Position, Momentum, and Continuous Spectra

Position and momentum are the observables that force the continuous machinery of the formalism. Neither has a normalizable eigenstate — a particle cannot be at exactly one point, nor have exactly one momentum, while remaining in $L^{2}$ — so the spectral theorem must be extended to continuous spectra. The eigenkets become delta-normalized, the sums become integrals, and the two observables turn out to be Fourier conjugates. Their non-commutation, expressed by the canonical commutator $[\overset{x}{^}, \overset{p}{^}] = i ℏ$ , is the algebraic root of the uncertainty principle and the starting point of nearly every quantization scheme.

Continuous bases

Discrete orthonormality $⟨ e_{i} ∣ e_{j} ⟩ = δ_{ij}$ has no direct continuous analogue, because a continuum of orthonormal vectors would require an uncountable sum equal to one. The resolution is to normalize the continuous eigenkets to a Dirac delta.

The Dirac delta $δ (ξ - ξ^{'})$ is not a function but a distribution, defined by $\int f (ξ^{'}) δ (ξ - ξ^{'}) d ξ^{'} = f (ξ)$ . It is the continuous Kronecker delta: it collapses an integral to the value of the integrand at the matching point, exactly as $δ_{ij}$ collapses a sum. Every discrete identity of the formalism carries over with $\sum_{i} \to \int d ξ$ and $δ_{ij} \to δ (ξ - ξ^{'})$ . A generic state expands as

∣ ψ ⟩ = \int ∣ ξ ⟩ ⟨ ξ ∣ ψ ⟩ d ξ = \int ψ (ξ) ∣ ξ ⟩ d ξ, ψ (ξ) = ⟨ ξ ∣ ψ ⟩,

with components $ψ (ξ)$ forming a wavefunction rather than a column. The price of the continuum is that $∣ ξ ⟩$ is not itself a physical state: its norm is $⟨ ξ ∣ ξ ⟩ = δ (0)$ , which is infinite. Continuous eigenkets are idealized basis elements, useful as intermediaries, and physical states are always normalizable superpositions of them.

A continuous basis labels its eigenkets by a real parameter; the orthonormality is a Dirac delta spike at coincident labels, the continuous counterpart of the Kronecker delta of a discrete basis.

The position operator

The position operator $\overset{x}{^}$ acts on a wavefunction by multiplication, $⟨ x ∣ \overset{x}{^} ∣ ψ ⟩ = x ψ (x)$ , and its eigenkets $∣ x_{0} ⟩$ satisfy $\overset{x}{^} ∣ x_{0} ⟩ = x_{0} ∣ x_{0} ⟩$ with continuous spectrum $x_{0} \in R$ . In the position representation the eigenket of $\overset{x}{^}$ at $x_{0}$ has wavefunction $⟨ x ∣ x_{0} ⟩ = δ (x - x_{0})$ , a spike at $x_{0}$ , confirming the delta normalization $⟨ x_{0} ∣ x_{0}^{'} ⟩ = δ (x_{0} - x_{0}^{'})$ . The completeness relation

\int ∣ x ⟩ ⟨ x ∣ d x = 1

is the identity inserted throughout wave mechanics to convert abstract kets into wavefunctions. Position is Hermitian: for real $x$ the multiplication operator satisfies $⟨ ϕ ∣ \overset{x}{^} ψ ⟩ = \int ϕ^{*} x ψ d x = ⟨ \overset{x}{^} ϕ ∣ ψ ⟩$ .

The momentum operator

Momentum is the generator of spatial translation. The translation operator $\hat{T} (a)$ defined by $\hat{T} (a) ∣ x ⟩ = ∣ x + a ⟩$ shifts a wavefunction, $⟨ x ∣ \hat{T} (a) ∣ ψ ⟩ = ψ (x - a)$ . For an infinitesimal shift $ε$ , Taylor expansion gives

ψ (x - ε) = ψ (x) - ε \partial_{x} ψ (x) + O (ε^{2}),

so $\hat{T} (ε) = 1 - ε \partial_{x}$ to first order. Writing the generator as a Hermitian operator, $\hat{T} (ε) = 1 - \frac{i}{ℏ} ε \overset{p}{^}$ , identifies

⟨ x ∣ \overset{p}{^} ∣ ψ ⟩ = - i ℏ \frac{\partial}{\partial x} ψ (x),

the momentum operator in the position representation. The factors are fixed by two requirements: $\overset{p}{^}$ must be Hermitian (the $- i ℏ$ makes the derivative self-adjoint on functions vanishing at infinity), and its eigenfunctions must be de Broglie waves. Solving $\overset{p}{^} u_{p} (x) = p u_{p} (x)$ gives $- i ℏ u_{p}^{'} = p u_{p}$ , hence

⟨ x ∣ p ⟩ = u_{p} (x) = \frac{1}{2 π ℏ} e^{i p x /ℏ},

a plane wave of wavelength $2 π ℏ/ p = h / p$ , the de Broglie relation recovered.

Momentum is the generator of translation: an infinitesimal spatial shift is the identity minus

\frac{i}{ℏ} ε \overset{p}{^}

, and exponentiating the generator produces a finite displacement of the wavefunction.

The normalization constant is fixed by requiring delta orthonormality:

⟨ p ∣ p^{'} ⟩ = \int ⟨ p ∣ x ⟩ ⟨ x ∣ p^{'} ⟩ d x = \frac{1}{2 π ℏ} \int e^{i (p^{'} - p) x /ℏ} d x = δ (p - p^{'}),

using the integral representation $\int e^{ik y} d y = 2 π δ (k)$ . Momentum has continuous spectrum $p \in R$ and its own completeness relation $\int ∣ p ⟩ ⟨ p ∣ d p = 1$ .

The canonical commutator

The defining algebraic relation of quantum mechanics follows by applying $[\overset{x}{^}, \overset{p}{^}]$ to an arbitrary wavefunction. In the position representation,

⟨ x ∣ [\overset{x}{^}, \overset{p}{^}] ∣ ψ ⟩ = x (- i ℏ \partial_{x} ψ) - (- i ℏ \partial_{x}) (x ψ) = - i ℏ x \partial_{x} ψ + i ℏ (ψ + x \partial_{x} ψ) = i ℏ ψ (x),

where the product rule on $\partial_{x} (x ψ) = ψ + x \partial_{x} ψ$ leaves only the term in which the derivative hits $x$ . Since this holds for every $ψ$ ,

[\overset{x}{^}, \overset{p}{^}] = i ℏ

the canonical commutation relation. It is the single equation from which the uncertainty principle, the ladder structure of the oscillator, and the correspondence with classical Poisson brackets all descend. The classical limit is explicit: the commutator divided by $i ℏ$ reproduces the Poisson bracket ${x, p} = 1$ , and Dirac's quantization rule ${A, B} \mapsto \frac{1}{i ℏ} [\hat{A}, \hat{B}]$ elevates every classical canonical pair to operators obeying this relation. In three dimensions it generalizes to $[\overset{x}{^}_{i}, \overset{p}{^}_{j}] = i ℏ δ_{ij}$ , with different components of position and momentum commuting, so a particle can have simultaneously definite $x$ -position and $y$ -momentum but never definite $x$ -position and $x$ -momentum. The trace argument of the spectral-theorem lesson shows why: no finite matrices can satisfy $[\overset{x}{^}, \overset{p}{^}] = i ℏ$ , so the operators that realize it are necessarily unbounded and act on an infinite-dimensional space.

The canonical commutator acts on a test function: applying position then momentum differs from momentum then position by exactly the term where the derivative strikes the multiplying coordinate, leaving

i ℏ

times the function.

Position and momentum as a Fourier pair

The two representations of a state are connected by the plane-wave overlap. Insert the momentum completeness into $ψ (x) = ⟨ x ∣ ψ ⟩$ ,

ψ (x) = \int ⟨ x ∣ p ⟩ ⟨ p ∣ ψ ⟩ d p = \frac{1}{2 π ℏ} \int \tilde{ψ} (p) e^{i p x /ℏ} d p,

and symmetrically, inserting position completeness into $\tilde{ψ} (p) = ⟨ p ∣ ψ ⟩$ ,

\tilde{ψ} (p) = \int ⟨ p ∣ x ⟩ ⟨ x ∣ ψ ⟩ d x = \frac{1}{2 π ℏ} \int ψ (x) e^{- i p x /ℏ} d x .

The position wavefunction and the momentum wavefunction are a Fourier-transform pair. Localizing one spreads the other: a narrow $ψ (x)$ needs a broad band of plane waves, so $\tilde{ψ} (p)$ is wide, and conversely. This is the Fourier statement that becomes the uncertainty principle once the widths are made precise. Both wavefunctions describe the identical abstract state $∣ ψ ⟩$ ; they are its components in two different continuous bases. Parseval's theorem, $\int ∣ ψ (x) ∣^{2} d x = \int ∣ \tilde{ψ} (p) ∣^{2} d p$ , expresses that the norm is the same computed in either basis, so $∣ \tilde{ψ} (p) ∣^{2}$ is the probability density in momentum.

Position and momentum wavefunctions are Fourier conjugates: a state sharply peaked in position is broad in momentum, and the same state viewed in the momentum basis narrows there only by spreading in position.

Expectation values in either representation

An expectation value can be computed in whichever basis is convenient, and the answer is the same. In the position basis, resolving the identity twice turns $⟨ ψ ∣ \hat{A} ∣ ψ ⟩$ into an integral of the wavefunction against the operator's kernel. For position and momentum the working forms are

⟨ \overset{x}{^} ⟩ = \int ψ^{*} (x) x ψ (x) d x, ⟨ \overset{p}{^} ⟩ = \int ψ^{*} (x) (- i ℏ \partial_{x}) ψ (x) d x,

or, in the momentum basis, with the roles exchanged and $\overset{x}{^} \to i ℏ \partial_{p}$ ,

⟨ \overset{p}{^} ⟩ = \int \tilde{ψ}^{*} (p) p \tilde{ψ} (p) d p, ⟨ \overset{x}{^} ⟩ = \int \tilde{ψ}^{*} (p) (i ℏ \partial_{p}) \tilde{ψ} (p) d p .

Position is diagonal in the position basis and momentum is diagonal in the momentum basis; each operator is a simple multiplication in its own representation and a derivative in the other. The symmetry $\overset{x}{^} \leftrightarrow \overset{p}{^}$ , $x \leftrightarrow p$ , $- i ℏ \partial_{x} \leftrightarrow + i ℏ \partial_{p}$ reflects the Fourier duality and the sign in the canonical commutator.

The momentum-space Schrödinger equation

The choice of representation extends to the dynamics. The time-dependent Schrödinger equation $i ℏ \partial_{t} ∣ ψ ⟩ = \hat{H} ∣ ψ ⟩$ becomes, in the momentum basis with $\hat{H} = \frac{p ^ ^{2}}{2 m} + V (\overset{x}{^})$ ,

i ℏ \frac{\partial ψ ~ ( p , t )}{\partial t} = \frac{p ^{2}}{2 m} \tilde{ψ} (p, t) + V (i ℏ \frac{\partial}{\partial p}) \tilde{ψ} (p, t) .

The kinetic term, a derivative operator in position space, is now a plain multiplication by $p^{2} /2 m$ , while the potential becomes a differential (or, for a generic $V$ , an integral) operator. Momentum space is therefore the natural setting whenever the kinetic energy dominates and the potential is simple in $p$ — the free particle, a uniform force, and scattering problems where the incident and scattered states are momentum eigenstates. The representations are exchanged by the same Fourier transform that relates the wavefunctions, so a problem awkward in one basis is often transparent in the other.

The continuous-spectrum apparatus — delta normalization, the Fourier pairing, and the canonical commutator — is the operator content behind wave mechanics. The next lesson turns $[\overset{x}{^}, \overset{p}{^}] = i ℏ$ into a quantitative limit on simultaneous knowledge through the generalized uncertainty principle.