Operators and observables

Linear operators on function spaces: adjoints, commutators, ladder operators, and canonical quantization.

An operator is a machine that turns a function into another function. Differentiation is one: hand it $f$, it returns $f'$. Multiplication by $x$ is another. Once functions are regarded as vectors in an infinite-dimensional space (linear algebra with the index running continuously), these operators are the exact analogue of matrices, and the whole apparatus of eigenvalues, adjoints, and the spectral theorem carries over. What is genuinely new — and what makes this the mathematics of quantum theory — is that operators, unlike numbers, generally do not commute: the order in which you apply them matters, and the amount by which it matters, the commutator, encodes the deepest structure of the theory.

This chapter is the operator formalism, developed as the last piece of mathematics before quantum field theory. The calculus of variations gave the classical field: a Lagrangian, an action, the field equations, and Noether’s conserved quantities. Quantization is the step that promotes those classical fields to operators, imposes a commutation relation, and discovers that each mode of the field is a harmonic oscillator whose energy comes in discrete quanta — particles. The machinery that makes that step precise — adjoints, commutators, ladder operators, and the exponential of an operator — is built here.

• 13.1 Operators on function spaces — functions as vectors; linear operators as infinite-dimensional matrices; Dirac’s bra–ket notation; why $\hat x$ and $\hat p$ do not commute.
• 13.2 Adjoints, self-adjointness, and observables — the adjoint $\hat A^\dagger$; Hermitian (self-adjoint) operators, their real spectra and orthogonal eigenfunctions; discrete versus continuous spectra; why observables are self-adjoint.
• 13.3 The commutator algebra and uncertainty — the commutator $[\hat A,\hat B]$; the canonical relation $[\hat x,\hat p]=i\hbar$; the uncertainty principle as its direct consequence.
• 13.4 Ladder operators and the harmonic oscillator — raising and lowering operators; solving the oscillator spectrum by algebra alone; the number operator; the ground state annihilated by $\hat a$.
• 13.5 Time evolution, generators, and canonical quantization — functions of operators; the Heisenberg equation; symmetries as generators (Noether, operator form); the quantization rule and the field mode as an oscillator — the doorway to QFT.

The chapter extends the spectral theorem of the linear-algebra chapter and supplies the operators behind the Schrödinger equation.