Chapter 22

The Harmonic Oscillator

In this chapter we’ll begin the study of the most important exactly solvable
physical system, the harmonic oscillator. Later chapters will discuss extensions
of the methods developed here to the case of fermionic oscillators, as well as free
quantum field theories, which are harmonic oscillator systems with an infinite
number of degrees of freedom.
For a finite number of degrees of freedom, the Stone-von Neumann theo-
rem tells us that there is essentially just one way to non-trivially represent the
(exponentiated) Heisenberg commutation relations as operators on a quantum
mechanical state space. We have seen two unitarily equivalent constructions
of these operators: the Schr ̈odinger representation in terms of functions on ei-
ther coordinate space or momentum space. It turns out that there is another
class of quite different constructions of these operators, one that depends upon
introducing complex coordinates on phase space and then using properties of
holomorphic functions. We’ll refer to this as the Bargmann-Fock representation,
although quite a few mathematicians have had their name attached to it for one
good reason or another (some of the other names one sees are Friedrichs, Segal,
Shale, Weil, as well as the descriptive terms “holomorphic” and “oscillator”).
Physically, the importance of this representation is that it diagonalizes the
Hamiltonian operator for a fundamental sort of quantum system: the harmonic
oscillator. In the Bargmann-Fock representation the energy eigenstates of such
a system are monomials, and energy eigenvalues are (up to a half-integral con-
stant) integers. These integers label the irreducible representations of theU(1)
symmetry generated by the Hamiltonian, and they can be interpreted as count-
ing the number of “quanta” in the system. It is the ubiquity of this example that
justifies the “quantum” in “quantum mechanics”. The operators on the state
space can be simply understood in terms of so-called annihilation and creation
operators which decrease or increase by one the number of quanta.