Quantum Mechanics for Mathematicians

A(α) =

∫

α(p)A(p)dp, A(α) =

∫

α(p)A(p)dp

with Poisson bracket relations written

{A(p),A(p′)}={A(p),A(p′}= 0, {A(p),A(p′)}=iδ(p−p′) (36.17)

To get observables, we would like to define quadratic products of operators
such as

N̂=

∫+∞

−∞

a(p)†a(p)dp

for the number operator,

P̂=

∫+∞

−∞

pa(p)†a(p)dp

for the momentum operator, and

Ĥ=

∫+∞

−∞

p^2

2 m

a(p)†a(p)dp

for the Hamiltonian operator. One way to make rigorous sense of these is
as limits of the operators 36.8, 36.9 and 36.10. Another is as bilinear forms
onS∗(H 1 )×S∗(H 1 ) forH 1 =S(R), sending pairs of states|φ 1 〉,|φ 2 〉to, for
instance,〈φ 1 |N̂|φ 2 〉(for details see [17], section 5.4.2).

36.4 Multi-particle wavefunctions

To recover the conventional formalism in which anN-particle state is described
by a wavefunction
ψN(p 1 ,p 2 ,···,pN)

symmetric in theNarguments, one needs to recall (see chapter 9) that the
tensor product of the vector space of functions on a setX 1 and the vector
space of functions on a setX 2 is the vector space of functions on the product
setX 1 ×X 2. The symmetric tensor product will be the symmetric functions.
Applying this to whatever spaceH 1 of functions onRwe choose to use, the
symmetric tensor productSN(H 1 ) will be a space of symmetric functions on
RN. For details of this construction, see for instance chapter 5 of [17].
From the point of view of distributional operatorsa(p),a†(p), given an ar-
bitrary state|ψ〉in the multi-particle state space, the momentum space wave-
function component with particle numberNcan be expressed as

ψN(p 1 ,p 2 ,···,pN) =〈 0 |a(p 1 )a(p 2 )···a(pN)|ψ〉

.