Wave Mechanics and the Position/Momentum Representation

The postulates are stated abstractly, in terms of a state vector $|\psi ⟩$ in a Hilbert space $\mathcal{H}$. For a particle moving in space, that abstract vector is made concrete by expanding it in the position basis, giving the familiar wavefunction $\psi (x)$ and turning the Schrödinger equation into a partial differential equation. This page makes that bridge — from Dirac kets to wavefunctions — and derives the first physical consequences: the probability current, Ehrenfest's theorem, and the classical limit.

Throughout, $\hslash$ is kept explicit and we work with a single spinless particle, $\mathcal{H}=L^2({\mathbb{R}}^3)$ (see preliminaries).

1. Position and Momentum Eigenkets

Introduce a continuum of position eigenkets $|x⟩$ satisfying

$$\hat{x}|x⟩=x|x⟩,⟨x|x^{\prime }⟩=\delta (x-x^{\prime }).$$

These are not elements of $\mathcal{H}$ (they are not square-integrable); they live in the larger rigged Hilbert space — see spectral-theorem.md for the rigorous framework. Treated formally, they furnish a resolution of the identity,

$$\hat{1}=\int dx|x⟩⟨x|,$$

which lets us represent any abstract ket by a function:

$$\boxed{\psi (x)\equiv ⟨x|\psi ⟩}$$

the wavefunction in the position representation. Normalization $⟨\psi |\psi ⟩=1$ becomes $\int |\psi (x){|}^{2}dx=1$, and by the Born rule $|\psi (x){|}^2$ is the probability density for finding the particle at $x$.

Momentum eigenkets are defined analogously, $\hat{p}|p⟩=p|p⟩$, with $⟨x|p⟩$ fixed by the canonical commutation relation below to be a plane wave:

$$⟨x|p⟩=\frac{1}{\sqrt{2\pi \hslash }}{e}^{ipx/\hslash }.$$

Consequently the momentum-space wavefunction $\tilde{\psi }(p)=⟨p|\psi ⟩$ is the Fourier transform of $\psi (x)$:

$$\tilde{\psi }(p)=\frac{1}{\sqrt{2\pi \hslash }}\int dx{e}^{-ipx/\hslash }\psi (x).$$

Position and momentum are thus two representations of the same abstract state, related by Fourier duality — the origin of the uncertainty relation.

2. The Canonical Commutation Relation

The single dynamical input that fixes wave mechanics is the canonical commutator

$$\boxed{[\hat{x},\hat{p}]=i\hslash \hat{1}.}$$

In the position representation, $\hat{x}$ acts by multiplication and $\hat{p}$ by differentiation:

$$\hat{x}\psi (x)=x\psi (x),\hat{p}\psi (x)=-i\hslash \frac{\partial }{\partial x}\psi (x).$$

One checks the commutator directly on a test function:

$$[\hat{x},\hat{p}]\psi =x(-i\hslash {\psi }^{\prime })-(-i\hslash )(x\psi {)}^{\prime }=i\hslash \psi .$$

That $\hat{x}$ and $\hat{p}$ cannot both have normalizable eigenstates, and that the relation cannot hold for finite matrices (trace of a commutator vanishes), is a structural fact requiring the infinite-dimensional $L^2({\mathbb{R}}^3)$.

3. The Schrödinger Equation as a PDE

For a particle of mass $m$ in a potential $V(x)$ the Hamiltonian is $\hat{H}={\hat{p}}^2/2m+V(\hat{x})$. Projecting Postulate 5 onto $⟨x|$ turns the abstract evolution law into the time-dependent Schrödinger equation:

$$i\hslash \frac{\partial \psi (x,t)}{\partial t}=-\frac{{\hslash }^2}{2m}{\nabla }^2\psi (x,t)+V(x)\psi (x,t).$$

Separating $\psi (x,t)=\varphi (x)e^{-iEt/\hslash }$ gives the time-independent (stationary-state) equation, an eigenvalue problem for $\hat{H}$:

$$-\frac{{\hslash }^2}{2m}{\nabla }^2\varphi +V(x)\varphi =E\varphi .$$

Its solutions — the energy eigenfunctions and spectrum — are the subject of the systems pages.

4. Probability Current and Continuity

Because evolution is unitary, total probability is conserved. Locally this is expressed by a continuity equation. Define the probability density and current

$$\rho (x,t)=|\psi {|}^2,\mathbf{j}(x,t)=\frac{\hslash }{2mi}({\psi }^{\ast }\nabla \psi -\psi \nabla {\psi }^{\ast })=\frac{\hslash }{m}\mathrm{Im}({\psi }^{\ast }\nabla \psi ).$$

Differentiating $\rho$ and using the Schrödinger equation (and its complex conjugate) gives

$$\boxed{\frac{\partial \rho }{\partial t}+\nabla \cdot \mathbf{j}=0.}$$

Integrating over all space, the flux vanishes at infinity for normalizable states, so $\frac{d}{dt}\int \rho dx=0$: normalization is preserved for all time. The current $\mathbf{j}$ is what carries probability through, e.g., a tunneling barrier.

5. Ehrenfest's Theorem

Expectation values obey equations that mirror classical mechanics. From the Heisenberg equation of motion (see heisenberg-picture.md), or directly by differentiating $⟨\hat{A}⟩=⟨\psi |\hat{A}|\psi ⟩$,

$$\frac{d}{dt}⟨\hat{A}⟩=\frac{i}{\hslash }⟨[\hat{H},\hat{A}]⟩+⟨\frac{\partial \hat{A}}{\partial t}⟩.$$

Applied to $\hat{x}$ and $\hat{p}$ this yields Ehrenfest's theorem:

$$\frac{d}{dt}⟨\hat{x}⟩=\frac{⟨\hat{p}⟩}{m},\frac{d}{dt}⟨\hat{p}⟩=-⟨\nabla V(\hat{x})⟩.$$

These are Hamilton's/Newton's equations for the means. Note the right-hand side is $⟨\nabla V⟩$, not $\nabla V(⟨x⟩)$ — the two agree only when $V$ is at most quadratic or the wavepacket is narrow.

6. The Classical Limit

Ehrenfest's theorem gives one route to classicality: when the potential varies slowly over the width of the wavepacket, $⟨\nabla V⟩\approx \nabla V(⟨x⟩)$ and the centroid follows a classical trajectory. A complementary route writes $\psi =\sqrt{\rho }{e}^{iS/\hslash }$ (the Madelung / WKB form): substituting into the Schrödinger equation gives, at leading order in $\hslash$, the classical Hamilton–Jacobi equation for the phase $S$, with quantum corrections organized as a power series in $\hslash$. This is the starting point of the WKB approximation and connects to the path integral, where the classical path is the stationary point of the action as $\hslash \to 0$.

See also

• Mathematical Preliminaries — Hilbert space, operators, $L^2$.
• Spectral theorem & unbounded operators — rigorous status of $|x⟩,|p⟩$.
• Uncertainty relations — the quantitative face of Fourier duality.
• Path Integral Formulation — the classical-limit / action perspective.