This preview shows pages 1–3. Sign up to view the full content.
Quantizing Radiation
Michael Fowler, 5/4/06
Introduction
In analyzing the photoelectric effect in hydrogen, we derived the rate of ionization of a hydrogen
atom in a monochromatic electromagnetic wave of given strength, and the result we derived is in
good agreement with experiment.
Recall that the interaction Hamiltonian was
()
( )
1
0
0
cos
.
2
ikr
t
t
e
Hk
r
t
A
mc
e
eeA
mc
ωω
ω
⋅−
− ⋅−
⎛⎞
=⋅
−
⋅
⎜⎟
⎝⎠
=+
GG
p
p
⋅
G
G
G
G
G
G
and we dropped the
it
e
term because it would correspond to the atom giving energy to the field,
and our atom was already in its ground state.
However, if we go through the same calculation
for an atom
not
initially in the ground state, then indeed an electromagnetic wave of appropriate
frequency will cause a transition rate to a lower energy state, and
e
is the relevant term.
But this is not the whole story. An atom in an excited state will eventually emit a photon and go
to a lower energy state, even if there is
zero
external field.
Our analysis so far does not predict
this—obviously, the interaction written above is only nonzero if
A
G
is nonzero!
So what are we
missing?
Essentially, the answer is that the electromagnetic field itself is quantized.
Of course, we know
that, it’s made up of photons.
Recall Planck’s successful analysis of radiation in a box: he
considered all possible normal modes for the radiation, and asserted that a mode of energy
could only gain or lose energy in amounts
=
.
This led to the correct formula for black body
radiation, then Einstein proved that the same assumption, with the same
, accounted for the
photoelectric effect.
We now understand that these modes of oscillation of radiation are just
simple harmonic oscillators, with energy
=
( )
1
2
n
+
=
, and, just as a mass on a spring oscillator
has fluctuations in the ground state,
0
x
=
but
2
0
x
≠
, for these electromagnetic modes
0
A
=
G
but
2
0
A
≠
G
.
These fluctuations in
A
G
mean the interaction Hamiltonian is momentarily nonzero, and therefore
can cause a transition.
Therefore, to find the spontaneous transition rate (as it’s called) for an atom in a zero (classically
speaking) electromagnetic field, we need to express the electromagnetic field in terms of normal
modes (we’ll take a big box), then quantize these modes as quantum simple harmonic oscillators,
introducing raising and lowering operators for each oscillator (these will be photon creation and
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
annihilation operators) then construct the appropriate quantum operator expression for
A
G
to put
in the electronradiation interaction Hamiltonian.
The bras and kets will now be quantum states of the electron
and
the radiation field, in contrast
to our analysis of the classical field above, where the radiation field didn’t change.
(Of course, it
did, really, in that it lost one photon, but in the classical limit there are infinitely many photons in
each mode, so that wouldn’t register.)
Our treatment follows Sakurai’s
Advanced Quantum Mechanics
, Chapter 2 (but we stay with
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '07
 MichaelFowler
 Radiation

Click to edit the document details