Theoretical and experimental justification for the Schrödinger equation

This article is at a postgraduate level. For a more general introduction to the topic see Introduction to quantum mechanics.

Classical electromagnetic waves

Nature of light

The quantum particle of light is called a photon. Light has both a wave-like and a particle-like nature. In other words, light can appear to be made of photons (particles) in some experiments and light can act like waves in other experiments. The dynamics of classical electromagnetic waves are completely described by Maxwell's equations, the classical description of electrodynamics. In the absence of sources, Maxwell's equations can be written as wave equations in the electric and magnetic field vectors. Maxwell's equations thus describe, among other things, the wave-like properties of light. When "classical" (coherent or thermal) light is incident on a photographic plate or CCD, the average number of "hits", "dots", or "clicks" per unit time that result is approximately proportional to the square of the electromagnetic fields of the light. By formal analogy, the wavefunction of a material particle can be used to find the probability density by taking its absolute-value squared. Unlike electromagnetic fields, quantum-mechanical wavefunctions are complex. (Often in the case of EM fields complex notation is used for convenience, but it is understood that in fact the fields are real. However, wavefunctions are genuinely complex.)

Maxwell's equations were completely known by the latter part of the nineteenth century. The dynamical equations for light were, therefore, well-known long before the discovery of the photon. This is not true for other particles such as the electron. It was surmised from the interaction of light with atoms that electrons also had both a particle-like and a wave-like nature. Newtonian mechanics, a description of the particle-like behavior of macroscopic objects, failed to describe very small objects such as electrons. Abductive reasoning was performed to obtain the dynamics of massive objects (particles with mass) such as electrons. The electromagnetic wave equation, the equation that described the dynamics of light, was used as a prototype for discovering the Schrödinger equation, the equation that describes the wave-like and particle-like dynamics of nonrelativistic massive particles.

Plane sinusoidal waves

Electromagnetic wave equation

The electromagnetic wave equation describes 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:

[math]\displaystyle{ \nabla^2 \mathbf{E} \ - \ { 1 \over c^2 } {\partial^2 \mathbf{E} \over \partial t^2} \ \ = \ \ 0 }[/math]

[math]\displaystyle{ \nabla^2 \mathbf{B} \ - \ { 1 \over c^2 } {\partial^2 \mathbf{B} \over \partial t^2} \ \ = \ \ 0 }[/math]

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

The magnetic field is related to the electric field through Faraday's law (cgs units)

[math]\displaystyle{ \nabla \times \mathbf{E} = - {1 \over c}\frac{ \partial \mathbf{B}} {\partial t} }[/math].

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)

[math]\displaystyle{ \mathbf{E} ( \mathbf{r} , t ) = \mid \mathbf{E} \mid \mathrm{Re} \left \{ |\zeta \rangle \exp \left [ i \left ( kz-\omega t \right ) \right ] \right \} \equiv \mid \mathbf{E} \mid \mathrm{Re} \left \{ |\phi \rangle \right \} }[/math]

for the electric field and

[math]\displaystyle{ \mathbf{B} ( \mathbf{r} , t ) = \hat { \mathbf{z} } \times \mathbf{E} ( \mathbf{r} , t ) }[/math]

for the magnetic field, where k is the wavenumber,

[math]\displaystyle{ \omega_{ }^{ } = c k }[/math]

is the angular frequency of the wave, and [math]\displaystyle{ c }[/math] is the speed of light. The hats on the vectors indicate unit vectors in the x, y, and z directions. In complex notation, the quantity [math]\displaystyle{ \mid \mathbf{E} \mid }[/math] is the amplitude of the wave.

Here

[math]\displaystyle{ |\zeta \rangle \equiv \begin{pmatrix} \zeta_x \\ \zeta_y \end{pmatrix} = \begin{pmatrix} \cos\theta \exp \left ( i \alpha_x \right ) \\ \sin\theta \exp \left ( i \alpha_y \right ) \end{pmatrix} }[/math]

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. The angles [math]\displaystyle{ \theta,\; \alpha_x,\; \mbox{and} \; \alpha_y }[/math] are the angle the electric field makes with the x axis and the two initial phases of the wave, respectively.

The quantity

[math]\displaystyle{ |\phi \rangle = \exp \left [ i \left ( kz-\omega t \right ) \right ] |\zeta \rangle }[/math]

is the state vector of the wave. It describes the polarization of the wave and the spatial and temporal functionality of the wave. For a coherent state light beam so dim that its average photon number is much less than 1, this is approximately equivalent to the quantum state of a single photon.

Energy density of classical electromagnetic waves

Energy in a plane wave

The energy per unit volume in classical electromagnetic fields is (cgs units)

[math]\displaystyle{ \mathcal{E}_c = \frac{1}{8\pi} \left [ \mathbf{E}^2( \mathbf{r} , t ) + \mathbf{B}^2( \mathbf{r} , t ) \right ] }[/math].

For a plane wave, converting to complex notation (and hence dividing by a factor of 2), this becomes

[math]\displaystyle{ \mathcal{E}_c = \frac{\mid \mathbf{E} \mid^2}{8\pi} }[/math]

where the energy has been averaged over a wavelength of the wave.

Fraction of energy in each component

The fraction of energy in the x component of the plane wave (assuming linear polarization) is

[math]\displaystyle{ f_x = \frac{ \mid \mathbf{E} \mid^2 \cos^2\theta }{ \mid \mathbf{E} \mid^2 } = \phi_x^*\phi_x }[/math]

with a similar expression for the y component.

The fraction in both components is

[math]\displaystyle{ \phi_x^*\phi_x + \phi_y^*\phi_y = \langle \phi | \phi\rangle = 1 }[/math].

Momentum density of classical electromagnetic waves

The momentum density is given by the Poynting vector

[math]\displaystyle{ \boldsymbol { \mathcal{P}} = {1 \over 4\pi c } \mathbf{E}( \mathbf{r}, t ) \times \mathbf{B}( \mathbf{r}, t ) }[/math].

For a sinusoidal plane wave traveling in the z direction, the momentum is in the z direction and is related to the energy density:

[math]\displaystyle{ \mathcal{P} c = \mathcal{E}_c }[/math].

The momentum density has been averaged over a wavelength.

References

• Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 047130932X. 
• Baym, Gordon (1969). Lectures on Quantum Mechanics. W. A. Benjamin. ISBN 978-0805306675. 
• Dirac, P. A. M. (1958). The Principles of Quantum Mechanics (Fourth ed.). Oxford. ISBN 0-19-851208-2.