Quantization of the Free Electromagnetic Field

APPENDIX Β

Quantization of the Free Electromagnetic Field

This appendix contains a brief description of the nonrelativistic quantization of a free electromagnetic field in the Schrödinger picture. The starting point is an expression for the energy of an electromagnetic field in the absence of charges and currents: U =

i / V [ i e o E ' + i(BV//o)],

(B.l)

in which Ε is the electric field strength and Β is the magnetic flux density. MKS units are used, with and the permittivity and permeability of free space, respectively. It is convenient to choose a gauge in which the scalar potential is zero so that both Ε and Β may be derived from a vector potential A : Ε = -dA/dt, Β = V XA. (B.2) We write A(r, t) as a Fourier transform. The procedure is quite similar to that followed in quantizing lattice vibrations in Section 2.3: A = c{μl'y%π')

Σ

£.(k)^.(k,Oexp(/kT)öf^Ä:.

(B.3)

ρ

It is to be noted that k is not restricted to a Brihouin zone. The quantity ζρ is a polarization vector, and qp is to become the normal coordinate. Since V · A = 0, we have k

· e,(k) 925

=

0.

926

APPENDIX Β

Then, {kxz.)qJk,t)Qxp(ik'r)d'k, Ρ

(ΒΑ)

J

Since the vector potential must be real (and ^^(k, 0 = Qp(-K

is real), we have t).

Also, (B.5)

Ε = -c(//¿'V8π^) Σ ί ^pQ{K t) exp(/k · r) d'k. Ρ

J

where the overdot indicates time derivative. The contribution to the energy from the magnetic field is ( - ο ν ΐ 2 8 π ' ) Σ d'rd'kd'k'ik pp' J

χ ε^) · (k' χ z,>)g,ik, 0^p'(k', 0

X exp[/(k + k') · rl

Now,

= ( 2 π ) - ^ ( ο ν 2 ) Σ d'k(k X ε^) · (k X t,.)q,(K pp' J (k X ε^) · (k X

t)g,i-k,

t).

= k\zp · ε^Ο = k^ δ^^.

Zp.)

Hence the magnetic energy is (with

= ω^)

UM = Η2π)-'

Σ\d'kω'\qJkJ)\\ Ρ

(B.6)

J

The energy in the electric field is similarly found to be C/E = ^{2π)-' Σ

(B.7)

d'k\qJk,t)\\

The system may be considered to be equivalent to a cohection of uncoupled simple harmonic oscihators (radiation oscihators). We proceed as in Section 2.3 by defining a Lagrangian and canonical momenta Pp(k, t) = ^^(k, / ) . The field energy ( B . l ) becomes the Hamiltonian H: Η = 1(2π)-^ Σ ρ

d^k{\pp{K O l ' + w^(k)k^(k, O l ' ] .

(B.8)

J

To quantize the theory, we may now introduce the usual commutation relations: [q*{k),p,.(\i')] = ihö,,.ö{k-k'). [9^(k), q,,(y')\

= [PpOí),Ppi\í')\

= 0,

etc.

(B.9)

Quantization of Free Electromagnetic Field

927

Creation and annihilation operators bl, b^ are introduced exactly in parallel with Section 2.3: bp(k) = {ω(k)/2hΫ'\(k)

+

{i/[2hw(k)Y''}ppik),

bl(k) = {ω{k)/2hΫ''g*{k)

- {i/[2hw(k)Y'']p*(k).

(B.IO)

These relations may be inverted to yield ^^(k) = [h/2ω(k)Ϋ'^[bp(k)

+

Pp(k) = (l/i)[hw{k)/2Y'^[bp{k)

bl{-k)l - bli-k)],

(B.ll)

The commutation relations for the bp are [bp(k)blik')]

= δρρ, o(k-k')

(B.12)

and [6,(k),ö;(k')] = [ö;(k), ö;,(k')] = 0. The substitution of ( B . l l ) enables us to write the total energy Η of (B.8) in a simple form: Η = (2π)-' Σ ρ

d'khwpmbl(k)bp{k)

+ i].

(B.13)

J

The first term in (B.13) evidently represents the number of photons of wave vector k and polarization and is denoted by «^(k): ripik) = bl(k)bpik).

(B.14)

For use in the main text, we require an expression for the vector potential in terms of creation and annihilation operators. This is obtained from (B.3) and ( B . l l ) : 1

A = €[{ημοΫ'^/8π']

Σ Ρ

= c[(Ä//o)''V8π^] Σ

i

[s/(2w(k))'^^][í>^(k) + ¿?J(-k)] exp(/k · r) d'k E/(2w(k))'^^][6.(k)exp(/kT)

+ blik) exp(-/k · r)] d^k.

(B. 15)

For use in Section 2.8, we wish to modify Eq. (B.15) by inserting the highfrequency dielectric constant κ^. This is done simply by replacing c by