#import "@preview/physica:0.9.2": *

#import "/book.typ": book-page
#show: book-page.with(title: "Hello, typst")

= The Polarization of photons

The discussion in the preceding section about the limit to the gentleness with which observations can be made and the consequent indeterminacy in the results of those observations dose not provide any quantitative basis for the building up of quantum mechanics.
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For this purpose a new set of accurate laws of nature is required.
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One of the most fundamental and most drastic of these is the #underline[Principle of Superposition of States.]
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We shall lead up to a general formulation of this through a consideration of some special cases,
taking first the example provided by the polarization of light.


It is known experimentally that when plane-polarized light is used for ejecting photo-electrons,
there is a preferential direction for the electron emission.
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Thus the polarization properties of light are closely connected with its corpuscular properties and one must ascribe a polarization to the photons.
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One must consider, for instance, a beam of light plane-polarized in a certain direction as consisting of photons each of which is plane-polarized in that direction and a beam of circularly polarized light as consisting of photons each circularly polarized.
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Every photon is in a certain #underline[state of polarization, ] as we shall say.
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The problem we must now consider is how to fit in these ideas with the known facts about the resolution of light into polarized components and the recombination of these components.

Let use take a definite case.
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Suppose we have a beam of light passing through a crystal of tourmaline,
which has the property of letting through only light plane-polarized perpendicular to its optic axis.
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Classical electrodynamics tells us what will happen for any given polarization of the incident beam.
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If this beam is polarized perpendicular to the optic axis,
it will all go through the crystal;
if parallel to the axis,
none of it will go through;
while if polarized at an angle $alpha$ to the axis,
a fraction $sin^2 alpha$ will go through.
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How are we understand these results on a photon basis?


A beam that is plane-polarized in a certain direction is to be pictured as made up of photons each plane-polarized in that direction.
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This picture leads to no difficulty in the cases when our incident beam is polarized perpendicular or parallel to the optic axis.
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We merely have to suppose that each photon polarized perpendicular to the axis passes unhindered and unchanged through the crystal,
while each photon polarized parallel to the axis is stopped and absorbed.
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A difficulty arises,
however, in the case of the obliquely polarized incident beam.
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Each of the incident photons is then obliquely polarized and it is not clear what will happen to such a photon when it reaches the tourmaline.

A question about what will happen to a particular photon under certain conditions is not really very precise.
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To make it precise one must imagine some experiment performed having a bearing on the question and inquire what will be the result of the experiment.
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Only questions about the results of experiments have a real significance and it is only such questions that theoretical physics has to consider.

In our present example the obvious experiment is to use an incident beam consisting of only a single photon and to observe what appears on the back side of the crystal.
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According to quantum mechanics the result of this experiment will be that sometimes one will find a whole photon,
of energy equal to the energy of the incident photon,
on the back side and other times one will find noting.
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When one find a whole photon,
it will be polarized perpendicular to the optic axis.
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One will never find only a part of a photon on the back side.
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If one repeats the experiment a large number of times,
one will find the photon on the back side in a fraction $sin^2 alpha$ of the total number of times.
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Thus we may say that the photon has a probability $sin^2 alpha$ of passing through the tourmaline and appearing on the back side polarized perpendicular to the axis and a probability $cos^2 alpha$ of being absorbed.
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These values for the probabilities lead to the correct classical results for a incident beam containing a large number of photons.


In this way we preserve the individuality of the photon in all cases.
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We are able to do this,
however, only because we abandon the determinacy of the classical theory.
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The result of an experiment is not determined,
as it would be according to classical ideas,
by conditions under the control of the experimenter.
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The most that can be predicted is a set of possible results,
with a probability of occurrence for each.


The foregoing discussion about the result of an experiment with a single obliquely polarized photon incident on a crystal of tourmaline answers all that can legitimately be asked about what happens to an obliquely polarized photon when it reached the tourmaline.
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Questions about what decides whether the photon is to go through or not and how it changes its direction of polarization when it does go through cannot be investigated b experiment and should be regarded as outside the domain of science.
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Nevertheless some further description is necessary in order to correlate the results of this experiment with the results of other experiments that might be performed with photons and to fit them all into a general scheme.
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Such further description should be regarded,
not as an attempt to answer questions outside the domain of science, 
but as an aid to the formulation of rules for expressing concisely the results of large numbers of experiments.


The further description provided by quantum mechanics runs as follows.
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It is supposed that a photon polarized obliquely to the optic axis may be regarded as being partly in the state of polarization parallel to the axis and partly in the state of polarization perpendicular to the axis.
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The state of oblique polarization may be considered as the results of some kind of superposition process applied to the two states of parallel and perpendicular polarization.
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This implies a certain special kind of relationship between the various states of polarization,
a relationship similar to that between polarized beams in classical optics,
but which is now to be applied,
not to beams,
but to states of polarization of one particular photon.
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This relationship allows any state of polarization to be resolved into.
or expressed as a superposition of,
any two mutually perpendicular states of polarization.


When we make the photon meet a tourmaline crystal,
we are subjecting it to an observation.
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We are observing whether it is polarized parallel or perpendicular to the optic axis.
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The effect of making this observation is to force the photon entirely into the state of parallel or entirely into the state of perpendicular polarization.
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It has to make a sudden jump from being partly in each of these two states to being entirely in one or other of them.
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Which of the two sates it will jump into cannot be predicted,
but is governed only by probability laws.
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If it jumps into the parallel state if gets absorbed and if it jumps into the perpendicular state it passes through the crystal and appears on the other side preserving the state of polarization.
