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 theorem 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 either coordinate space or momentum space. It turns out that there is another class of quite diferent 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 constant) integers. These integers label the irreducible representations of the U(1) symmetry generated by the Hamiltonian, and they can be interpreted as counting 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.
Chapter contents
- 22.1 The harmonic oscillator with one degree of freedom
- 22.2 Creation and annihilation operators
- 22.3 The Bargmann-Fock representation
- 22.4 Quantization by annihilation and creation operators
- 22.5 For further reading
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正文：英文 · OCR 机器稿 · 待校对
核对状态：OCR 机器稿 · 待校对
原书 PDF · 印刷页 244、245、246、247、248、249、250、251、252、253
来源版本：2025-10-20
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