Quantum Mechanics for Mathematicians

with

cn=

∫+∞

−∞

ψn(q)ψ(q,0)dq

(note that theψnare real-valued). At later times, the wavefunction will be

ψ(q,t) =

∑∞

n=0

cnψn(q)e−

i~Ent

=

∑∞

n=0

cnψn(q)e−i(n+

(^12) )~ωt

22.2 Creation and annihilation operators

It turns out that there is a quite easy method which allows one to explicitly find
eigenfunctions and eigenvalues of the harmonic oscillator Hamiltonian (although
it’s harder to show it gives all of them). This also leads to a new representation
of the Heisenberg group (of course unitarily equivalent to the Schr ̈odinger one
by the Stone-von Neumann theorem). Instead of working with the self-adjoint
operatorsQandP that satisfy the commutation relation

[Q,P] =i~ 1

we define

a=

√

mω

2 ~

Q+i

√

1

2 mω~

P, a†=

√

mω

2 ~

Q−i

√

1

2 mω~

P

which satisfy the commutation relation

[a,a†] = 1

Since

Q=

√

~

2 mω

(a+a†), P=−i

√

mω~

2

(a−a†)

the Hamiltonian operator is

H=

P^2

2 m

+

1

2

mω^2 Q^2

=

1

4

~ω(−(a−a†)^2 + (a+a†)^2 )

=

1

2

~ω(aa†+a†a)

=~ω

(

a†a+

1

2

)

The problem of finding eigenvectors and eigenvalues forH is seen to be
equivalent to the same problem for the operator

N=a†a