The goal is to imagine how we could select quantum state with an experimental apparatus. The Stern Gerlach experiment suggests a method of choosing a specific spin state.

An electron beam can be split because the electron has an additional property known as spin. The difference in the force on the electron and the direction of spin is used to pull a beam with a mixture of electrons of different spins apart. A simplified diagram of the apparatus is shown above. An electron beam entering the apparatus is pulled apart and one spin orientation is pulled up while the other is pulled down. An absorber placed so as to block the path of the down going electrons is depicted above. With the absorber in place the apparatus can be used to select electrons with a specific spin orientation.

We will consider a similar apparatus but for photons.

Interpret as the probability. This will be used to make predictions when we set up N identical experiments and predict the percentage of incident particles that are observed after the beam passes through the filter.

We can of course choose any set of axises. Therefore

where , are just rotated by an angle  wrt ,

Finally let us ask what happens if we were to change the phase of the , waves by 90o ? The electric field rotates about the z-axis.  We can introduce this as an alternative formulation in the following way.

We would like therefore to consider three bases which are available to describe a photon propagating in the z-direction.

 linearly polarized ,  (choose   linear polarized rotated by circularly polarized

To simplify the notation we will use the following

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

The vector nature of the photon field is revealing its spin structure. Photons are spin 1 particles. Indeed the E&M field carries angular momentum and the basic quantum of the field carries it as well. Interestingly, the vector structure of the E&M field is now a consequence of the intrinsic spin structure of the photon. Later when we discuss spin we will see that spin is related to an internal structure that requires a new basis just as we have defined here in terms of polarization. We will also discover that the general spin 1 particle can have three states of polarization but the free photon because it is masseless has only two. Finally one can look at a general field, for example the static Coulomb field around a charge. For general fields the photon is not massless. It is referred to as a virtual photon. It has the possibility of carrying all three states of polarization. The electric field for example can have components along the propagation direction (longitudinally polarized). Quantum development of the photon is usually done within a relativistic framework. So we will go no further at this point except to use the two-state nature of the photon to discuss quantum behavior.

The way we shall proceed is by imagining projectors and analyzers that are made up of the filters described above.  For light we actually have simple filters that select polarization.  Any reasonable optics lab will have beam splitters that can split a laser beam into any chosen basis.

A general state that is some linear combination of  will be transmitted undisturbed through the open system.

A specific state |A> or |B> will projected onto the output from any input. Put in   a|A> for a blocked –B filter and the amount a will emerge. Any |B> component will be blocked.

This filter can be rotated so that the axes that determine the meaning of   can be changed from  to  with new basis states

Consider a system that has two quantum states.

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 filter can be used as a filter that selects all or part of the incident beam PROJECTOR or as an ANALYZER 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.   How much of the beam will be transmitted for the following situations.

 Prepared state F 1 Selected state Measured state Result type blocking x-y none x-y none x-y Block x x-y Block x X-Y none R-L none R-L Block R R-L Block R X-Y Block Y X-Y Block Y

There are no surprises if all the filters are set as described by case 1 to select a specific state, for example, .  All the particle are transmitted once the state I prepared by the first filter. So 100 % of the prepared beam is measured.

 Prepared state F 1 Selected state Measured state Result type blocking 1 x-y none 100%

Measuring the orthogonal state again we are probably not surprised will have a 0% of the beam measured.  These results are the same no matter what analyzer is used as filter 2.

 Rule 1 Open channels in the analyzer transmit the incident beam unchanged.

Using a different basis for filter 3 provides results that are based on the amplitudes squared.

Result of using the  on the projected states

Count photons emerging from the final filter.

• The above amplitudes build up as a result of the ensemble. We can reduce the intensity so that one photon is in the system at a time. (Because photons are bosons we can simultaneously put millions of photons through the system at once. Thus we can simply measure the intensity of a transmitted laser beam.)
• We see 1 or 0 photons for any measurement (low intensity result). Photon must interfere with itself.
•

Now let us examine the blocked analyzer.

 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 A1č amp(enter at A1). Then we find the amplitudes to proceed from A1 to B1 č amp(A1čB1).  The amplitude for any path is the product of amplitudes.

Path 1:

AMP1= amp(A1) amp(A1čB2) amp(B2čC1) amp(C1čD1) amp(D1čX)

Path 2:

AMP2= amp(A1) amp(A1čB3) amp(B3čC2) amp(C2čD3) amp(D3čX)

1. 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+…..

1. Amplitude squared č 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.

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

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

č 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 A1čB2čC1čD3čX.

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.

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

 http://hyperphysics.phy-astr.gsu.edu/hbase/waves/emwavecon.html#c1

So to describe a photon in QM we would expect to introduce a wavefunction that provides us with the location of the photons and a traveling wave solution similar to the particle description. Light does have some special properties due to the fact that it is massless it must travel at a constant velocity c. But we can think the photon, for example, as having a momentum state  or position state  just as a particle.

How does the photon relate to the observed macroscopic field.  In QM we know that the observations are performed by looking at ensembles in order to extract what can be known about the wavefunction. For Fermions nature requires us to repeat the experiment multiple times because there is no possibility [Pauli exclusion principle] to design an experiment with simultaneous multiple Fermions in the same state. However for Boson a field can be built that has n identical photons.  A state prepared in this way exhibits the ensemble average simultaneously. Consider a laser beam. It is, in an ideal sense, a set of photons [billions] all in the same state. If you send it through slits you see part of the beam pass through each slit. For quantum mechanics we need to imagine that each individual photon must go through both slits but when we view the experiment we instantaneously see the ensemble average by placing a white card in the path of the beam or observe immediately the interference pattern on a screen. Although the underlying rules for photon and electron are the same the Fermion/Boson nature of these particles has provided a very different view of the two phenomena matter/E&M. One that until the advent of QM was assumed to be of a different character.

So now we are going to require that somehow each individual photon carry the basic structure of the fields. Each photon, in some sense, must be a traveling wave with both  andfields present or alternatively a field with 4 components through the potential formalism  .

Let us go back to the basic structure of the traveling wave and consider that it must be labeled by the electric field vector. If we pick a direction of propagation  then can be described by two independent possibilities, for example, . What does this mean that it carries the vector nature of the field? Somehow the basic building block, the photon, has some kind of internal structure that provides an overall direction. This structure is two dimensional in nature. We see that there must be two independent types of photons an x and a y photon.  Now any general field can be described as a linear combination of the these two photons.  Let us turn back to classical physics to discuss this feature of light.

Let us assume a traveling sine wave solution with an Electric field pointing in the x direction. There is then an independent field solution with its vector pointing in the y direction. A general direction for the field can be reached simply by adding these two solutions in the same way that usual vector components are added. The way that the E field depends on position via the sinusoidal z dependence is preserved in the sum. A wave with an electric field oscillating with a x-direction as a function of z plus an equal wave oscillating up and down in the y direction as a sinusoidal function of z give a vector pointing at 45o to the x-direction and oscillating back and forth as as function of z.

So we will characterize the photon as having an “internal” polarization. We fix the direction of propagation and then have an independent 2-state degree of freedom that allows us to define two different types of photons, x-polarized and y-polarized.  Should we want to describe a new photon with a polarization of a different direction we simply combine the two states.

First we consider what is a photon?  One beneficial aspect of QM is that it unifies its treatment of particles and fields. In classical physics one has a description of matter and a separate description of the Electric and Magnetic field. In QM the same formalism is used to describe both phenomena. Matter is different than fields because of the properties possessed by the different entities.  One finds different types of matter also.  For example, there are quarks and leptons that comprise the known forms of matter. The quarks and leptons are further divided by properties such as mass, charge, and color.  So the photon and the electron are two elements of quantum theory but with different characteristics but which are confined by the same behavior. Let us look at the critical differences:

 property electron photon mass small (0.5 MeV=mass energy) 0 (free photons) spin 1/2 1 statistics Fermion Boson Pauli property one per state many per state

The type of photon we will be discussing is the free photon. In classical physics Maxwell’s equation can describe a field in a region where there are no charges or currents.

These equations are normally reworked to form a wave equation for the electric and magnetic field.

Light is then understood as a traveling Electric and magnetic disturbance. In particular it is interesting to look for the special solutions to the wave equation that exhibit sinusoidal behavior.  This is a basis to describe all traveling waves in terms of a mixture of light of various colors (frequency).

The magnetic field accompanies the electric field. The magnetic field is perpendicular. The relative position of the magnetic field will determine the direction of orientation. Both  and are perpendicular to the direction of propagation. So for every there is a known . One can then characterize a traveling light wave by its frequency, direction and Electric field vector.

One might also remember that the E&M can be described using two potentials. is the vector potential which is used to obtain . And V is a scalar potential that is introduced to describe .

The traditional quantum formulation is to build a 4-vector from these two fields