Photon dynamics in the double-slit experiment

The dynamics of photons in the double-slit experiment describes the relationship between classical electromagnetic waves and photons, the quantum counterpart of classical electromagnetic waves, in the context of the double-slit experiment. The dynamics of a photon can be completely described by the classical Maxwell's equations with only a reinterpretation of the classical field as a probability amplitude for the photon.

Classical description of the double-slit experiment

Electromagnetic wave equations

The electromagnetic wave equations are a simplified version of Maxwell's equations which describe the propagation of electromagnetic waves through a medium or in a vacuum. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

:$$abla^2 mathbf{E} - { 1 over c^2 } {partial^2 mathbf{E} over partial t^2} = 0

:$$abla^2 mathbf{B} - { 1 over c^2 } {partial^2 mathbf{B} over partial t^2} = 0

where "c" is the speed of light in the medium. In a vacuum, c = 2.998 x 10⁸ meters per second, which is the speed of light in free space.

The magnetic field is related to the electric field through the Maxwell correction to Ampere's Law

:$$abla imes mathbf{B} = {1 over c} frac{ partial mathbf{E {partial t} .

Plane wave solution of the electromagnetic wave equation

The plane sinusoidal solution for an electromagnetic wave traveling in the z direction is (cgs units and SI units)

:$$mathbf{E} ( mathbf{z} , t ) = egin{pmatrix} E_x^0 cos left ( kz-omega t + alpha_x ight ) \ E_y^0 cos left ( kz-omega t + alpha_y ight ) \ 0 end{pmatrix} = E_x^0 cos left ( kz-omega t + alpha_x ight ) hat {mathbf{x ; + ; E_y^0 cos left ( kz-omega t + alpha_y ight ) hat {mathbf{y

for the electric field and

:$$mathbf{B} ( mathbf{z} , t ) = hat { mathbf{z} } imes mathbf{E} ( mathbf{z} , t ) = egin{pmatrix} -E_y^0 cos left ( kz-omega t + alpha_x ight ) \ E_x^0 cos left ( kz-omega t + alpha_y ight ) \ 0 end{pmatrix} = -E_y^0 cos left ( kz-omega t + alpha_y ight ) hat {mathbf{x ; + ; E_x^0 cos left ( kz-omega t + alpha_x ight ) hat {mathbf{y

for the magnetic field, where k is the wavenumber,

:$$omega_{ }^{ } = c k

is the angular frequency of the wave, and $$c is the speed of light. The hats on the vectors indicate unit vectors in the x, y, and z directions.

The plane wave is parameterized by the amplitudes

:$$E_x^0 = mid mathbf{E} mid cos heta

:$$E_y^0 = mid mathbf{E} mid sin heta

and phases

:$$alpha_x^{ } , alpha_y

where

:$$heta stackrel{mathrm{def{=} an^{-1} left ( { E_y^0 over E_x^0 } ight ) .

and

:$$mid mathbf{E} mid^2 stackrel{mathrm{def{=} left ( E_x^0 ight )^2 + left ( E_y^0 ight )^2 .

The solution can be written concisely as

:$$mathbf{E} ( mathbf{z} , t ) = mid mathbf{E} mid mathrm{Re} left { |psi angle exp left [ i left ( kz-omega t ight ) ight ] ight }

where

:$$ |psi angle stackrel{mathrm{def{=} egin{pmatrix} psi_x \ psi_y end{pmatrix} = egin{pmatrix} cos heta exp left ( i alpha_x ight ) \ sin heta exp left ( i alpha_y ight ) end{pmatrix}

is the Jones vector in the x-y plane. The notation for this vector is the bra-ket notation of Dirac, which is normally used in a quantum context. The quantum notation is used here in anticipation of the interpretation of the Jones vector as a quantum state vector.

pherical and cylindrical wave solutions of the electromagnetic wave equation

pherical waves

The solution for spherical waves emanating from the origin is

:$$mathbf{E} ( mathbf{r} , t ) = mid mathbf{E} ( mathbf{r_0} , t ) mid left ( { r_0 over r} ight ) mathrm{Re} left { |psi angle exp left [ i left ( kr-omega t ight ) ight ] ight }

where $$r is the distance from the origin and $$r_0 is some distance from the origin at which the electric field $$mathbf{E} ( mathbf{r_0} , t ) is measured.

Again, the magnetic field is related to the electric field by

:$$mathbf{B} ( mathbf{r} , t ) = hat { mathbf{r} } imes mathbf{E} ( mathbf{r} , t )

where the unit vector is in the radial direction.

Cylindrical waves

The cylindrical solutions of the wave equation for waves emanating from an infinitely long line are Bessel functions. For large distances from the line, the solution reduces to

:$$mathbf{E} ( mathbf{r} , t ) = mid mathbf{E} ( mathbf{r_0} , t ) mid left ( { r_0 over r} ight )^{1/2} mathrm{Re} left { |psi angle exp left [ i left ( kr-omega t ight ) ight ] ight }

:$$mathbf{B} ( mathbf{r} , t ) = hat { mathbf{r} } imes mathbf{E} ( mathbf{r} , t )

where $$r is now the distance from the line. This solution falls off as the square root of distance while the spherical solution falls off as the distance.

Huygens' principle

Huygen's principle states that each point of an advancing wave front is in fact the center of a fresh disturbance and the source of a new train of waves; and that the advancing wave as a whole may be regarded as the sum of all the secondary waves arising from points in the medium already traversed.

This means that a plane wave impinging on two nearby slits in a barrier can be thought of as two coherent sources of light emanating from each of the slits. If the slits are very long compared with the distance at which the waves are observed, then the waves are cylindrical waves. If the slits are very short compared with the distance they are observed, then the waves are spherical waves. In either case the electric field for the wave emanating from each slit is proportional to

:$$mathrm{Re} left { |psi angle exp left [ i left ( kr-omega t ight ) ight ] ight } stackrel{mathrm{def{=} mathrm{Re} left { |phi angle ight } .

Interference

Consider two slits separated by a distance d. Place a screen a distance L from the slits. The distance from slit 1 to a point x on the screen is

:$$r_1 = sqrt{ L^2 + x^2 }

and the distance from slit 2 to the point x on the screen is

:$$r_2 = sqrt{ L^2 + (x-d)^2 } .

For large L and small x compared with L, the difference between the two distances is approximately

:$$Delta r approx {xd over r_1} approx {xd over L} .

The electric field at point x is given by the superposition of the states of the waves from each of the slits and is proportional to the real part of

:$$ |phi_1 angle + |phi_2 angle = |psi angle left { exp left [ i left ( kr_1 -omega t ight ) ight ] + exp left [ i left ( kr_2 -omega t ight ) ight ] ight } = |phi_1 angle exp left [ i left ( kDelta r -omega t ight ) ight ] .

The total electromagnetic energy striking the screen at point x is proportional to the square of the electric field and is therefore proportional to

:$$cos^2 left ( k Delta r ight ) approx cos^2 left ( 2 pi {xd over lambda } ight )

where $$lambda is the wavelength of the light. The fields from the two slits constructively interfere and form antinodes when the phase is equal to multiples of $$pi

:$$2 pi {xd over lambda } = n pi quad n=0,1,2,cdots

or

:$$x_n = { n lambda over 2 d } quad n=0,1,2,cdots .

The waves destructively interfere and form nodes halfway in between the antinodes.

Quantum description of the double-slit experiment

The treatment to this point has been classical. It is a testament, however, to the generality of Maxwell's equations for electrodynamics that the treatment can be made quantum mechanical with only a reinterpretation of classical quantities. The reinperpretation is that the state vectors

:$$mid phi angle

in the classical description of the double-slit experiment become quantum state vectors in the description of photons.

Energy and momentum of photons

The reinterpretation is based on the experiments of Max Planck and the interpretation of those experiments by Albert Einstein.

The important conclusion from these early experiments is that electromagnetic radiation is composed of irreducible packets of energy, known as photons.

Energy

The energy of each packet is related to the angular frequency of the wave by the relation

:$$epsilon = hbar omega

where $$hbar is an experimentally determined quantity known as Planck's constant. If there are $$N photons in a box of volume $$V , the energy in the electromagnetic field is

:$$N hbar omega

and the energy density is

:$$ {N hbar omega over V}

The energy of a photon can be related to classical fields through the correspondence principle which states that for a large number of photons, the quantum and classical treatments must agree. Thus, for very large $$N , the quantum energy density must be the same as the classical energy density

:$$ {N hbar omega over V} = mathcal{E}_c = frac{mid mathbf{E} mid^2}{8pi} .

The number of photons in the box is then

:$$N = frac{V }{8pi hbar omega}mid mathbf{E} mid^2 .

Momentum

The correspondence principle also determines the momentum of the photon. The momentum density is

:$$mathcal{P}_c = {N hbar omega over cV} = {N hbar k over V}

which implies that the momentum of a photon is

:$$hbar k .

The nature of probability in quantum mechanics

Probability for a single photon

There are two ways in which probability can be applied to the behavior of photons; probability can be used to calculate the probable number of photons in a particular state, or probability can be used to calculate the likelihood of a single photon to be in a particular state. The former interpretation violates energy conservation. The latter interpretation is the viable, if nonintuitive, option. Dirac explains this in the context of the double-slit experiment:

Probability amplitudes

The probability for a photon to be in a particular polarization state depends on the fields as calculated by the classical Maxwell's equations. The state of the photon is proportional to the field. The probability itself is quadratic in the fields and consequently is also quadratic in the quantum state of the photon. In quantum mechanics, therefore, the state or probability amplitude contains the basic probability information. In general, the rule for combining probability amplitudes look very much like the classical rules for composition of probabilities: [The following quote is from Baym, Chapter 1]

:

# The probability amplitude for two successive probabilities is the product of amplitudes for the individual possibilities. ...
# The amplitude for a process that can take place in one of several indistinguishable ways is the sum of amplitudes for each of the individual ways. ...
# The total probability for the process to occur is the absolute value squared of the total amplitude calculated by 1 and 2.

ee also

*Theoretical and experimental justification for the Schrödinger equation
*Stern–Gerlach experiment
*Wave-particle duality
*Photon polarization
*Photoelectric effect

References

*cite book |author=Jackson, John D.|title=Classical Electrodynamics (3rd ed.)|publisher=Wiley|year=1998|id=ISBN 0-471-30932-X
*cite book |author=Baym, Gordon |title=Lectures on Quantum Mechanics|publisher=W. A. Benjamin|year=1969|id=ISBN 0-8053-0667-6
*cite book |author=Dirac, P. A. M. |title=The Principles of Quantum Mechanics, Fourth Edition|publisher=Oxford|year=1958|id=ISBN 0-19-851208-2