Quantum Mechanics for Mathematicians

Our space of solutions is the space of all sets of complex numbersα(pj). In
principle we could take this space as our dual phase space and quantize using
the Schr ̈odinger representation, for instance taking the real parts of theα(pj)
as position-like coordinates. Especially since our dual phase space is already
complex, it is much more convenient to use the Bargmann-Fock method of
quantization. Recalling the discussion of section 26.4, we will need both a dual
phase spaceMand its conjugate spaceM, which means that we will need to
consider not just solutions of the Schr ̈odinger equation, but of its conjugate

−i

∂

∂t

ψ=−

1

2 m

∂^2

∂x^2

ψ (36.6)

which is satisfied by conjugatesψof solutionsψof the usual Schr ̈odinger equa-
tion. We will takeM=H 1 to be the space of Schr ̈odinger equation solutions
36.5.M=H 1 will be the space of solutions of 36.6, which can be written

+∑Λ 2 Lπ

j=−Λ 2 Lπ

α(pj)e−ipjxei

p^2 j

2 mt

for some complex numbersα(pj).
A basis forMwill be given by the

A(pj) =

{

α(pk) = 0 k 6 =j

α(pk) = 1 k=j

with conjugatesA(pj) a basis forM. The Poisson bracket onM⊕Mwill be
determined by the following Poisson bracket relations on basis elements

{A(pj),A(pk)}={A(pj),A(pk)}= 0, {A(pj),A(pk)}=iδjk (36.7)

Bargmann-Fock quantization gives as state space a Fock spaceFD, whereDis
the number of values ofpj. This is (ignoring issues of completion) the space
of polynomials in theDvariablesA(pj). One has a pair of annihilation and
creation operators

a(pj) =

∂

∂A(pj)

, a(pj)†=A(pj)

for each possible value ofj, which indexes the possible valuespj. These operators
satisfy the commutation relations

[a(pj),a(pj)†] =δjk

In the occupation number representation of the Fock space, orthonormal
basis elements are
|···,npj− 1 ,npj,npj+1,···〉

with annihilation and creation operators acting by

apj|···,npj− 1 ,npj,npj+1,···〉=

√

npj|···,npj− 1 ,npj− 1 ,npj+1,···〉