Quantum Mechanics for Mathematicians

For each of these descriptions ofFd+we have basis elements we can take to
be orthonormal, providing an inner product onFd+. We also have a set ofd
annihilation and creation operatorsaFj,aF†jthat are each other’s adjoints, and
satisfy the canonical anticommutation relations

[aFj,aF†k]+=δjk

We will describe the basis state|n 1 ,n 2 ,···,nd〉as one containingn 1 quanta of
type 1,n 2 quanta of type 2, etc., and a total number of quanta

n=

∑d

j=1

nj

Analogously to the bosonic case, a multi-particle fermionic theory can be
constructed usingFd+, by takingVJ+ =H 1. This is a fermionic version of
second quantization, with the multi-particle state space given by quantization
of a pseudo-classical dual phase spaceVof solutions to some wave equation. The
formalism automatically implies the Pauli principle (no more than one quantum
per state) as well as the antisymmetry property for states of multiple fermionic
quanta that is a separate postulate in our earlier description of multiple particle
states as tensor products.

36.2 Multi-particle quantum systems of free par-

ticles: finite cutoff formalism

To describe multi-particle quantum systems in terms of quanta of a harmonic
oscillator system, we would like to proceed as described in section 36.1, taking
solutions to the free particle Schr ̈odinger equation (discussed in chapters 10
and 11) as the single-particle state space. Recall that for a free particle in one
spatial dimension such solutions are given by complex-valued functions onR,
with observables the self-adjoint operators for momentum

P=−i

d

dx

and energy (the Hamiltonian)

H=

P^2

2 m

=−

1

2 m

d^2

dx^2

Eigenfunctions for bothPandHare the functions of the form

ψp(x)∝eipx

forp∈R, with eigenvaluespforP and p

2
2 mforH. Recall that these eigen-
functions are not normalizable, and thus not in the conventional choice of state
space asL^2 (R).