TSP-4-1-photons

ter some algebra, the above integral can be turned into summation over all
possible wave numbers k,
"
#
X H=
k
✏0 2
|E(k)|2
+
1 2µ0
|B(k)|2
.
(3)
This is still classical – we just look at the same Hamiltonian at a di↵erent
(11)
k
n
80
where 1/8 arises because only the positive octant of the space is included.
• Planck radiation law for the spectral density
We are now ready to derive the Planck radiation law. The total energy of the photons in the cavity is
angle. In quantum theory, the fields E(k) and B(k) are operators and do not
commute with each other, i.e. there is uncertainty relation between them.
Though not completely correct, it is inspiring to compare the above
hni
=
h✏i ~!
=
1 exp(~!/⌧ )
, 1
(8)
known as the Planck distribution function. At a given temperature ⌧ , the average number of the low-frequency (~! ⌧ ⌧ ) photons is huge, hni ⇡
(7)
In the high temperature limit, ~! ⌧ ⌧ , the average energy h✏i ⇡ ⌧ as described by the equipartition of energy in the classical regime. Because
each photon carries energy quantum ~!, the average number of photons in thermal equilibrium is
k
✏
k
=
n
k
~! , k
(5)
where n k = 0, 1, 2, ... are integers. The energy quantization of the electromagnetic fields was first proposed by Einstein with the groundbreaking notion of photons (originally named as light quanta).
infinitesimally close to each other. Therefore, one can replace the discrete
sum by an integral in the space of wave numbers,
X (· · · )
=
X (· · · )
=
1
Z
1
4⇡n2dn(· · · ),
• Planck distribution function
Let us study the thermodynamics of a single mode with frequency ! first,
i.e. just one type of photons with energy n~!, where n = 0, 1, 2, ... is the photon number. The partition is rather straightforward to compute,
- HH0064tion Law
Hsiu-Hau Lin hsiuhau.lin@ (Oct 23, 2012)
Quantum theory begins with the
for thermal radi-
Planck radiation law
creation/annihilation operators,
H
=
X "✓
~! k
b†1kb1k
+
◆ 1 2
+
✓ b†2kb2k
+
◆# 1, 2
(4)
k
where ! = c|k| = ck is the dispersion relation in vacuum. The two distinct k
!
save us from the famous ultraviolet catastrophe in classical theory.
• photons: quantization of electromagnetic fields
The Planck radiation law can explained by the quantization of electromag-
Hamiltonian
with
the
simple
harmonic
oscillator,
H
=
1 2m
p2 +
k 2
x2
.
The
-HH0064- photons and Planck radiation law
2
similarity is clear. Loosely speaking, E(k) and B(k) can be viewed as con-
netic fields, i.e. the photons. According to Maxwell equations, the Hamilto-
nian for electromagnetic fields in vacuum is
Z H=
"
#
d3r
✏0 |E(r)|2 2
+
1 2µ0
|B
(r)|2
ation at di↵erent frequencies,
u !
=
~ ⇡2c3
!3 exp(~!/⌧ )
, 1
(1)
where u is the spectral density defined as the radiated energy per unit !
volume per unit frequency range. For a given temperature ⌧ , the spectral density for large frequency is exponentially suppressed, u ⇠ !3e ~!/⌧ , and
.
(2)
I(t1/ispcVon)vPenkieEnt(kto)edke·rcoamnpdotsheethsaemfieeldrselianttioonthfeoirr
Fourier components, E(r) = the magnetic field. Because
the field is real, its Fourier components are related, E( k) = E⇤(k). Af-
average energy.
• mode counting for di↵erent wave numbers
To derive the Planck radiation law, we need to count the modes for di↵erent wave numbers properly. For electromagnetic fields confined within a perfectly conducting cubic cavity (of length L), the wave number is quantized,
per unit volume per unit frequency range,
U
=
2
⇥
X h✏ i
k
=
X 2⇥
~! k
exp(~! /⌧ )
, 1
(12)
k
k
k
-HH0064- photons and Planck radiation law
4
where ! = ck = n⇡c/L and the factor of two comes from two polarizations k
Z = 1 + e ~!/⌧ + e 2~!/⌧ + · · · = 1
1 exp(
~!/⌧ ) .
(6)
The average energy of the photon system in thermal equilibrium is
h✏i
=
⌧2
@ log Z @⌧
=
~! exp(~!/⌧ )
1.
jugate variables to each other, just like x and p. The analogy turns out
to be correct by the more advanced theory named quantum electrodynam-