THIS set of notes like most others is a compilation of notes from other lectures on similar topics.  This leads to perhaps a jumpy description of the material.  For example the ability to express quantum states as a linear combination of other quantum states requires some discussion and some mathematical detail. These notes visit and revisit this idea and the supporting detail several times.  If one section seems difficult to understand then as you read further you may encounter a more understandable discussion of the same material.  This can be disconcerting if one expects all discussions to be continuations of previous material rather than a rehash of previous material but from a slightly different perspective.

 

A basic review of QM can be found in Feynman volume III.  Also visit this good web site that describes the Feynman approach using spin filters.  [lecture 15 = Feynman diagrams]

 

Another excellent review by Abner Shimony appears in the New Physics. He breaks QM into a set of basic postulates summarized here. He also discusses the problem of entanglement and measurement. A brief summary is provided.

 

1

“Associated with every physical system is a complex linear vector space V, such that each vector of unit length represents a state of the system.

2

“There is a one to one correspondence between the set of eventualities (observables) concerning the system and the set of subspaces of the vector space associate with the system, such that if e is an eventuality (observable) and E Is the subspace that corresponds to it, then e is true in a state |S> if and only if any vector that represents S belong to E; and is false in the state S if and only if any vector that represents S belongs to E(orthogonal) .”   A states described by a vector that has components in both subspaces represents a state with an unspecified value for this observable e.

3

“If |S> is a state and e is an eventuality (observable) corresponding to the subspace E, then the probability that e will turn out to be true if the initially the system is in state |S> and an operation is performed to actualize (measure) it.

v is a unit vector representing S … and PE is the projection.”

 

4

“If 1 and 2 are two physical systems, with which the vector spacesV1 and V2 are associated, then the composite system 1 + 2 consisting of 1 and 2 is associated with the tensor product V1 x V2.”

5

“If a system is in a nonreactive environment between 0 -> t, then there is a linear operator U(t) such that  U(t) |v> represents the state of the system at time t if |v> represents the state of the system at time 0. Furthermore, ||U(t) v||2 = || v||2 for all v in the vector space.”

 

 

 

The author (Shimony) is an expert on quantum measurement, EPR, and Schrodinger cat paradoxes.  He carefully explores the basic structure of quantum theories.

 

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This page is under continuing development.

 

| a,b,c,d  >

symbol for the state of a QM system with parameters that have values of a,b,c,d

 

For a classical system how do we define a particle’s state?

 

Simple system consists of a point mass.

mass

x,y,z

vx, vy, vz

 

                        | mass,x,y,z,vx,vy,vz  >

 

 

 

Lets add charge as well.

 

| mass,x,y,z,vx,vy,vz, Q  >

 

Given a particle in a potential well there is an infinite set of possible orbits. The mass and charge of course pick certain orbits by choosing the strength of the potential and of course the inertial response of the particle. The position and velocity specify which of these classical orbits the particle will follow. Thus given the value of these variables and of course the forces (potential), the orbit is determined and the location and velocity at any time can be found.

 

Parameters are continuous (position and velocity) or discrete (mass and charge).

 

 

For a quantum system we will define a state in terms of a subset of observables as we did classically. However these observables will have peculiar properties. There will be classical observables such as momentum and position that cannot be simultaneously used to specify a system.

 

For every observable we define an operator.

[Operators and states are defined in some sense together. An operator has meaning in the way it changes a state. Rotation can be an operation. We understand the rotation operator by what it does to objects. We can even have a mathematical representation of what we mean by rotating using a column vector and a rotation matrix. What is a rotation? Well if you take this book and ….(you can fill in this answer). Mathematically the column and matrix show all of the features that rotations possess. The abstract nature of a transformation and a mathematical representation that has the same character both have important roles in the way particle states are constructed.  Another question that will be relevant is once I have identified and operation what does it do to all the states of my system.]

 

So, for every observable we define an operator and insist that the operation, state and value for the observable behave according to an eigenvector equation.

 

.

 

is an operator for the observable a.

 

The state  is specified by a value of the observable a.

 

Note not all operators have to correspond to observables. Since observables need to be real only a subset of the operators in the theory will be associated with a measurable quantity.

 

The most general way that an operator can act on a state is to transform it to a new state.

 

.

 

This should not be a surprise.  Using rotation as a model for operators, we expect that in general a vector, such as position, will become a different position after rotation. However some vectors may not be changed. Vectors pointing along the axis of a rotation are not changed.  Thus the rotation about an axis does not change the vectors pointing along the axis and so these vectors would satisfy and eigenequation for this specific transformation or operation.

 

 

Our states will be elements of a vector space.  That means several important things. Most of the properties of vector spaces can be understood by considering the well know space of 3-vectors. 

  • There is a mathematical process called the dot product.
  • Vectors have components that determine (STRENGTH) the amount in any direction.
  • Vectors can be expressed in terms of a basis.
  • Basis vectors need to be complete. You need to pick basis vectors that allow you to specify all directions. Usually this is x,y,z.
  • There are infinite possibilities for the basis vectors.
  • You can create mathematical relationships that define how the components change. (Representations).

 

In place of the dot product we introduce the inner product

 

.

 

Some care needs to be taken in how you define this product but we will ignore some of the details for now.  The easiest way to treat the inner product is to first choose a basis set that spans the space.

 

Basis vectors are .

 

Therefore any general state can be written as

 

.

 

  =  is the amplitude and tells us how much  there is in our state .

 

Choosing orthogonal and normal basis vectors means that

 

 

 

An interesting application of this general formalism is to construct your states with respect to eigenstates of position. You find the states that have the character that each state describes a particle at a specific location x.

 

Basis:

 

General state

 

If you examine how various operators behave on the state and express the mathematics in terms of the components . You arrive at a representation of QM called the Schrodinger representation.

 

 

A critical feature of the mathematics of operators is that they may depend on the order of application.  The standard example of this property is the rotation of an object like a blackboard eraser. Try rotating the eraser several times. Vary the order of the rotations. You find the final orientation of the eraser indeed depends on the order in which the rotations are carried out. We express this feature or relationship between operators as the commutator.

 

 

If the commutator is zero then the operations can be performed in any order because

 

Some standard observable and there operators are:

 

 

For now we note  is the commutator for position and momentum. This equation determines the relationship between the observables and makes position and momentum mutually exclusive in the sense that both cannot be simultaneously precisely measured for a quantum system.

 

List of important quantities:

Operators

States

Eigenstates

Examples of operators: H, P, X, R, J

Inner product

Vector Space

Dual Vector (Bra-vectors)

Superposition

Basis state

Completeness

Orthonormal

Spin

Groups

Poincare transformation

·       translations of space and time

·       rotations

·       boosts

·       reflections (parity and time reversal)

 

Fourier transform of sound waves

Fundamental character (What is it?)

Symmetry

Unitary transformations (very simple overview)

Generators as observables (very simple overview)

Irreducible representations (very simple overview, comparing ˝, ˝ , with 1,0)

 

*********************************************************

    

Waves are:

    • Local
    • Spread out
    • Add via interference rules (constructive and destructive)
    • Are derivable assuming Newton’s laws and an elastic medium

 

So the bottom line is that although waves have interesting behavior there properties are not surprising.

 

 

We now make a giant leap and require that waves move from the phenomenological or perhaps derivable to the fundamental.

 

We require the basic elements in our theory, particle and interactions, to have characteristics similar to particles and characteristics similar to waves. These properties are simply “the way it is”.  We will take the perspective that it is not justifiable in terms of some overriding principle merely valid based on observation.  Taking this point of view is verified in practice to an almost undeniable degree.  Quantum mechanics, the theory that merges these features successfully has very few doubters and stands an very firm ground.

 

Particle nature: BULLETS

-        observations reveal that the particle of the theory always show up in the measurement as distinct chunks. You will never measure 1/2 of an electron or a partial photon.  You will always get either one or none!  To understand this we will need to be clear about the measurement process.  It will be important to make predictions based on what you measure.

Wave nature: Interference

-        The classic way to begin to understand the wave nature of  QM is the interference particles and waves impinging on two slits.  There will be an interference pattern in both cases.  Waves from the classical view point contribute an amplitude

 

There are two types of idealized problems that are usually introduced to provide examples of quantum behavior. 

  • Two slit experiment:  Feynman uses this example to introduce QM. There are several great websites that discuss the ideas.
  • Filters: Again Feynman devotes considerable time to the topic and several interesting web sites can be found under Stern Gerlach Spin filters e. g.

 

 

Double slit accumulation

http://www.upscale.utoronto.ca/PVB/Harrison/DoubleSlit/Flash/Histogram.html

 

http://www.upscale.utoronto.ca/GeneralInterest/QM.html

 

filters;

http://faraday.physics.utoronto.ca/PVB/Harrison/SternGerlach/Flash/SGInteractive.html

 

 

 

 

 

Our goal is to obtain an overview of the essential ingredients in QM. Here are some salient aspects:

 

  • Quanta: Observations or measurements will record many discrete quantities. For example, a detector will detect an electron or detect no electron but it will not detect 0.5 electrons.  The quantum nature of particles allows for a probability of detection that can span continuously the values from 0-> 1 but independent of the probability an actual measurement finds that an entire electron either present or not present.
  • Amplitudes: To account for the wave nature of QM amplitudes are used to describe “how much”. For example, for a two slit experiment, one needs to know the amplitude for the particle to pass through slit 1 and arrive at location y on the screen.  Amplitudes are used to describe:
    • the contribution of various sequences of events or paths to a final result,
    • the content of a state.

 

 

To determine the probability of an event one squares the amplitude.

 

  • Vector spaces: In order to incorporate the nature of QM correctly the states are viewed as constituting a vector space.  One might notice that classically light exhibits wave behavior.  A very interesting example of how light behaves is the impact of polarizing filters. An interesting result is the transmission of light through two perpendicular filters (zero transmission) when a third filter is inserted in between the two perpendicular ones.  In this case the transmission need not be zero. Although the result might be surprising it is understandable because of the vector nature of the electric field.  Quantum states are vectors in Hilbert space.  Some states are fundamentally different from other states (orthogonal) a particle at location x1 is not at all the same as a particle at location x2. However a particle at location x1 does and a particle of definite momentum do share or overlap properties. Both of these states may result in a particle being detected at x1. The behavior of QS is encapsulated in the mathematics of vector spaces.
    • Vector addition: The sum of two QS with weight factors (amplitudes) is a valid QS. You can any state A to any state B and the result is a valid QS.

 

 

    • Basis: There exists sets of QS that are linearly independent and from which any conceivable QS can be built using the above addition rule.
    • Inner product: An inner product can be introduced by adding a dual space and a product rule.

 

Familiar vector space of 3-d does not typically use the notion of a dual space when introducing the inner or dot product. However 4-d spacetime can be conveniently cast into the above form and the minus sign for the time component, when calculating length, can be introduced by allowing the space and its dual to have opposite signs for the t-component.

    • Orthonormal: using the inner product one can impose the condition of normality and orthogonality for a basis.

 

These are somewhat formal mathematical rules but they are essential.  If one can learn to formulate QM as vector space with amplitudes and bases, then the behavior of QS can be extracted and a certain intuition can be built.

 

Consider the earth orbiting around the sun. We know the interaction and can solve for the orbit.

 

From the force you calculate the earth’s orbit and with initial condition you know where it is at all times.

 

 

Given the Hamitonian H one can calculate the quantum wave function that is equivalent to the full QS.

 

Know how the atom will behave in terms of the wave function.

 

Suppose the problem is too difficult to solve because the interaction is more complicated. (Prelude to Feynman diagrams and the notion of propagation.)

 

 

For the two slit experiment you propagate a particle to slit 1 (the interaction) and then to the screen.

For this I can write down an amplitude to reach the screen by nature of the interaction.

 

 

One difference in QM is that I need to find all possible ways to reach the ending state

 

 

I can imagine a more complex system with a series of interactions.

My goal would be to calculate the amplitude to travel along any given path and then sum over all possible paths.  Feynman diagrams diagrammatically represent this deeply complicated mathematical process.  A first order diagram in some sense shows the interaction as a single step process, as if you can get a good result by correcting the trajectory one time classically. High order diagrams are similar to the multiple corrections required to find the asteroids final trajectory.  In addition to the expansion of the interaction Feynman diagrams represent the multitude of ways that quantum systems can travel. The amplitudes for each trajectory will be included and interference may result.

 

There are several aspects of QM we will need to understand as general concepts.

 

Quantum states form a vector space (Hilbert space).

  • A QS can be written as a linear combination of other QS.
    • Tuning forks can be represented as a string of local pressures (snaps)
    • Snaps can be decomposed using the Fourier transform (tuning forks)
  • Certain sets of states can form  a basis
  • As vectors one can compute an inner product.
    • Orthogonality: No overlap between QS. This implies that the states are chosen to be completely distinct in some way. There is a label that can be used to completely distinguish state 1 and state 2.
    • Overlap: State can share some characteristic. Two states may both have some likelihood of being at some location.

 

Operators may represent observables so their behavior and interpretation tells us how to think about particle properties.

 

 

8888888888888888888888

 

There are two notions we need to develop.

 

1)     Quantum states are like vectors: 

a.      Addition:

                                                    i.    

                                                  ii.    

b.     inner product: 

c.      basis: there are a set of complete states that completely define any QS in that all QS can be written as a sum over these states with appropriate amplitudes.

d.     orthogonal and normal = orthonormal: given an inner product vectors can be normalized and orthogonalized s.t.

 

 

 

To clarify the notion of basis we consider the the filter shown below. We imagine a measurable property of a system such as the spin of an electron or the polarization of a photon. We build an idealized system that can separate the two states and then recombine them. In addition we can choose to completely block one component of the state by inserting a blocking mechanism.

 

http://csma31.csm.jmu.edu/physics/giovanetti/particlePhysics/Stern.htm

1)     If the state is unblocked any entering QS will exit with no change.  with no detectable change. This is almost perfectly attainable with beam optics. States of polarization can be split and recombined and any tests done on the beam before the split are reproduced after the open filter [on the beam after the split].

 

2)     If a blocking part is inserted then the Filter picks out the specific amount of the unblocked state and absorbs all of the blocked state.

·      

 

These are the amplitudes i.e.  is an amplitude.

But a measurement is expressed as a probability.

You can imagine that you do a host of experiments with photons:

  • unpolarized beam
  • beam polarized “up”
  • beam polarized “down”
  • beam partially polarized

 

You make measurements with a filter device in its various states and count the number of photons before and after or measure intensity before and after.  This step is important because the character of a quantum state is affected by the measurement. It is like observing the hit on a screen in the double slit experiment. You imagine that the final state is the interference pattern but when you measure the location you get one and only one value. If you view the counts over time you get an intensity distribution that matches the interference pattern.  Thus quantum measurement is complicated.

 

Now we can rotate filters around the beam axis.  The basis states that were defined by the original filter orientation are now changed to new basis states.  Indeed, for polarized photons, we simply change the orientation of “up” and “down”. This is a simple coordinate transformation and the definition of polarization is arbitrary in this respec so there is nothing surprising about the interpretation.

  • New state “up rotated” and “down rotated” are complete in that we can express all aspects of polarization wrt this new orientation.
  • Using a closed rotated filter will pick out the new states.
  • The angle of rotation is arbitrary and new unique states are define for angle less than 360O beyond 360 we are repeating.

 

Finally we can arrange filters of various orientations in succession and predict the outcome of an intensity measurement.  The correct method is to decompose states into the appropriate basis states and then project when the state is blocked. The transmitted state will be unchanged if unblocked.

 are one basis

 are a different basis

but the state  is the same and all measurable aspects are repeated using identical basis representations.

 

Representation is an important vocabulary word. It implies that there is something that has an intrinsic character and that character is maintained in the way we express it. Both bases based on different orientations of our open filter carry the entire character of the QS.

 

For the photon which has spin 1 and two polarization states on can choose two open filter orientations that allow one to discuss the direction of the electric field along an x,y direction or 45 O rotated system x’,y’. In addition we can choose states of circular polarization and we can imagine designing a filter which split the beam based on circular polarization. Therefore consider these three bases.

 

 

http://www.upscale.utoronto.ca/GeneralInterest/Harrison/SternGerlach/SternGerlach.html

 

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.

 

 

 

 

 

The intial state is:

  1. Decomposed into the basis of the filter

  1. examine the state of the filter
    • B blocked

  1. Express this in any basis

 

This predicts that the state  which was not present initially is now present because you have eliminated some state.  This is perhaps a bit surprising.

 

Lets look at the polarized filters

 

 

 

 

 

 

 

 

 


initial  pol.         filter orientation                      Notice the new field has X&Y components                   

 

 

 

 

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.

 

prob. of finding A starting with X * prob. find X in a state Ač

prob. of finding B starting with X * prob. find X in a state B č

 

If you block B you eliminate any contribution from B therefore

č transmitted X

If you block A you eliminate any contribution from A therefore

č transmitted X

 

If both are open then you don’t get the sum of the above but get an additional term

 

č interference

 

Let us examine this for the measurement of Y

If you block B you eliminate any contribution from B therefore

č transmitted Y

If you block A you eliminate any contribution from A therefore

č transmitted Y

 

If both are open then you don’t get the sum of the above but get an additional term

 

č interference

 

This is key because the amount of Y transmitted should be zero.

 

 

 

 

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

 

 

=1

 

 

Prepared state

Analyzed using

Measured state

Result

Filter 1

Filter 2

final filter

 

 

type

blocking

 

 

x-y

none

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

 

 

 

 

 

 


More details about light-----------------------------

 

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.

Image:Polarisation (Circular).svg

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