Advanced Solid State Physics

Aas the amplitude of the plane wave, with a linear dispersion relationship (still classical) of

ω=c|k|.

To quantize the wave equation it is necessary to construct a Lagrangian by inspection. First it is
possible to write the wave equation for every component like

−c^2 k^2 As=

∂^2 As

∂t^2

because of Fourier space (∇ →ik). By looking long enough at this equation the Lagrangian which
satisfies the Euler-Lagrange equations (3) is found to be

L=

A ̇^2 s

2

−

c^2 k^2

2

A^2 s

The conjugate variable is

∂L

∂A ̇s

=A ̇s.

Again the Hamiltonian can be derived with a Legendre transformation:

H=A ̇sA ̇s−L=

A ̇^2 s

2

+

c^2 k^2

2

A^2 s

For quantization the conjugate variable is replaced

A ̇s→−i~ ∂

∂As

,

which gives the Schrödinger equation

−

~^2

2

∂^2 Ψ

∂A^2 s

+

c^2 k^2

2

A^2 sΨ =EΨ.

This equation is mathematically equivalent to the harmonic oscillator. In this case also the eigenvalues
are the same:

Es=~ωs(js+

1

2

)

withωs=c|ks|andjs= 0, 1 , 2 ,...,jsis the number of photons in modes.