Chapter 27

The Fermionic Oscillator

In this chapter we’ll introduce a new quantum system by using a simple varia-
tion on techniques we used to study the harmonic oscillator, that of replacing
commutators by anticommutators. This variant of the harmonic oscillator will
be called a “fermionic oscillator”, with the original sometimes called a “bosonic
oscillator”. The terminology of “boson” and “fermion” refers to the principle
enunciated in chapter 9 that multiple identical particles are described by tensor
product states that are either symmetric (bosons) or antisymmetric (fermions).
The bosonic and fermionic oscillator systems are single-particle systems, de-
scribing the energy states of a single particle, so the usage of the bosonic/fermion-
ic terminology is not obviously relevant. In later chapters we will study quantum
field theories, which can be treated as infinite dimensional oscillator systems.
In that context, multiple particle states will automatically be symmetric or an-
tisymmetric, depending on whether the field theory is treated as a bosonic or
fermionic oscillator system, thus justifying the terminology.

27.1 Canonical anticommutation relations and the fermionic oscillator

Recall that the Hamiltonian for the quantum harmonic oscillator system ind
degrees of freedom (setting~=m=ω= 1) is

H=

∑d

j=1

1

2

(Q^2 j+Pj^2 )

and that it can be diagonalized by introducing number operatorsNj =a†jaj
defined in terms of operators

aj=

1

√

2

(Qj+iPj), a†j=

1

√

2

(Qj−iPj)