To simplify the notation we will use the following

Photon is always traveling along the z-direction.
	Choose 60 degree angle
	Circularly polarization

This exercise will reinforce the role of amplitudes and probability. For classical E&M waves the total field at any point is the sum of the field vectors of the waves impinging on that point while the intensity of the light is proportional to the field squared. Thus the sum of the vectors (vector addition) provides for destructive and constructive interference and plays the role of an amplitude while intensity plays the role of probability. Two coherent beams can arrive at a point where you would find no total beam intensity despite the intensities of each beam not being zero. In this case the electric (and magnetic) fields cancel at the point of observation.

A beam splitter and beam combiner could be used on an optical bench to split and recombine a laser beam based on polarization.

For each of the above bases an apparatus can be built so that any state will be split into its components and then recombined. The arrows indicate that the blocking elements can either be inserted or removed. A and B represent the above states x,y; X,Y; or R,L

Such filters can be used as a filter that selects all or part of the incident beam PROJECTOR which prepares the beam or as an ANALYZER that transmits beam in various combinations or as a MEASURER by measuring the transmission for a given state either A or B. (Measure intensity with one of the paths blocked).

We place three of these systems in a row:

The first filter will prepare the state. You can assume that beam prepared by the first state is normalized to 100%. The second filter will select components and the third filter will measure. In each case the filter can be chosen to work in any of the bases.

The x-y is related to the X-Y by simply rotating the x-y filter by theta to become an X-Y filter. We will not focus on the specific details for splitting and recombining. Just assume the ideal situation.

		Amplitude Squared
TOTAL
TOTAL
TOTAL
=1

1. Amplitude for any path is a product of amplitudes at each step.

The diagram above illustrates that there are many possible paths through the systems. Along each path I can define amplitudes for each sector of the path. We start with an amplitude to arrive at A1è amp(enter at A1). Then we find the amplitudes to proceed from A1 to B1 è amp(A1èB1). The amplitude for any path is the product of amplitudes.

Path 1:

AMP1= amp(A1) amp(A1èB2) amp(B2èC1) amp(C1èD1) amp(D1èX)

Path 2:

AMP2= amp(A1) amp(A1èB3) amp(B3èC2) amp(C2èD3) amp(D3èX)

2. The amplitude for a measured state is the sum of all amplitudes that reach this state.

The amplitude for measuring a particle at X is AMPX=sum over all the ways to reach X

AMPX = AMP1+AMP2+…..

3. Amplitude squared è Probability=AMPX². Interference occurs when amplitudes add constructively or destructively.

RULE 2

The system above can be thought of as having a set of states that span the space and are labeled as indicted in the drawing above.

è state where the particle goes through opening A1 and proceeds to all possible points.

è state where the particle goes through opening A2 and proceeds to all possible points.

è state where the particle goes through opening B2 from any conceivable starting point and proceeds to all possible points.

All the A’s span the space since the particle must go through an A opening. All the B’s and C’s span the space for the same reason.

would be any state that goes through A1 and B2.

would be the amplitude to go A1èB2èC1èD3èX.

If the states are truly complete then

Thus particles that impinge on walls must not be considered. But within this framework any state can be represented by any basis and a particle that goes through any one of the three available paths through a wall can be described by an amplitude for each of these paths.

Looking at the R and L states

In the derivation of amplitudes so far I used the usual rules for the relationship between the primed and unprimed variables that one finds with coordinate transformations. These worked fine.

I now want to find the relationship between all the states. This is easily represented in tabular form.

	1	0
	0	1
			1	0
			0	1
					1	0
					0	1

Let us examine some amplitudes. We won’t assume that we know the overall complex factor for each amplitude. This will be important since for circularly polarized light there will be some complex factors.

The projection of circular to linear must be an overall factor of 1/2. In the formalism for x-y and R-L we explicitly included the . Our arguments would in general lead us to the same statement for all linearly polarized states. The circularly must include an equal amount with a phase shift.

	for all projections of a circularly polarized state onto any plane polarized state
	same as above. The magnitude needs to be 1/2.

Therefore the amplitudes between circularly polarized states and linear polarized states must in general be a product of a complex phase and . Choose some values.

This is a general form for these amplitudes withare all real arbitrary phases.

The above equation will work if .

If the primed and unprimed axes are the same.

If thus .

All of the remaining amplitudes can be determined by these types of evaluation.

	1	0
	0	1
			1	0
			0	1
					1	0
					0	1

These phase conventions are the ones worked out in French and Taylor. They are somewhat different than the ones that I began with

Adding and multiplying amplitudes can be seen in a rather straightforward manner in this two-state system. The role of the relative phase and the overall phase of the states is also demonstrated in the final section.