Quantum Mechanics for Mathematicians

on the harmonic oscillator state space. This gives the same construction of all
SU(2)⊂U(2) irreducible representations that we studied in chapter 8. The
cased= 3 corresponds to the physical example of an isotropic quadratic central
potential in three dimensions, with the rotation group acting on the state space
as anSO(3) subgroup of the subgroupU(3)⊂Sp(6,R) of symmetries com-
muting with the Hamiltonian. This gives a construction of angular momentum
operators in terms of annihilation and creation operators.

25.1 Multiple degrees of freedom

Up until now we have been working with the simple case of one physical degree of
freedom, i.e., one pair (Q,P) of position and momentum operators satisfying the
Heisenberg relation [Q,P] =i 1 , or one pair of adjoint operatorsa,a†satisfying
[a,a†] = 1. We can easily extend this to any numberdof degrees of freedom by
taking tensor products of our state spaceF, anddcopies of our operators, each
acting on a different factor of the tensor product. Our new state space will be

H=Fd=F ⊗···⊗F︸ ︷︷ ︸

d times

and we will have operators

Qj,Pj j= 1,...,d

satisfying
[Qj,Pk] =iδjk 1 , [Qj,Qk] = [Pj,Pk] = 0

HereQjandPj act on thej’th term of the tensor product in the usual way,
and trivially on the other terms.
We define annihilation and creation operators then by

aj=

1

√

2

(Qj+iPj), a†j=

1

√

2

(Qj−iPj), j= 1,...,d

These satisfy:

Definition(Canonical commutation relations).The canonical commutation re-
lations (often abbreviated CCR) are

[aj,a†k] =δjk 1 , [aj,ak] = [a†j,a†k] = 0

In the Bargmann-Fock representationH=Fdis the space of holomorphic func-
tions indcomplex variableszj(with finite norm in theddimensional version
of 22.4) and we have

aj=

∂

∂zj

, a†j=zj

The harmonic oscillator Hamiltonian forddegrees of freedom will be

H=

1

2

∑d

j=1

(Pj^2 +Q^2 j) =

∑d

j=1

(

a†jaj+

1

2

)

(25.1)