Quantum Mechanics for Mathematicians

• Space translation (q→q+a). On states one has

|ψ〉→e−iaP|ψ〉

which in the Schr ̈odinger representation is

e−ia(−i

dqd)

ψ(q) =e−a

dqd

ψ(q) =ψ(q−a)

So, the Lie algebra action is given by the operator−iP=−dqd. Note that

this has opposite sign to the time translation. On operators one has

O(a) =eiaPOe−iaP

or infinitesimally

d

da

O(a) = [O,−iP]

• The classical expressions for angular momentum quadratic inqj,pj, for
  example
  l 1 =q 2 p 3 −q 3 p 2
  under quantization go to the self-adjoint operator

L 1 =Q 2 P 3 −Q 3 P 2

and−iL 1 will be the skew-adjoint operator giving a unitary representation

of the Lie algebraso(3). The three such operators will satisfy the Lie

bracket relations ofso(3), for instance

[−iL 1 ,−iL 2 ] =−iL 3

A.4 Complex structures and Bargmann-Fock quan-

tization

We define complex coordinates on phase space by

zj=

1

√

2

(qj−ipj), zj=

1

√

2

(qj+ipj)

The standard choice of complex structure on phase spaceMis given by

J 0

∂

∂qj

=−

∂

∂pj

, J 0

∂

∂pj

=

∂

∂qj

and on coordinate basis vectorsqj,pjof the dual spaceMby

J 0 qj=pj, J 0 pj=−qj

The complex coordinates satisfy

J 0 zj=izj, J 0 zj=−izj