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.

Classical description of the double-slit experiment

Electromagnetic wave equations

Main article: Electromagnetic wave equation

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:

$\nabla^2 \mathbf{E} \ - \ { 1 \over c^2 } {\partial^2 \mathbf{E} \over \partial t^2} \ \ = \ \ 0$

$\nabla^2 \mathbf{B} \ - \ { 1 \over c^2 } {\partial^2 \mathbf{B} \over \partial t^2} \ \ = \ \ 0$

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

$\nabla \times \mathbf{B} = {1 \over c} \frac{ \partial \mathbf{E}} {\partial t}$.

Plane wave solution of the electromagnetic wave equation

Main article: Sinusoidal plane-wave solutions of the electromagnetic wave equation

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

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

for the electric field and

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

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.

Electromagnetic radiation can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This diagram shows a plane linearly polarised wave propagating from left to right.

The plane wave is parameterized by the amplitudes

$E_x^0 = \mid \mathbf{E} \mid \cos \theta$

$E_y^0 = \mid \mathbf{E} \mid \sin \theta$

and phases

$\alpha_x^{ } , \alpha_y$

where

$\theta \ \stackrel{\mathrm{def}}{=}\ \tan^{-1} \left ( { E_y^0 \over E_x^0 } \right )$.

and

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

The solution can be written concisely as

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

where

$|\psi\rangle \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} \psi_x \\ \psi_y \end{pmatrix} = \begin{pmatrix} \cos\theta \exp \left ( i \alpha_x \right ) \\ \sin\theta \exp \left ( i \alpha_y \right ) \end{pmatrix}$

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.

Spherical and cylindrical wave solutions of the electromagnetic wave equation

Spherical waves

Main article: Wave equation

The solution for spherical waves emanating from the origin is

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

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

$\mathbf{B} ( \mathbf{r} , t ) = \hat { \mathbf{r} } \times \mathbf{E} ( \mathbf{r} , t )$

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

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

$\mathbf{B} ( \mathbf{r} , t ) = \hat { \mathbf{r} } \times \mathbf{E} ( \mathbf{r} , t )$

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

Main article: Huygens–Fresnel principle

Wave Diffraction in the manner of Huygens.

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

$\mathrm{Re} \left \{ |\psi\rangle \exp \left [ i \left ( kr-\omega t \right ) \right ] \right \} \ \stackrel{\mathrm{def}}{=}\ \mathrm{Re} \left \{ |\phi\rangle \right \}$.

Interference

Main article: Interference (wave propagation)

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

$|\phi_1\rangle + |\phi_2\rangle = |\psi\rangle \left \{ \exp \left [ i \left ( kr_1 -\omega t \right ) \right ] + \exp \left [ i \left ( kr_2 -\omega t \right ) \right ] \right \} = 2|\phi_1\rangle \exp \left [ i k {\Delta r \over 2 } \right ].$

Thomas Young's sketch of two-slit diffraction of light.

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

$\cos^2 \left ( k \Delta r \right ) \approx \cos^2 \left ( \pi {xd \over L \lambda } \right )$

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 $\pi$

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

or

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

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

Semi-classical description of the double-slit experiment

It is possible to explain why a photographic film darkens in "spots" at very low light intensities while keeping the electromagnetic field classical (described by the Maxwell equations). In order to get quantitatively correct results, the electrons in the photographic plate or screen must be treated according to the Schroedinger equation of quantum mechanics. Because the atoms in the screen or photographic plate are treated quantum-mechanically but the light treated classically, such analyses are said to be semi-classical. This type of reasoning predicts the correct results when thermal light sources are used, and follows the same lines as the semi-classical photoelectric effect^[1].

Quantum description of the double-slit experiment

Main article: Photon

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 reinterpretation is that the state vectors

$\mid \phi \rangle$

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 $\epsilon$ of each packet is related to the angular frequency $\omega$ of the wave by the relation

$\epsilon = \hbar \omega$

where $\hbar$ is an experimentally determined quantity known as Planck's constant divided by 2 pi.

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 (complex notation, cgs units)

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

The number of photons in the box is then

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

Momentum

Double-slit experiment when performed with electrons. The results are similar for photons. The figures show the buildup over time of electron collisions with the screen.

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 (see Matter wave ) is

$\hbar k$.

(since $\hbar = {{h}\over{2\pi}}$ and $k = {{2\pi}\over{\lambda}}$ this reduces to ${h}\over{\lambda}$

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 is applicable for coherent states and statistical mixtures of such, as thermal light, while the latter is to be used for a single-photon Fock state as defined in quantum optics. Dirac explains the situation in terms of one photon, although this was well before our modern understanding of quantum optics. For the double-slit experiment:

“	Some time before the discovery of quantum mechanics people realized that the connection between light waves and photons must be of a statistical character. What they did not clearly realize, however, was that the wave function gives information about the probability of one photon being in a particular place and not the probable number of photons in that place. The importance of the distinction can be made clear in the following way. Suppose we have a beam of light consisting of a large number of photons split up into two components of equal intensity. On the assumption that the beam is connected with the probable number of photons in it, we should have half the total number going into each component. If the two components are now made to interfere, we should require a photon in one component to be able to interfere with one in the other. Sometimes these two photons would have to annihilate one another and other times they would have to produce four photons. This would contradict the conservation of energy. The new theory, which connects the wave function with probabilities for one photon gets over the difficulty by making each photon go partly into each of the two components. Each photon then interferes only with itself. Interference between two different photons never occurs.	”
—Paul Dirac, The Principles of Quantum Mechanics, Fourth Edition, Chapter 1

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 photon probability density of a single-photon Fock state is related to the expectation value of the energy density of the equivalent E and B fields.

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]

1. The probability amplitude for two successive probabilities is the product of amplitudes for the individual possibilities. ...
2. 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. ...
3. The total probability for the process to occur is the absolute value squared of the total amplitude calculated by 1 and 2.

Wave function of the photon

Superficially, it may look like 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, however this notion is fraught with danger and ultimately leads to contradictions. One should not simply assume that the electromagnetic fields are a wave-function for the photon. For one thing, they are real and thus contain both positive and negative frequency components, which cannot be reconciled with the requirement for Schroedinger wavefunctions which are complex, positive frequency only. In addition, the electromagnetic fields are observable (e.g. with an oscilloscope) while Schroedinger wavefunctions are not observable, even in principle. Clearly, then, the fields are not wavefunctions, are physical, observable fields, rather than merely what you take the modulus-square of to obtain the probability of finding a photon somewhere. The existence of some "wavefunction of the photon" is not a fully settled issue.^[2]

See also

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

References

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