13.5 Time evolution, generators, and canonical quantization

The pieces are in place: operators as observables, self-adjointness, the commutator, the ladder. This final lesson assembles them into the step the whole chapter was built for — canonical quantization, the recipe that turns a classical theory into a quantum one — and shows how it opens onto quantum field theory. Along the way, functions of operators give time evolution, and commutators reappear as the operator form of Noether’s theorem.

Functions of operators

An operator can be fed into a power series just as a number can. The most important is the exponential,

e^{\hat A} \;=\; \sum_{n=0}^\infty \frac{\hat A^n}{n!},

which converges for the operators we use and inherits the algebra of the exponential — with one caveat: $e^{\hat A}e^{\hat B} = e^{\hat A + \hat B}$ only if $[\hat A,\hat B]=0$ . When they do not commute, the correction is itself built from commutators (the Baker–Campbell–Hausdorff formula), a reminder that the commutator governs everything.

The exponential of a self-adjoint operator is central because it generates continuous transformations. The time-evolution operator is

\hat U(t) = e^{-i\hat H t/\hbar},

and $|\psi(t)\rangle = \hat U(t)|\psi(0)\rangle$ solves the Schrödinger equation (Foundations 6.8) $i\hbar\,\partial_t|\psi\rangle = \hat H|\psi\rangle$ : energy generates evolution in time.

The Heisenberg equation

Equivalently, one can hold the states fixed and let the operators evolve. Differentiating $\hat A(t) = \hat U^\dagger \hat A\,\hat U$ gives the Heisenberg equation of motion:

\boxed{\;\frac{d\hat A}{dt} = \frac{i}{\hbar}\,[\hat H, \hat A] \;+\; \frac{\partial \hat A}{\partial t}.\;}

▶ Commutator with H is the operator time-derivative Derivation

With $\hat A(t) = \hat U^\dagger(t)\,\hat A\,\hat U(t)$ and $\hat U = e^{-i\hat H t/\hbar}$ , differentiate using $d\hat U/dt = -\tfrac{i}{\hbar}\hat H\hat U$ and $d\hat U^\dagger/dt = +\tfrac{i}{\hbar}\hat U^\dagger\hat H$ :

\frac{d\hat A}{dt} = \frac{i}{\hbar}\hat U^\dagger\hat H\hat A\hat U - \frac{i}{\hbar}\hat U^\dagger\hat A\hat H\hat U = \frac{i}{\hbar}\hat U^\dagger[\hat H,\hat A]\hat U = \frac{i}{\hbar}[\hat H,\hat A(t)],

(plus $\partial\hat A/\partial t$ if $\hat A$ depends on time explicitly). The rate of change of any observable is its commutator with the Hamiltonian. This is the exact quantum echo of the classical statement that the time-derivative of a quantity is its Poisson bracket with the Hamiltonian, $\dot A = \{A, H\}$ — a correspondence we exploit in a moment.

Two immediate consequences: if $\hat A$ commutes with $\hat H$ , then $d\hat A/dt = 0$ and $\hat A$ is conserved; and applied to $\hat x$ and $\hat p$ , the Heisenberg equation reproduces Newton’s laws in operator form, $\dot{\hat x} = \hat p/m$ and $\dot{\hat p} = -\partial_x\hat V$ .

Symmetries as generators — Noether, in operator form

A self-adjoint operator $\hat G$ generates a family of transformations $e^{-i\hat G s/\hbar}$ as the parameter $s$ runs. The examples are the fundamental ones:

$\hat p$ generates spatial translations: $e^{-i\hat p a/\hbar}$ shifts a wavefunction by $a$ .
$\hat H$ generates time translation (above).
$\hat L$ (angular momentum) generates rotations.

Now the connection to Noether’s theorem becomes an identity. A transformation is a symmetry when its generator commutes with $\hat H$ , $[\hat G,\hat H]=0$ — and by the Heisenberg equation that is exactly the statement that $\hat G$ is conserved. Symmetry $\Leftrightarrow$ its generator commutes with $\hat H$ $\Leftrightarrow$ a conservation law. Translation symmetry $\Rightarrow$ momentum conserved; time-translation symmetry $\Rightarrow$ energy conserved; rotational symmetry $\Rightarrow$ angular momentum conserved. The variational Noether theorem of Chapter 12 and this operator statement are the same law in the classical and quantum languages.

Canonical quantization

The parallel between the classical Poisson bracket and the quantum commutator is the whole recipe. To quantize a classical system:

Take its classical Hamiltonian formulation — coordinates $q_i$ and conjugate momenta $p_i$ , obtained from the Lagrangian of Chapter 12.
Promote each $q_i, p_i$ to a self-adjoint operator.
Replace the classical Poisson bracket by the commutator via

\boxed{\;\{A, B\}_{\text{PB}} \;\longrightarrow\; \frac{1}{i\hbar}[\hat A,\hat B].\;}

The fundamental bracket $\{q, p\} = 1$ becomes $[\hat q,\hat p] = i\hbar$ — the canonical commutation relation, now derived as the quantization of a classical structure rather than postulated. Everything in this chapter has been the machinery this one substitution sets in motion.

where

$\{A,B\}_{\text{PB}}$: classical Poisson bracket of two phase-space functions
$\hat U(t) = e^{-i\hat H t/\hbar}$: time-evolution operator; Ĥ generates time translation
$\hat G$: a self-adjoint generator; e^{-iĜs/ℏ} is the transformation it generates

The doorway to quantum field theory

Apply the recipe not to a particle but to a field and quantum field theory is born. The classical acoustic field of Sound 4.8, or the electromagnetic field, has a Lagrangian density, a conjugate momentum field, and — after a Fourier decomposition (Ch 7) — a description as infinitely many independent oscillators, one per mode $k$ . Canonical quantization promotes each mode to a quantum oscillator with its own ladder operators obeying

[\hat a_{k},\, \hat a_{k'}^\dagger] = \delta_{kk'}.

Then, exactly as in 13.4, $\hat a_k^\dagger$ creates a quantum of mode $k$ — a particle — and $\hat a_k$ destroys one. The field itself becomes an operator, a superposition over modes of creation and annihilation operators, and its excitations above the vacuum $|0\rangle$ are the particles of the theory: photons for light, phonons for sound (the quantized lattice waves of Sound 4.6), and the electrons, quarks, and gauge bosons of the Standard Model for the matter and force fields. That is where this chapter has been heading: the operator formalism is the language in which a classical field, handed over by the calculus of variations, becomes a quantum theory of particles.

What we use this for

The operator formalism is the mathematical substrate of quantum theory across the bookshelf and beyond:

The Schrödinger equation (Foundations 6.8) — the Hamiltonian operator, its spectrum, and time evolution $e^{-i\hat H t/\hbar}$ .
Phonons and quantized sound — the lattice of oscillators in Sound 4.6, quantized mode by mode, are phonons; heat capacity, sound attenuation, and thermal transport in solids are their statistics.
The spectral theorem in action (Foundations 4.6) — self-adjoint operators, real spectra, and complete eigenbases, now for the observables of quantum mechanics.
Quantum field theory — the creation/annihilation algebra above is the starting point of every modern theory of particles and fields.

⏳ The history — Dirac's q-numbers and the quantization of the field

Paul Dirac, reading Heisenberg’s 1925 paper, noticed that the non-commuting quantum quantities — his “q-numbers” — obeyed the same algebra as the classical Poisson bracket, and in a 1925 paper proposed the correspondence $\{A,B\} \to \frac{1}{i\hbar}[\hat A,\hat B]$ that became canonical quantization. His 1930 Principles of Quantum Mechanics set the operator-and-bra–ket formalism used ever since. Then, in 1927, Dirac quantized the electromagnetic field itself — treating its modes as oscillators and their excitations as photons — inventing quantum field theory and giving the first account of the emission and absorption of light as the raising and lowering of field oscillators. The Born–Heisenberg–Jordan “three-man paper” of 1925 had already quantized a field of oscillators in the abstract; Dirac made it physics. The line from Lagrange’s action to a photon runs straight through the operator calculus of this chapter.

That closes the chapter. From functions-as-vectors to the creation of a particle, the operator formalism is the final piece: with the calculus of variations supplying the classical field and this chapter supplying its quantization, the road into quantum field theory is open.