by the arrival of the photon. As proposed by Einstein in his 1905 paper, a beam of light was no longer to be viewed as a propagating electromagnetic field but rather had to be conceived of as a collection of individual energy quanta. Therefore, the property of polarization we have assigned to an electromagnetic field must now be assigned to each photon separately. Specifically, we can still talk of a photon with a polarization which is perpendicular to the optical axis of the crystal, parallel to that axis, or forming an angle $\alpha$ with it. But we can no longer talk about the latter as a mixture of two orthogonal linear polarizations. Indeed, to do so would imply that, upon traversing the crystal, the photon would have to split in two with only the perpendicular portion thereof making it through to the other side, and, given the individuality of the photon, that is not possible. Therefore, using Dirac’s words:

Quantum mechanics assumes that a photon polarized obliquely to the optical axis can be considered partly in a state of polarization parallel to this axis and partly in a state of polarization orthogonal to it. The state of oblique polarization can thus be considered to be the result of a special process of superposition applied to the two states of polarization perpendicular and parallel.

From this strange superposition of polarization states, the photon must somehow emerge on the other side of the tourmaline crystal, if it emerges at all, in a state of polarization perpendicular to the crystal’s optical axis. How such a superposition may reasonably happen to the physical properties of an object of experience, and how the superposition may then collapse upon one of its terms as a consequence of observing the property, has remained a mystery ever since Einstein first suggested the photon. As far as quantum mechanics is concerned, however, this mystery will have to remain a mystery because the very question has been forbidden by the gatekeeper postulate according to which no question is legitimate which cannot be posed experimentally. All that quantum mechanics is allowed to say is, again in Dirac’s words, that:

the photon has a probability ${\mathrm{s}\mathrm{i}\mathrm{n}}^2\alpha$ of making it through the tourmaline and of appearing on the other side of the crystal with polarization orthogonal to the optical axis and a probability ${\mathrm{c}\mathrm{o}\mathrm{s}}^2\alpha$ of it being absorbed.

This special process of superposition and reduction to one of its terms are a manifestation of the resistance we encounter in trying to take what is fundamentally a syndosis of various states of polarization assigned to an object of explanation as a synthesis of same. The object of explanation we call light has thus become a strange object of experience, one which is conceived to exist in two or more different states of polarization at the same time, a superposition from which an observation will force it into one of its terms. Its strangeness reflects the true ontology of light according to which the property of polarization we assign to it is a syndosis of all possible polarizations, but expressed in a language which conceives of light as an object of experience objectively possessed of all its physical properties; what was contingent thus gets founded in ontology.