Instead of using two real coordinates to describe points in the phase space
(and having to introduce a reality condition when using complex exponentials),
one can instead use a single complex coordinate, which we will choose as
z(t) =
√
mω
2
(
q(t)−
i
mω
p(t)
)
Then the equation of motion is a first-order rather than second-order differential
equation
z ̇=iωz
with solutions
z(t) =z(0)eiωt (22.1)
The classical trajectories are then realized as complex functions oft, and paramet-
rized by the complex number
z(0) =
√
mω
2
(
q(0)−
i
mω
p(0)
)
Since the Hamiltonian is quadratic in thepandq, we have seen that we can
construct the corresponding quantum operator uniquely using the Schr ̈odinger
representation. ForH=L^2 (R) we have a Hamiltonian operator
H=
P^2
2 m
+
1
2
mω^2 Q^2 =−
~^2
2 m
d^2
dq^2
+
1
2
mω^2 q^2
To find solutions of the Schr ̈odinger equation, as with the free particle, one
proceeds by first solving for eigenvectors ofHwith eigenvalueE, which means
finding solutions to
HψE=
(
−
~^2
2 m
d^2
dq^2
+
1
2
mω^2 q^2
)
ψE=EψE
Solutions to the Schr ̈odinger equation will then be linear combinations of the
functions
ψE(q)e−
~iEt
Standard but somewhat intricate methods for solving differential equations
like this show that one gets solutions forE=En= (n+^12 )~ω,na non-negative
integer, and the normalized solution for a givenn(which we’ll denoteψn) will
be
ψn(q) =
(
mω
π~ 22 n(n!)^2
) (^14)
Hn
(√
mω
~
q
)
e−
mω 2 ~q^2
(22.2)
whereHnis a family of polynomials called the Hermite polynomials. The
ψnprovide an orthonormal basis forH(one does not need to consider non-
normalizable wavefunctions as in the free particle case), so any initial wavefunc-
tionψ(q,0) can be written in the form
ψ(q,0) =
∑∞
n=0
cnψn(q)