Tarea

1. Instituto de F´ısica, Universidad de Antioquia
Homework 2
Quantization of the Electromagnetic Field
Alejandro Correa L´opeza∗
.
a
Universidad de Antioquia.
August 12, 2014
1 Simple Harmonic Oscillator
Let’s examine the simple harmonic oscillator using the
Wilson–Sommerfeld–Ishiwara quantization rule. This
rule states that the periodic motion has a quantized or
discrete phase space. The area in phase space can only
change by discrete jumps,in units of h. A necessary con-
dition for the application of this method is that each
generalized coordinate qk and its conjugate momentum
pk must be periodic functions of time. Then, the action
integral taken over one cycle of the motion is quantized;
that is,
pkdqk = nkh. (1)
To illustrate this method,consider a one-dimensional
simple harmonic oscillator. The equation of motion is
m¨x + kx = 0.
If ω2
= k/m, then
¨x + ω2
x = 0.
This equation has solutions of the form
x = x0 sin ωt.
Then,
px = m ˙x = mωx0 cos ωt
The equation (1) becomes
nh = pxdx =
T
0
mω2
x2
0 cos2
ωtdt
= mωx2
0
2π
0
cos2
θdθ
= mωπx2
0.
Thus,
x2
0 =
nh
mωπ
,
namely, the amplitudes are quantized. The energy states
are
E = T + V =
1
2
m ˙x2
+
1
2
kx2
=
1
2
mω2
x2
0,
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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2. Quantum Theory of Radiation Homework 2
and the overall Hamiltonian is
H = Hd3
r. (3)
On the other hand, the Poynting vector
S =
c
4π
E × B (4)
is related to the momentum density by a factor of c2
, so
the field carries a total momentum given by
c2
p = Sd3
r. (5)
The time evolution of EM field is given by the Maxwell
equations, say
× E = −
1
c
∂B
∂t
, (6)
× B =
1
c
∂E
∂t
, (7)
and the spatial configuration of the EM field is restricted
by the Gauss laws in absence of charge sources
· E = 0, , (8)
· B = 0, . (9)
This laws can be combined to yield the wave equation
in a vacuum
2
E =
1
c2
∂2
E
∂t2
. (10)
For a single mode k, solutions of (10) can be separated
into 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
2
E0 + k2
E0 = 0. (11)
These functions may be complex, so by convention E is
the 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 of
E. The spatial Fourier transform of a function f = f(r)
is
˜f(k) = (2π)−3/2
f(r)e−ik·r
d3
r. (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 ˜B
can be combined to give
αk
˜E0(k) = ˜E(k, t) + iek × ˜B(k, t) (17)
It is clear that E takes the role of x and B takes the
role of p in the harmonic oscillator. Furthermore, αkis
a 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 field
given by (12) satisfying (8) yield the simplification
( × E0)2
d3
r = k2
E2
0(r)d3
r, (18)
( × E∗
0)2
d3
r = k2
E∗2
0(r)d3
r, (19)
| × E0|2
d3
r = k2
|E0(r)|2
d3
r. (20)
Also, the spatial function E0 can be chosen to have the
next normalization rule
|E0(r)|2
d3
r = 2π ck, (21)
without any loss of generality. This is because the
normalization choice comes from a particular choice of
boundary 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 not
depend 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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3. Instituto de F´ısica Universidad de Antioquia
Using the results of (18) and (21), plugging (12) and
(22) into the Hamiltonian density in (2) and integrating
overall, yields
Hk =
ck
2
(α∗
kαk + αkα∗
k). (23)
This result is exactly of the simple harmonic oscillator,
without having assumed anything about the EM field
configuration 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 quantum
state, and converting observable quantities into Hermi-
tian operators. In the case of EM field, now states are
given by vectors |Φ in a Hilbert space rather than by
values 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 the
quantum EM Hamiltonian is
Hk = ck a†
kak +
1
2
. (24)
Because the operators ak and a†
k plays the same roles
as creation and annihilation operators as in harmonic
oscillator, they satisfy a commutation relation
[ak, a†
k] = 1. (25)
Given the dependence of E and B on ak and a†
k, the
field operators E(r) and B(r) can then found to satisfy
a 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 operators
for a single mode. In general,many different modes k will
exist in space, with the details depending on boundary
conditions. 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∗
replaced
by a and a†
hold for a given mode k; the overall EM
fields 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) = eik·r
ek,
where ek is a unit vector dependent on k that specifies
the direction of E.
The normalization condition only works in a finite vol-
ume V even though the results are more generally true
even in a vacuum of infinite size, so the electric field co-
efficient can be chosen from the normalization condition
as
=
2π ck
V
.
The volume V is arbitrary and it can be chosen for
simplicity asa cubic box of side length L with periodic
boundary 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) = eik·r
ek
yields the condition k · ek = 0. This has two linearly
independent solutions ek,j satisfying ek,i · ek,j = δij for
i, j = 1, 2. These units vectors describe the two indepen-
dent polarizations of light. Now, the EM field modes are
no longer indexed only by the wave vector k, but must
also include the polarizations j. Hence, the EM fields
for a given mode (k, j) are
Ek,j(r, t) =
2π ck
L3
e−ik·r
a†
j(k) + eik·r
aj(k) ek,j,
Bk,j(r, t) =
2π ck
L3
e−ik·r
a†
j(k) + eik·r
aj(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 accounting
for 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 literally
create and annihilate photons in a mode (k, j) given that
nk,j is the number of photons in that mode and |nk,j is
the corresponding eigenstate. Furthermore, satisfy the
commutation relations
[ak,j, a†
k ,j ] = δk,k δj,j , (28)
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4. Quantum Theory of Radiation Homework 2
as the allowed values of k are discretized by the periodic
boundary conditions.
The vacuum state, |0 , can be defined as nk,j = 0
for all (k, j). The overall energy and linear momentum
in that state then become
0|H|0 =
k,j
ck
2
. (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 Field
Quantization and Applications to the Casimir Ef-
fect. MIT Department of Physics. May 2003.
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