Quantum Mechanics for Mathematicians

These will be linear combinations of pairs of either creation or annihilation
operators, so will change the eigenvalue ofHor :H: by±2, mapping

Hn→Hn±^2

and in particular taking| 0 〉to either 0 or a state inH^2.
his a basis element for theu(1) inu(2) =u(1)⊕su(2). For thesu(2) part, on
basis elementsXj=−i
σj
2 the moment map 25.3 gives the following quadratic
polynomials

μX 1 =

1

2

(z 1 z 2 +z 2 z 1 ), μX 2 =

i

2

(z 2 z 1 −z 1 z 2 ), μX 3 =

1

2

(z 1 z 1 −z 2 z 2 )

This relates two different but isomorphic ways of describingsu(2): as 2 by 2
matrices with Lie bracket the commutator, or as quadratic polynomials, with
Lie bracket the Poisson bracket.
Quantizing using the Bargmann-Fock representation give a representation
ofsu(2) onH

Γ′BF(X 1 ) =−

i

2

(a† 1 a 2 +a† 2 a 1 ), Γ′BF(X 2 ) =

1

2

(a† 2 a 1 −a† 1 a 2 )

Γ′BF(X 3 ) =−

i

2

(a† 1 a 1 −a† 2 a 2 )

Comparing this to the representationπ′ofsu(2) on homogeneous polynomials
discussed in chapter 8, one finds that Γ′BFandπ′are the same representation.
The inner product that makes the representation unitary is the one of equation
8.2. The Bargmann-Fock representation extends thisSU(2) representation as
a unitary representation to a much larger group (H 5 oMp(4,R)), with all
polynomials inz 1 ,z 2 now making up a single irreducible representation ofH 5.
The fact that we have anSU(2) group acting on the state space of thed= 2
harmonic oscillator and commuting with the action of the HamiltonianHmeans
that energy eigenstates can be organized as irreducible representations ofSU(2).
In particular, one sees that the spaceHnof energy eigenstates of energyn+ 1
will be a single irreducibleSU(2) representation, the spin n 2 representation of
dimensionn+ 1 (son+ 1 will be the multiplicity of energy eigenstates of that
energy).
Another physically interesting subgroup here is theSO(2)⊂ SU(2) ⊂
Sp(4,R) consisting of simultaneous rotations in the position and momentum
planes, which was studied in detail using the coordinatesq 1 ,q 2 ,p 1 ,p 2 in section
20.3.1. There we found that the moment map was given by

μL=l=q 1 p 2 −q 2 p 1

and quantization by the Schr ̈odinger representation gave a representation of the
Lie algebraso(2) with
UL′=−i(Q 1 P 2 −Q 2 P 1 )