Quantum Mechanics for Mathematicians

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)