Quantum Mechanics for Mathematicians

22.1 The harmonic oscillator with one degree of freedom

An even simpler case of a particle in a potential than the Coulomb potential
of chapter 21 is the case ofV(q) quadratic inq. This is also the lowest-order
approximation when one studies motion near a local minimum of an arbitrary
V(q), expandingV(q) in a power series around this point. We’ll write this as

h=

p^2

2 m

+

1

2

mω^2 q^2

with coefficients chosen so as to makeωthe angular frequency of periodic motion
of the classical trajectories. These satisfy Hamilton’s equations

p ̇=−

∂V

∂q

=−mω^2 q, q ̇=

p

m

so
q ̈=−ω^2 q

which will have solutions with periodic motion of angular frequencyω. These
solutions can be written as

q(t) =c+eiωt+c−e−iωt

forc+,c−∈Cwhere, sinceq(t) must be real, we havec−=c+. The space of
solutions of the equation of motion is thus two real dimensional, and abstractly
this can be thought of as the phase space of the system.
More conventionally, the phase space can be parametrized by initial values
that determine the classical trajectories, for instance by the positionq(0) and
momentump(0) at an initial timet(0). Since

p(t) =mq ̇=mc+iωeiωt−mc−iωe−iωt=imω(c+eiωt−c+e−iωt)

we have

q(0) =c++c−= 2 Re(c+), p(0) =imω(c+−c−) =− 2 mωIm(c+)

so

c+=

1

2

q(0)−i

1

2 mω

p(0)

The classical phase space trajectories are

q(t) =

(

1

2

q(0)−i

1

2 mω

p(0)

)

eiωt+

(

1

2

q(0) +i

1

2 mω

p(0)

)

e−iωt

p(t) =

(

imω

2

q(0) +

1

2

p(0)

)

eiωt+

(

−imω

2

q(0)−

1

2

p(0)

)

e−iωt