Quantum Mechanics for Mathematicians

B.16 Chapter 36

Problem 1:
When the single-particle state spaceH 1 is a complex vector space with Her-
mitian inner product, one has an infinite dimensional case of the situation of
section 26.4. In this case one can write annihilation and creation operators act-
ing on the multi-particle state spacesS∗(H 1 ) or Λ∗(H 1 ) in a basis independent
manner as follows:

a†(f)P±(g 1 ⊗g 2 ⊗···⊗gn) =

√

n+ 1P±(f⊗g 1 ⊗g 2 ⊗···⊗gn)

a(f)P±(g 1 ⊗g 2 ⊗···⊗gn) =

1

√

n

∑n

j=1

(±1)j+1〈f,gj〉P±(g 1 ⊗g 2 ···⊗ĝj⊗···⊗gn)

Heref,gj∈ H 1 ,P±is the operation of summing over permutations used in
section 9.6 that produces symmetric or antisymmetric tensor products, andĝj
means omit thegjterm in the tensor product.
Show that, forforthonormal basis elements ofH 1 , these annihilation and
creation operators satisfy the CCR (+ case) or CAR (−case).

Problem 2:
In the fermionic case of problem 1, show that the inner product on Λ∗(H 1 )
that, for an orthonormal basise 1 ,···,enofH 1 , makes theei 1 ∧ei 2 ∧···∧eik
orthonormal fori 1 < i 2 <···< ikcan be written in an basis independent way
as
〈f 1 ∧f 2 ∧···∧fk,g 1 ∧g 2 ∧···∧gk〉= detM

whereMis thekbykmatrix withlmentry〈fl,gm〉.
WhenH 1 is a space of wavefunctions (in position or momentum space),
then takingfjto be some single-particle wavefunctions, and thegjto be delta-
functions in position or momentum space, this construction is known as the
“Slater determinant” construction giving antisymmetric wavefunctions. For the
bosonic case, a similar construction exists for symmetric tensor products, using
instead of the determinant of the matrix, something called the “permanent” of
the matrix.

B.17 Chapters 37 and 38

Problem 1:
Show that if one takes the quantum field theory Hamiltonian operator to be

Ĥ=

∫∞

−∞

Ψ̂†(x)

(

−

1

2 m

d^2

dx^2

+V(x)

)

Ψ(̂x)dx

the field operators will satisfy the conventional Schr ̈odinger equation for the
case of a potentialV(x).

Problem 2: