The rest of the energy eigenfunctions can be found by computing|n〉=
a†
√
n···
a†
√
2a†
√
1| 0 〉=
1
√
n!(mω
2 ~)n 2 (
q−~
mωd
dq)n
ψ 0 (q)To show that these are the eigenfunctions of equation 22.2, one starts with the
definition of Hermite polynomials as a generating function
e^2 qt−t2
=∑∞
n=0Hn(q)
tn
n!(22.3)
and interprets theHn(q) as the Taylor coefficients of the left-hand side att= 0,
deriving the identity
Hn(q) =(
dn
dtne^2 qt−t2)
|t=0=eq2(
dn
dtne−(q−t)2)
|t=0=(−1)neq2(
dn
dqne−(q−t)2)
|t=0=(−1)neq(^2) dn
dqn
e−q
2
=e
q 22
(
q−d
dq)n
e−q 22Takingqto
√mω
~ qthis can be used to show that|n〉=ψn(q) is given by 22.2.
In the physical interpretation of this quantum system, the state|n〉, with
energy~ω(n+^12 ) is thought of as a state describingn“quanta”. The state
| 0 〉is the “vacuum state” with zero quanta, but still carrying a “zero-point”
energy of^12 ~ω. The operatorsa†andahave somewhat similar properties to
the raising and lowering operators we used forSU(2) but their commutator is
different (the identity operator), leading to simpler behavior. In this case they
are called “creation” and “annihilation” operators respectively, due to the way
they change the number of quanta. The relation of such quanta to physical
particles like the photon is that quantization of the electromagnetic field (see
chapter 46) involves quantization of an infinite collection of oscillators, with the
quantum of an oscillator corresponding physically to a photon with a specific
momentum and polarization. This leads to a well known problem of how to
handle the infinite vacuum energy corresponding to adding up^12 ~ωfor each
oscillator.
The first few eigenfunctions are plotted below. The lowest energy eigenstate
is a Gaussian centered atq= 0, with a Fourier transform that is also a Gaussian
centered atp= 0. Classically the lowest energy solution is an oscillator at rest
at its equilibrium point (q=p= 0), but for a quantum oscillator one cannot
have such a state with a well-defined position and momentum. Note that the