13.1 Operators on function spaces

The linear-algebra chapter treated vectors with finitely many components and matrices that act on them. If we let the index run continuously, a vector becomes a function and a matrix becomes an operator. This one change of viewpoint — functions are vectors, operators are matrices — carries the entire machinery of eigenvalues, inner products, and the spectral theorem into the setting where quantum mechanics lives.

Functions as vectors

A vector $\mathbf v$ is a list of numbers $v_i$ indexed by $i = 1,\dots,n$ . A function $f$ is a list of numbers $f(x)$ indexed by the continuous label $x$ . The dictionary is exact: the discrete index $i$ becomes the continuous label $x$ , and sums over the index become integrals. The inner product of the linear-algebra chapter, $\langle \mathbf u, \mathbf v\rangle = \sum_i u_i^* v_i$ , becomes

\langle f, g\rangle \;=\; \int f^*(x)\, g(x)\; dx,

the same object with the sum replaced by an integral. Functions with a finite such norm form a vector space — a Hilbert space — with lengths, angles, and orthogonality just as in finite dimensions. Everything from Chapter 4 applies; only the dimension is infinite.

Operators as infinite-dimensional matrices

A linear operator $\hat A$ takes a function and returns a function, respecting linearity, $\hat A(\alpha f + \beta g) = \alpha \hat A f + \beta \hat A g$ . Three that recur:

Differentiation $\hat D = \dfrac{d}{dx}$ : $\;\hat D f = f'$ .
Multiplication by $x$ , written $\hat X$ : $\;(\hat X f)(x) = x\,f(x)$ .
The Laplacian $\nabla^2$ , and integral operators $(\hat K f)(x) = \int K(x,x')\,f(x')\,dx'$ built from a kernel $K(x,x')$ .

where

$\hat A$: a linear operator — a map from functions to functions
$\langle f, g\rangle = \int f^* g\, dx$: inner product on the function space
$\hat X$: position operator: multiply by x
$\hat D = d/dx$: differentiation operator

The matrix analogy is literal. A matrix acts by $(A\mathbf v)_i = \sum_j A_{ij}v_j$ ; an integral operator acts by $(\hat K f)(x) = \int K(x,x')f(x')\,dx'$ — the kernel $K(x,x')$ is the “continuous matrix” $A_{ij}$ with rows labelled by $x$ and columns by $x'$ . In any chosen orthonormal basis $\{e_n\}$ of the function space (the Fourier modes, say, or the oscillator eigenfunctions of 13.4), an operator becomes an honest matrix with entries $A_{mn} = \langle e_m, \hat A\, e_n\rangle$ , and operator algebra becomes matrix algebra.

Dirac’s bra–ket notation

The vector viewpoint calls for notation that suppresses the coordinate. Write a function as a ket $|\psi\rangle$ (an abstract vector), its conjugate as a bra $\langle\psi|$ , and the inner product as the bracket $\langle\phi|\psi\rangle$ . An operator acts on a ket to give a ket, $\hat A|\psi\rangle$ . The wavefunction $\psi(x)$ is recovered as the component of $|\psi\rangle$ along the position basis, $\psi(x) = \langle x|\psi\rangle$ — exactly as $v_i = \langle e_i, \mathbf v\rangle$ picks out a component. The notation is not new mathematics; it is Chapter 4’s inner-product geometry with a coordinate-free spelling, and it makes operator manipulations read cleanly.

Order matters

Here the infinite-dimensional world reveals its defining feature. Numbers commute — $x\cdot D$ and $D\cdot x$ would be equal if they were numbers. Operators, like matrices, generally do not. Apply $\hat X$ then $\hat D$ to a function, then in the other order, and compare.

[X̂, D̂] f = −f (max residual 3e-4)

bump centre x₀ = 0.00 bump width w = 0.80

Multiply-by-x and differentiate do not commute: swapping their order changes the result by exactly −f, so [X̂, D̂] = −1 (the identity operator). Multiply D̂ by −iℏ to get the momentum operator and this becomes the canonical commutation relation [x̂, p̂] = iℏ.

The two orderings give different curves, and — as the interactive verifies numerically — their difference is exactly $-f$ :

(\hat X \hat D)f = x f', \qquad (\hat D \hat X)f = \frac{d}{dx}(x f) = f + x f', \qquad (\hat X\hat D - \hat D\hat X)f = -f.

So $\hat X\hat D - \hat D\hat X = -\hat 1$ , the identity operator (up to sign). The failure of two operators to commute is measured by their commutator $[\hat A,\hat B] = \hat A\hat B - \hat B\hat A$ , and here $[\hat X, \hat D] = -\hat 1$ — a nonzero operator. That single fact, dressed with the constants of quantum mechanics, becomes the canonical commutation relation $[\hat x, \hat p] = i\hbar$ and with it the uncertainty principle (13.3).

Position and momentum

The two operators the rest of the chapter turns on are the position and momentum operators. Position multiplies:

\hat x\,\psi(x) = x\,\psi(x).

Momentum differentiates. Its form is fixed by demanding that a plane wave $e^{ikx}$ — which carries definite momentum $p = \hbar k$ (de Broglie) — be an eigenfunction with eigenvalue $p$ :

\hat p = -i\hbar\frac{d}{dx}, \qquad \hat p\, e^{ikx} = -i\hbar\,(ik)\,e^{ikx} = \hbar k\, e^{ikx} = p\, e^{ikx}.

where

$\hat x\,\psi = x\psi$: position operator — multiplication by the coordinate
$\hat p = -i\hbar\, d/dx$: momentum operator — the generator of translations (13.5)
$\hbar$: reduced Planck constant J·s

These are precisely $\hat X$ and $-i\hbar\hat D$ from above, so their commutator follows immediately from the interactive’s result — the computation of the next-but-one lesson. First, though, the next lesson asks which operators are allowed to represent measurable quantities, and finds the answer in the adjoint.