Instituto de Fısica, Universidad de Antioquia

Homework 2

Quantization of the Electromagnetic Field

Alejandro Correa Lopeza∗.

aUniversidad de Antioquia.

August 12, 2014

1 Simple Harmonic Oscillator

Let’s examine the simple harmonic oscillator using theWilson–Sommerfeld–Ishiwara quantization rule. Thisrule states that the periodic motion has a quantized ordiscrete phase space. The area in phase space can onlychange by discrete jumps,in units of h. A necessary con-dition for the application of this method is that eachgeneralized coordinate qk and its conjugate momentumpk must be periodic functions of time. Then, the actionintegral taken over one cycle of the motion is quantized;that is, ∮

pkdqk = nkh. (1)

To illustrate this method,consider a one-dimensionalsimple harmonic oscillator. The equation of motion is

mx+ kx = 0.

If ω2 = k/m, then

x+ ω2x = 0.

This equation has solutions of the form

x = x0 sinωt.

Then,px = mx = mωx0 cosωt

The equation (1) becomes

nh =

∮pxdx =

∫ T

0

mω2x20 cos2 ωtdt

= mωx20

∫ 2π

0

cos2 θdθ

= mωπx20.

Thus,

x20 =nh

mωπ,

namely, the amplitudes are quantized. The energy statesare

E = T + V =1

2mx2 +

1

2kx2

=1

2mω2x20,

or,En = nω} = nhν.

2 Classical Electromagnetic Field

In CGS units, the Hamiltonian density of the electro-magnetic (EM) field is given by

H =1

8π(E2 + B2) (2)

∗alejandro@gfif.udea.edu.co

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Quantum Theory of Radiation Homework 2

and the overall Hamiltonian is

H =

∫Hd3r. (3)

On the other hand, the Poynting vector

S =c

4πE×B (4)

is related to the momentum density by a factor of c2, sothe field carries a total momentum given by

c2p =

∫Sd3r. (5)

The time evolution of EM field is given by the Maxwellequations, say

∇×E = −1

c

∂B

∂t, (6)

∇×B =1

c

∂E

∂t, (7)

and the spatial configuration of the EM field is restrictedby the Gauss laws in absence of charge sources

∇ ·E = 0, , (8)

∇ ·B = 0, . (9)

This laws can be combined to yield the wave equationin a vacuum

∇2E =1

c2∂2E

∂t2. (10)

For a single mode k, solutions of (10) can be separatedinto temporal and spatial functions αk(t) and E0(r), re-spectively. Plugging the single mode temporal and spa-tial functions into (10) requires that αk(t) = αk(0)e−ickt

(where k = kek and ω(k) = ck) as in the harmonic oscil-lator, while E0(r) must satisfy the Helmholtz equation

∇2E0 + k2E0 = 0. (11)

These functions may be complex, so by convention E isthe real part of the product of α and E0 given by

E(r, t) = α∗k(t)E∗0(r) + αk(t)E0(r). (12)

Now, its helpful to take the spatial Fourier transform ofE. The spatial Fourier transform of a function f = f(r)is

f(k) = (2π)−3/2∫f(r)e−ik·rd3r. (13)

Plugging (12) into (13) yields

E(r, t) = α∗k(t)E∗0(r) + αk(t)E0(r). (14)

By other side, the spatial Fourier transform of (7) is

ick× B =∂E

∂t, (15)

and plugging (14) into (15) yields

ek × B(k, t) = i[α∗k(t)E∗0(r)− αk(t)E0(r)

], (16)

which gives the spatial Fourier transform of the mag-netic field in a different direction. The fields E and Bcan be combined to give

αkE0(k) = E(k, t) + iek × B(k, t) (17)

It is clear that E takes the role of x and B takes therole of p in the harmonic oscillator. Furthermore, αkisa compact representation of the EM field (E, B) anal-ogous to the harmonic oscillator representation (x, p),and the former can indeed be measured deterministi-cally in classical electrodynamics.

Returning now to (12),the combinations of the spa-tial functions E0 satisfying (11) and the overall fieldgiven by (12) satisfying (8) yield the simplification∫

(∇×E0)2d3r = k2∫

E20(r)d3r, (18)∫

(∇×E∗0)2d3r = k2∫

E∗20(r)d3r, (19)∫|∇ ×E0|2d3r = k2

∫|E0(r)|2d3r. (20)

Also, the spatial function E0 can be chosen to have thenext normalization rule∫

|E0(r)|2d3r = 2π}ck, (21)

without any loss of generality. This is because thenormalization choice comes from a particular choice ofboundary conditions, and those choices matter for de-riving the Hamiltonian and momentum of the EM field,but the results of those two quantities ultimately do notdepend on that choice.In real space, plugging (12) into (6) obtain

B(r, t) =i

k[α∗k(t)∇×E∗0(r)− αk(t)∇×E0(r)] . (22)

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Instituto de Fısica Universidad de Antioquia

Using the results of (18) and (21), plugging (12) and(22) into the Hamiltonian density in (2) and integratingoverall, yields

Hk =}ck2

(α∗kαk + αkα∗k). (23)

This result is exactly of the simple harmonic oscillator,without having assumed anything about the EM fieldconfiguration besides its existence in a vacuum.

2.1 Quantized EM Field

2.1.1 Canonical Quantization

In single-particle mechanics, canonical quantization en-tails using vectors |Φ〉 in a Hilbert space as quantumstate, and converting observable quantities into Hermi-tian operators. In the case of EM field, now states aregiven by vectors |Φ〉 in a Hilbert space rather than byvalues of the EM field. Instead, those fields and asso-ciated quantities, like the Hamiltonian and momentum,are now operators, i.e., the variables αk and α∗k are pro-

moted to the operators ak and a†k. This means that thequantum EM Hamiltonian is

Hk = }ck(a†kak +

1

2

). (24)

Because the operators ak and a†k plays the same rolesas creation and annihilation operators as in harmonicoscillator, they satisfy a commutation relation

[ak, a†k] = 1. (25)

Given the dependence of E and B on ak and a†k, thefield operators E(r) and B(r) can then found to satisfya canonical commutation

[Ei(r), Bj(r′)] = i}δijδ3(r − r′). (26)

2.1.2 General Modes Superpositions

In the last section, it was considered the fields operatorsfor a single mode. In general,many different modes k willexist in space, with the details depending on boundaryconditions. Each mode specifies a new harmonic oscil-lator, into which any number nk photons can be added.The electric and magnetic field operators given in (12)and (22) and the Hamiltonian in (24) with α α∗ replacedby a and a† hold for a given mode k; the overall EMfields and Hamiltonian are the sums of the individual

mode operators. To see this, it necessary to make fur-ther use of the spatial mode function E0(r) = εeik·rek,where ek is a unit vector dependent on k that specifiesthe direction of E.The normalization condition only works in a finite vol-ume V even though the results are more generally trueeven in a vacuum of infinite size, so the electric field co-efficient can be chosen from the normalization conditionas

ε =

√2π}ckV

.

The volume V is arbitrary and it can be chosen forsimplicity asa cubic box of side length L with periodicboundary conditions.These conditions restricted the al-lowed of k to be kj = (2π/L)Nj (Nj ∈ Z). Furthermore,the charge-free Gauss law in (8) with E0(r) = εeik·rek

yields the condition k · ek = 0. This has two linearlyindependent solutions ek,j satisfying ek,i · ek,j = δij fori, j = 1, 2. These units vectors describe the two indepen-dent polarizations of light. Now, the EM field modes areno longer indexed only by the wave vector k, but mustalso include the polarizations j. Hence, the EM fieldsfor a given mode (k, j) are

Ek,j(r, t) =

√2π}ckL3

[e−ik·ra†j(k) + eik·raj(k)

]ek,j ,

Bk,j(r, t) =

√2π}ckL3

[e−ik·ra†j(k) + eik·raj(k)

]k× ek,j .

The EM operators accounting for all modes are

E(r, t) =∑k,j

Ek,j(r, t)

B(r, t) =∑k,j

Bk,j(r, t)

Using these in conjunction with the normalization con-dition leads to the fact that the Hamiltonian accountingfor all modes are simply the respective sums of the indi-vidual mode Hamiltonian

H =∑k,j

Hk,j =∑k,j

}ck(a†k,jak,j +

1

2

). (27)

The creation and annihilation operators now literallycreate and annihilate photons in a mode (k, j) given thatnk,j is the number of photons in that mode and |nk,j〉 isthe corresponding eigenstate. Furthermore, satisfy thecommutation relations

[ak,j , a†k′,j′ ] = δk,k′δj,j′ , (28)

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Quantum Theory of Radiation Homework 2

as the allowed values of k are discretized by the periodicboundary conditions.

The vacuum state, |0〉, can be defined as nk,j = 0for all (k, j). The overall energy and linear momentumin that state then become

〈0|H|0〉 =∑k,j

}ck2. (29)

References

[1] Chaddha, G.S. Quantum Mechanics. New Dehli:New Age international. pp. 8-9. ISBN 81-224-1465-6.

[2] Prasahanth S. Venkataram. Electromagnetic FieldQuantization and Applications to the Casimir Ef-fect. MIT Department of Physics. May 2003.

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