Quantum Mechanics for Mathematicians

and the induced Hilbert space structure on such tensor products. We will
adopt this point of view here, with details available in the references of
section 36.6.
In the continuum normalization, an arbitrary solution to the free particle
Schr ̈odinger equation is given by

ψ(x,t) =

1

√

2 π

∫∞

−∞

α(p)eipxe−i

2 pm^2 t

dp (36.11)

Att= 0

ψ(x,0) =

1

√

2 π

∫∞

−∞

α(p)eipxdp

which is the Fourier inversion formula, expressing a functionψ(x,0) in terms

of its Fourier transform,α(p) =ψ ̃(x,0)(p). We see that the functionsα(p)
parametrize initial data andH 1 , the solution space of the free particle Schr ̈odinger
equation, can be identified with the space of suchα.
Using the notationA(α) to denote the element ofH 1 determined by initial
dataα, in the Fock space description of multi-particle states as symmetric tensor
products ofH 1 we have the following annihilation and creation operators (these
were discussed in the finite dimensional case in section 26.4)

a†(α)P+(A(α 1 )⊗···⊗A(αn)) =

√

n+ 1P+(A(α)⊗A(α 1 )⊗···⊗A(αn)) (36.12)

a(α)P+(A(α 1 )⊗···⊗A(αn)) =

1

√

n

∑n

j=1

〈α,αj〉P+(A(α 1 )⊗···⊗Â(αj)⊗···⊗A(αn)) (36.13)

(theÂ(αj) means omit that term in the tensor product, andP+is the sym-
metrization operator defined in section 9.6) satisfying the commutation relations

[a(α 1 ),a(α 2 )] = [a†(α 1 ),a†(α 2 )] = 0

[a(α 1 ),a†(α 2 )] =〈α 1 ,α 2 〉=

∫

α 1 (p)α 2 (p)dp

(36.14)

Different choices for which space of functionsαto take asH 1 lead to different
problems. Three possibilities are:

• H 1 =L^2 (R)
  This choice allows for an isomorphism betweenH 1 and its dual, using the
  inner product
  〈α 1 ,α 2 〉=

∫

α 1 (p)α 2 (p)dp

As in the single-particle case the problem here is that position

α(p) =

1

√

2 π

eipx

′