Quantum Mechanics for Mathematicians

•

{wj,wk}=jklll

(

− 2 h

m

)

This is the most surprising relation, and it has no simple geometrical

explanation (although one can change variables in the problem to try and

give it one). It expresses a highly non-trivial relationship between the

Hamiltonianhand the two sets of symmetries generated by the vectors

l,w.

Thewj are cubic in theq andpvariables, so the Groenewold-van Hove
no-go theorem implies that there is no consistent way to quantize this system
by finding operatorsWj,Lj,Hproviding a representation of the Lie algebra
generated by the functionswj,lj,h(taking Poisson brackets). Away from the
locush= 0 in phase space, the functionhcan be used to rescale thewj, defining

kj=

√

−m

2 h

wj

and the functionslj,kjthen do generate a finite dimensional Lie algebra. Quan-
tization of the system can be performed by finding appropriate operatorsWj,
then rescaling them using the energy eigenvalue, giving operatorsLj,Kjthat
provide a representation of a finite dimensional Lie algebra on energy eigenspaces.
A choice of operatorsWjthat will work is

W=

1

2 m

(L×P−P×L) +e^2

Q

|Q|

where the last term is the operator of multiplication bye^2 qj/|q|. By elaborate
and unenlightening computations theWjcan be shown to satisfy the commu-
tation relations corresponding to the Poisson bracket relations of thewj:

[Wj,H] = 0

[Lj,Wk] =i~jklWl

[Wj,Wk] =i~jklLl

(

−

2

m

H

)

as well as
L·W=W·L= 0

The first of these shows that energy eigenstates will be preserved not just by the
angular momentum operatorsLj, but by a new set of non-trivial operators, the
Wj, so will be representations of a larger Lie algebra thanso(3). In addition,
one has the following relation betweenW^2 ,Hand the Casimir operatorL^2

W^2 =e^41 +

2

m

H(L^2 +~^21 ) (21.2)