34  Canonical commutation relations

35 Canonical commutation relations

In quantum mechanics, physical quantities such as position, momentum, and energy are represented by linear operators acting on quantum states. Unlike ordinary numbers, operators do not in general commute: the order in which two operators are applied can matter. Given two operators \(\hat{A}\) and \(\hat{B}\), their commutator is defined as \[ [\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}. \] Thus \([\hat{A},\hat{B}]=0\) precisely when the two operators commute. Commutators play a fundamental role throughout quantum mechanics, both in expressing the mathematical structure of the theory and in deriving properties of quantum systems.

The most important example in elementary quantum mechanics is the canonical commutation relation between position and momentum. In one spatial dimension, \[ [\hat{x},\hat{p}]=i\hbar\hat{1}, \] where \(\hat{1}\) denotes the identity operator. In the position representation, the two operators act on a wavefunction \(\psi\) according to \[ (\hat{x}\psi)(x)=x\psi(x),\qquad (\hat{p}\psi)(x)=\frac{\hbar}{i}\frac{d\psi(x)}{dx}. \] The commutation relation follows directly from these definitions and the product rule for differentiation. It is important to understand this as an operator identity: it states that applying \([\hat{x},\hat{p}]\) to any suitable wavefunction has the same effect as multiplying that wavefunction by \(i\hbar\).

Canonical commutation relations also appear naturally when quantum systems are described using creation and annihilation operators. For bosons, creation and annihilation operators \(\hat{a}_p^\dagger\) and \(\hat{a}_p\) satisfy the canonical commutation relations \[ [\hat{a}_p,\hat{a}_q^\dagger]=\delta_{pq}\hat{1},\qquad [\hat{a}_p,\hat{a}_q]=0,\qquad [\hat{a}_p^\dagger,\hat{a}_q^\dagger]=0. \] These relations encode the algebraic structure associated with particles obeying Bose–Einstein statistics.

For fermions, ordinary commutators are replaced by anticommutators. The anticommutator is defined by \[ \{\hat{A},\hat{B}\}=\hat{A}\hat{B}+\hat{B}\hat{A}, \] and fermionic creation and annihilation operators satisfy the canonical anticommutation relations \[ \{\hat{a}_p,\hat{a}_q^\dagger\}=\delta_{pq}\hat{1},\qquad \{\hat{a}_p,\hat{a}_q\}=0,\qquad \{\hat{a}_p^\dagger,\hat{a}_q^\dagger\}=0. \] In particular, setting \(p=q\) gives \[ (\hat{a}_p^\dagger)^2=(\hat{a}_p)^2=0. \] Thus a fermionic one-particle state cannot be occupied more than once. This is the algebraic manifestation of the Pauli exclusion principle and is one reason anticommutation relations are central to electronic-structure theory.

Commutation and anticommutation relations allow us to manipulate products of operators without referring to a particular matrix representation. For example, if \[ [\hat{A},\hat{B}]=\hat{C}, \] then we can interchange the order of \(\hat{A}\) and \(\hat{B}\) according to \[ \hat{A}\hat{B}=\hat{B}\hat{A}+\hat{C}. \] Similarly, canonical (anti)commutation relations can be used repeatedly to rearrange strings of creation and annihilation operators. Such manipulations occur throughout quantum chemistry, in particular in second quantization and many-body theory. The exercises in this chapter provide practice with these algebraic rules, beginning with the position–momentum commutator and progressing to bosonic and fermionic creation and annihilation operators.

Exercises

Exercise 35.1 (The canonical commutation relation in one dimension) Consider wavefunctions of one spatial variable, \[ \psi\in L^2(\mathbb{R})=\mathcal{H}. \]

A linear operator \(\hat{A}\) maps wavefunctions to wavefunctions, \[ \hat{A}:\mathcal{H}\to\mathcal{H},\qquad \psi\mapsto\hat{A}\psi. \]

Strictly speaking, some operators are defined only on suitable subspaces of \(\mathcal{H}\).

The position and momentum operators are defined by their action on a wavefunction: \[ (\hat{x}\psi)(x)=x\psi(x) \] and \[ (\hat{p}\psi)(x)=\frac{\hbar}{i}\frac{d\psi(x)}{dx}. \]

Products of operators mean composition, \[ (\hat{A}\hat{B})\psi=\hat{A}(\hat{B}\psi), \] and the commutator of two operators is \[ [\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}. \]

  1. In the expression \[ (\hat{x}\psi)(x)=x\psi(x), \] the letter \(x\) occurs in several different roles. Explain the difference between the operator \(\hat{x}\), the argument \(x\) on the left-hand side, and the factor \(x\) on the right-hand side. What are the input and output of the operator \(\hat{x}\)? Why is it mathematically misleading to write simply \(\hat{x}=x\)?

  2. Show that the position operator \(\hat{x}\) is linear. That is, show that for wavefunctions \(\psi,\varphi\) and scalars \(a,b\in\mathbb{C}\), \[ \hat{x}(a\psi+b\varphi)=a\hat{x}\psi+b\hat{x}\varphi. \]

  3. Let \(\psi\) be an arbitrary sufficiently differentiable wavefunction. Evaluate \[ ([\hat{x},\hat{p}]\psi)(x) \] by first expanding the definition of the commutator and then applying the definitions of \(\hat{x}\) and \(\hat{p}\). Use the product rule when necessary.

  4. Hence show that \[ [\hat{x},\hat{p}]=i\hbar\hat{1}, \] where \(\hat{1}\) is the identity operator. Explain why the right-hand side is written as \(i\hbar\hat{1}\) rather than simply as the number \(i\hbar\).