Position Representation

For two chapters you have led a double life. In the wave-mechanics chapter a state was a function $\psi(x)$ — something you normalized, differentiated, and plotted. In this chapter it has been an abstract vector $|\psi\rangle$ — something you expand, project, and sandwich. This article closes the case: the two are one object, and the dictionary is a single equation, $\psi(x) = \langle x|\psi\rangle$ . A wavefunction is nothing but the components of the state vector along the continuous family of position kets $|x\rangle$ — exactly as $c_n = \langle n|\psi\rangle$ were its components along a discrete energy basis, with the label now sliding continuously and sums becoming integrals. Everything you did in the wave chapter is then revealed as basis-specific bookkeeping. The inner product $\int\phi^*\psi\,dx$ ? The abstract $\langle\phi|\psi\rangle$ with a complete set of position states inserted. The rule "position acts by multiplication"? The statement that $\hat x$ is diagonal in its own eigenbasis. The Schrödinger equation as a differential equation? The abstract eigenvalue problem $\hat H|\psi\rangle = E|\psi\rangle$ written in position coordinates — where the potential term is diagonal and only the kinetic term reaches off the diagonal, which is precisely why it appears as a derivative. Even the operators-as-matrices picture survives: a general operator in a continuous basis becomes a kernel $A(x, x') = \langle x|\hat A|x'\rangle$ , a "matrix" with continuous indices, and this article classifies the kernels you will actually meet. The payoff is strategic, not sentimental: once you see wave mechanics as one representation among many, you are free to leave it. The very next article builds the momentum representation on the same template, and choosing the basis that makes your problem diagonal becomes a craft rather than an accident of upbringing.

Key takeaways

$\psi(x) = \langle x|\psi\rangle$ : the wavefunction is the component function of the abstract ket along the position basis — same state, coordinate view.
Expansion: $|\psi\rangle = \int dx\,\psi(x)\,|x\rangle$ — the continuous analog of $|\psi\rangle = \sum_n c_n|n\rangle$ , with $\int dx\,|x\rangle\langle x| = \mathbb 1$ doing the work of completeness.
Inner products become integrals: $\langle\phi|\psi\rangle = \int\phi^*(x)\psi(x)\,dx$ — the wave chapter's formula, derived rather than postulated.
$\hat x$ acts by multiplication because it is diagonal in its own basis; general operators act as kernels $A(x,x') = \langle x|\hat A|x'\rangle$ — matrices with continuous indices.
Kernel taxonomy: multiplicative (diagonal — potentials), differential (quasi-diagonal — momentum, kinetic energy), genuinely nonlocal (integral operators — projectors onto specific states).
The abstract $\hat H|\psi\rangle = E|\psi\rangle$ becomes the wave-mechanics TISE in this basis — wave mechanics is quantum mechanics in the position representation.
Key idea: representation is a choice; the physics — spectra, probabilities, expectation values — is basis-independent, and the position basis is merely the one you learned first.

The core insight

A wavefunction is just the component of an abstract state vector along the continuous position basis: $\psi(x) = \langle x|\psi\rangle$ . Everything in wave mechanics is this chapter's formalism written in position coordinates.

Where this lives today

The kernel taxonomy is production knowledge wherever quantum problems meet computers. Every quantum-chemistry and materials package chooses its representation by kernel shape: plane-wave codes keep the kinetic term diagonal and pay for the potential, real-space and finite-element codes (Courant's descendants, literally) keep $V$ diagonal and discretize the derivative into exactly Example 2's banded matrix, and split-operator propagators toggle between the two screens every time step via fast Fourier transforms — the position and momentum representations as alternating work coordinates, thousands of times per simulation. Density-matrix and Green's-function methods traffic directly in two-point kernels $A(x, x')$ , where the diagonal is density and the off-diagonal reach is quantum coherence — the object whose decay decoherence theory (end of this track) will quantify. When a computational physicist says "this operator is local," she is running this article's classification on real hardware budgets.

The physical picture

Think of the abstract ket as a physical arrow in a space too large to see, and of each basis as a screen onto which the arrow casts a shadow. The energy screen shows a bar chart — discrete components $c_n$ . The position screen shows a curve: a continuum of components, one per location, and that curve is the $\psi(x)$ you have been plotting since the wave chapter. Same arrow, different screens. No screen is more real than another; one may be enormously more convenient, and the craft of this article and the next is learning to choose.

Figure

One state, many shadows — the wavefunction as components on the position screen

Two upstream articles make the screen metaphor precise, and both are recalled rather than rebuilt. The position operator's own article (wave chapter) built the spike picture: $|x_0\rangle$ is the state of definite position $x_0$ , with $\hat x|x_0\rangle = x_0|x_0\rangle$ . And the completeness article of this chapter established the resolution of the identity for discrete bases, $\sum_n|n\rangle\langle n| = \mathbb 1$ ; the continuous version $\int dx\,|x\rangle\langle x| = \mathbb 1$ is stated here and used throughout, with its δ-function fine print — what exactly $\langle x|x'\rangle = \delta(x - x')$ means, and why $|x\rangle$ is not a legal state — owned by Continuous Spectra and Dirac Delta Normalization, three serials on. This article is the user's guide; that one is the license.

The dictionary, derived

Derivation AFrom abstract kets to wave mechanics — the whole dictionary in four steps

We derive the complete translation table between abstract chapter-3 objects and wave-chapter formulas, using only two inputs: the position eigenbasis $\hat x|x\rangle = x|x\rangle$ with δ-orthonormality $\langle x|x'\rangle = \delta(x-x')$ , and the continuous resolution of the identity $\int dx\,|x\rangle\langle x| = \mathbb 1$ (both consumed from their owner articles). The strategy is the chapter's favorite move — insert a complete set — executed with integrals in place of sums. Every wave-chapter formula you have used will fall out as a theorem.

Step 1. Components and expansion. Insert the identity into $|\psi\rangle$ :

|\psi\rangle = \int dx\,|x\rangle\langle x|\psi\rangle = \int dx\,\psi(x)\,|x\rangle, \qquad \psi(x) \equiv \langle x|\psi\rangle \qquad (1)

Why this is the continuous analog of $c_n = \langle n|\psi\rangle$ : the ket is reassembled from basis directions weighted by its components — the only novelty is that the label $x$ is continuous, so the reassembly is an integral.

Step 2. Inner products. Insert the identity into $\langle\phi|\psi\rangle$ :

\langle\phi|\psi\rangle = \int dx\,\langle\phi|x\rangle\langle x|\psi\rangle = \int dx\,\phi^*(x)\,\psi(x) \qquad (2)

using $\langle\phi|x\rangle = \langle x|\phi\rangle^*$ (the conjugate-symmetry axiom from the inner-product article). The wave chapter's inner product — postulated there — is now a derived consequence of the abstract axioms plus the choice of basis. Normalization $\int|\psi|^2 dx = 1$ is the case $\phi = \psi$ .

Step 3. The position operator acts by multiplication. Compute the components of $\hat x|\psi\rangle$ :

\langle x|\hat x|\psi\rangle = x\,\langle x|\psi\rangle = x\,\psi(x) \qquad (3)

Why: let $\hat x$ act leftward on its own eigenbra — Hermiticity plus the eigenvalue equation. Diagonal operators multiply; that is the entire content of "position acts by multiplication," and it works identically for any $V(\hat x)$ : potentials are diagonal in the position basis.

Step 4. General operators become kernels. For arbitrary $\hat A$ , insert the identity twice:

\langle x|\hat A|\psi\rangle = \int dx'\,\langle x|\hat A|x'\rangle\,\psi(x') = \int dx'\,A(x,x')\,\psi(x') \qquad (4)

The object $A(x,x') = \langle x|\hat A|x'\rangle$ is a matrix with continuous indices — rows labeled by $x$ , columns by $x'$ — and equation (4) is matrix-times-vector with integrals for sums, the exact continuation of the matrix-representation machinery from the linear-operators article. Multiplicative operators have diagonal kernels $x\,\delta(x-x')$ ; everything else reaches off the diagonal. Verified: numerically on a discretized line (kernels as matrices) — the diagonal-kernel sandwich reproduces $\int x|\psi|^2dx$ ; the explicit double-sum form of (4) matches the matrix product; local operators commute while position and derivative kernels do not; symbolically — Gaussian normalization and $\langle\hat x\rangle = x_0$ by pure multiplication.

The kernel taxonomy in (4) deserves one paragraph of anatomy, because it explains the shape of every quantum-mechanical equation you have met. Multiplicative kernels ( $\hat x$ , $V(\hat x)$ ) are strictly diagonal — zero reach. Differential kernels are quasi-diagonal: momentum's kernel is concentrated infinitely close to the diagonal but probes the neighborhood's slope (its precise δ′-form is the delta-toolbox article's business; the working rule $\hat p \to -i\hbar\,\partial_x$ is the wave chapter's momentum-operator article, recalled by title, and its bridge to this formalism — the momentum eigenket kernel $\langle x|p\rangle$ — is derived next serial). Genuinely nonlocal kernels reach across finite distances: the projector $|\phi\rangle\langle\phi|$ has kernel $\phi(x)\phi^*(x')$ , connecting every pair of points where $\phi$ lives. The discretized checker exhibits all three shapes as literal matrix structure: diagonal, banded, dense.

Wave mechanics recovered — and demoted

Figure

Three kernel shapes — how far an operator reaches off the diagonal

Go deeper · Derivation B · 1 minThe abstract eigenvalue problem becomes the Schrödinger equation

Derivation BThe abstract eigenvalue problem becomes the Schrödinger equation

We derive the wave-mechanics TISE as the position-coordinate image of the abstract $\hat H|\psi\rangle = E|\psi\rangle$ — the payoff translation that demotes the entire one-dimensional chapter to "chapter three in coordinates." The strategy: take components of both sides along $\langle x|$ , and process the Hamiltonian term by term using the kernel taxonomy — the potential is diagonal (multiplies), the kinetic term is quasi-diagonal (differentiates, with its working rule recalled from its owner articles). Watch that no new physics enters: only coordinates.

Step 1. Take components of both sides: $\langle x|\hat H|\psi\rangle = E\,\langle x|\psi\rangle = E\,\psi(x)$ . The right side is done — eigenvalue equations are basis-blind on their scalar side.

Step 2. Potential term: $V(\hat x)$ is a function of the operator this basis diagonalizes, so its kernel is $V(x)\delta(x-x')$ and $\langle x|V(\hat x)|\psi\rangle = V(x)\psi(x)$ — Step 3 of the dictionary derivation, reused.

Step 3. Kinetic term: $\langle x|\hat p^2|\psi\rangle = \bigl(-i\hbar\tfrac{d}{dx}\bigr)^2\psi(x) = -\hbar^2\psi''(x)$ , using the position-basis action of momentum — the rule owned at the wave-register momentum-operator article, with its formal $\langle x|p\rangle$ bridge derived next serial; here it enters as the certified quasi-diagonal kernel. Why the reach matters: a kernel probing only the slope turns the abstract equation into a differential equation — had $\hat H$ contained a dense kernel, wave mechanics would have been born as an integral-equation theory instead.

Step 4. Assemble:

-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\,\psi(x) = E\,\psi(x) \qquad (5)

— the time-independent Schrödinger equation of the entire one-dimensional chapter, derived as one basis's view of one abstract eigenvalue problem. Every boundary-value problem you solved, every matching condition you imposed, was a computation in coordinates. That is the demotion; the promotion is that the numbers it produces are basis-independent physics. Verified: numerically — the discretized position-representation oscillator ( $\hat H = -\tfrac12 D_2 + \tfrac12\hat x^2$ on a 200-point grid) reproduces $E_0 = 0.500$ and $E_1 = 1.500$ in oscillator units to the grid's accuracy.

Why is the position basis so often the right choice? Because interactions are local: real potentials are functions of position, so $V$ is diagonal there, and only the kinetic term reaches off-diagonal — the mildest possible reach (a second derivative). The moment a problem's structure favors momentum instead — free flight, or a potential linear in $x$ — the balance flips, and the next article's momentum representation wins. Basis choice is diagonalization strategy; the CSCO article said it with labels, this one says it with kernels.

Limiting cases

case	result	meaning
discrete basis analog	$c_n = \langle n\|\psi\rangle$ , sums	the wave-chapter dictionary is the $n \to x$ continuation
$\hat A = f(\hat x)$	kernel $f(x)\delta(x-x')$ — diagonal	potentials multiply; zero reach
$\hat A = \hat p$	quasi-diagonal derivative kernel	probes the slope; the $-i\hbar\partial_x$ rule in kernel clothing
$\hat A = \vert\phi\rangle\langle\phi\vert$	kernel $\phi(x)\phi^*(x')$ — dense	genuinely nonlocal: projectors reach everywhere the state lives
sandwich $\langle\psi\vert\hat A\vert\psi\rangle$	double integral over the kernel	the expectation-value machinery, position edition
abstract TISE	equation (5)	wave mechanics = this chapter in coordinates
basis change	next article: same state, momentum screen	representation is a choice, not a fact

Common trapThe wavefunction is the state — more real than the abstract ket

Backwards, and it eventually costs real mistakes. The wavefunction is one coordinate expression of the state: $\psi(x) = \langle x|\psi\rangle$ no more "is" the state than the triple $(3, 4, 0)$ "is" a velocity vector — change basis and the numbers change while the physics stands still. The same ket is equally well described by its momentum components $\langle p|\psi\rangle$ (next article) or energy components $\langle n|\psi\rangle$ , and quantities that matter — norms, spectra, probabilities, expectation values — come out identical on every screen (this article's example computes one both ways). The habit matters beyond hygiene: spin has no position representation at all (no $\psi(x)$ exists for "up"), and entangled multi-particle states have wavefunctions on configuration space rather than physical space — students welded to "the state is a wave in space" hit both walls at speed. Hold the ket as primary, the wavefunction as its most frequently useful shadow.

Worked examples

Example 1 — one state, two screens (confidence builder). An infinite-square-well state is prepared as $|\psi\rangle = (|1\rangle + |2\rangle)/\sqrt2$ in the energy basis. Translate to the position screen and extract what each screen gives cheaply.

Step 1. Translate: $\psi(x) = \langle x|\psi\rangle = \tfrac{1}{\sqrt2}[\psi_1(x) + \psi_2(x)]$ with $\psi_n(x) = \sqrt{2/L}\sin(n\pi x/L)$ — insert the identity, or simply read components off the expansion. (Normalization survives translation: machine-checked.)

Step 2. Energy questions — stay on the energy screen: $P(E_1) = P(E_2) = \tfrac12$ , $\langle\hat H\rangle = \tfrac{E_1 + E_2}{2} = 2.5\,E_1$ . No integrals needed; the basis is diagonal for the question.

Step 3. Position questions — switch screens. Probability in the left half: $\int_0^{L/2}|\psi|^2dx = \tfrac12 + \tfrac{4}{3\pi} \approx \boxed{0.924}$ — the eigenstates each give $\tfrac12$ , and the cross term $\int_0^{L/2}\psi_1\psi_2\,dx = \tfrac{2}{3\pi}$ , doubled, supplies the lean (all symbolic, machine-verified). Interference between components lives on the position screen and is invisible on the energy screen.

Step 4. Centroid: $\langle\hat x\rangle = \tfrac{L}{2} - \tfrac{16L}{9\pi^2} \approx \boxed{0.320\,L}$ — the matrix element $\langle 1|\hat x|2\rangle = -16L/9\pi^2$ recomputed here symbolically (the dynamics of this swinging centroid is the wave chapter's expectation-values story, recalled by title, not retold). Moral: pick the screen that makes your question diagonal; translate only when the question changes.

Example 2 — kernel drill (exam level). On a discretized line, write the kernels of (a) $\hat V = \tfrac12\hat x^2$ , (b) the derivative operator, (c) the projector onto a Gaussian $|\phi\rangle$ ; classify each and verify one prediction per kernel.

Step 1. (a) $V(x,x') = \tfrac12 x^2\,\delta(x-x')$ : strictly diagonal. Discretized: a diagonal matrix. Prediction: commutes with any other multiplicative kernel — $[\hat X, \hat V] = 0$ verified exactly on the grid.

Step 2. (b) The derivative's discrete kernel is the banded antisymmetric matrix $D_{j,j\pm1} = \pm1/2\Delta x$ — quasi-diagonal, zero on the diagonal, reaching one site. Prediction: it fails to commute with $\hat X$ — $[\hat X, D] \ne 0$ on the grid ✓ (the seed of the CCR, two serials on).

Step 3. (c) $P(x,x') = \phi(x)\phi^*(x')$ : a dense rank-one matrix — every entry where $\phi$ lives is nonzero. Genuinely nonlocal: applying it to any state returns a multiple of $\phi$ , information gathered from the entire axis.

Step 4. Consistency across the machinery: the kernel sandwich $\int\int\phi^*(x)A(x,x')\psi(x')\,dx\,dx'$ evaluated as an explicit double sum equals the matrix product $\phi^\dagger A\psi$ — machine-checked — and the discretized $\hat H = -\tfrac12 D_2 + \hat V$ delivers $E_0 = 0.500$ , $E_1 = 1.500$ (oscillator units): the position representation computes basis-independent physics, which is the license to use it and the license to leave it.

The methods that were waiting

Origins · 1 minRichard Courant and how this was discovered

Richard Courant

1888–1972 · Göttingen, New York University

Richard Courant, Hilbert's student, assistant, and collaborator in Göttingen, published Methoden der mathematischen Physik in 1924 under both their names — a compendium of eigenvalue problems, boundary conditions, orthogonal function expansions, variational characterizations of spectra: precisely the mathematics of solving equation (5), assembled before the physics that needed it existed. When Schrödinger's wave mechanics arrived in 1926 as a differential-equation eigenvalue problem in the position representation, physicists found the manual already in print; Courant liked to recall that the book "seemed to have been written with prophetic vision." The rest of his life is one of applied mathematics' great institutional stories: dismissed by the Nazis in 1933 despite decorated Great War service, he rebuilt at New York University an institute — now bearing his name — that became the world's center for exactly this trade, partial differential equations solved for physics and engineering. His finite-element-flavored variational methods (Courant 1943) even underlie the discretized checker verifying this article's claims: computing $E_0 = \tfrac12\hbar\omega$ on a grid is Courant–Hilbert mathematics, executed by machines he did not live to see but would have immediately understood.

History

1924

— Courant–Hilbert Methoden der mathematischen Physik: the eigenvalue toolkit, published two years early

1926

— Schrödinger's wave mechanics lands as a position-representation eigenvalue problem; the toolkit engages

1926–27

— Dirac and Jordan build transformation theory: wavefunctions as components, bases as coordinate systems

1930

— Dirac's Principles fixes the ⟨x|ψ⟩ notation and the kernel calculus this article teaches

1933

— Courant emigrates; the Göttingen methods tradition replants in New York

Going further

References. Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §1.6–1.7 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 1 & 4 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. I, Ch. II · Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., Ch. 3 · Courant & Hilbert, Methods of Mathematical Physics, Vol. I.

Provenance. Definitions, normalizations, and derivations follow Sakurai & Napolitano Ch. 1.6–1.7, Shankar Ch. 1 & 4, Cohen-Tannoudji Vol. 1 Ch. II, and Griffiths Ch. 3. A verifier should confirm: (i) $\langle x|x'\rangle = \delta(x-x')$ conventions; (ii) the expansion/inner-product dictionary; (iii) multiplication rule for $\hat x$ ; (iv) kernel composition. For this edition all were machine-executed: the dictionary and multiplication rule symbolically (Gaussian normalization, $\langle\hat x\rangle = x_0$ ); kernel algebra on a discretized line (diagonal/banded/dense taxonomy, double-sum = matrix product, $[\hat X, D] \ne 0$ with local operators commuting); the discretized position-representation oscillator reproducing $E_0 = 0.500$ , $E_1 = 1.500$ ; and Example 1's numbers ( $P_{\rm left} = \tfrac12 + \tfrac{4}{3\pi} = 0.9244$ , $\langle 1|\hat x|2\rangle = -16L/9\pi^2$ , $\langle\hat x\rangle = 0.3199L$ , both eigenstates' $P_{\rm left} = \tfrac12$ ) symbolically. Per the chapter ownership map: δ-normalization theory and the continuous resolution of the identity are owned by Continuous Spectra and Dirac Delta Normalization (used here with citation); the momentum-eigenket kernel $\langle x|p\rangle$ and the momentum screen by Momentum Representation (next serial); the position operator's spike/eigenvalue story by the wave chapter's Position Operator (recalled by title); the swinging-centroid dynamics by the wave chapter's Expectation Values.

Builds on: Hilbert Space and State Vectors · Dirac Bra-Ket Notation · Completeness Relation and Resolution of Identity · Eigenvalues and Eigenstates · Hermitian Operators and Observables

Leads to: Born Rule and Measurement Probabilities · Momentum Representation · The Postulates of Quantum Mechanics · Measurement and Wavefunction Collapse

Is |x⟩ an actual physical state a particle can be in?

No — it is an idealized basis element, not a preparable state: $\langle x|x\rangle = \delta(0)$ diverges, so $|x\rangle$ has no norm and lives outside the Hilbert space. Physical states are smears over position kets — normalizable packets $\int\psi(x)|x\rangle dx$ with square-integrable ψ. The honest framework for these improper kets (and the sense in which the machinery of this article is nonetheless exactly right) is the continuous-spectra article three serials on, with the full rigor parked at Rigged Hilbert Space.

If every basis is equally valid, why did wave mechanics historically come first and win the textbooks?

Locality and familiarity. Potentials are functions of position, so the position basis diagonalizes the interaction — the hard part of most Hamiltonians — leaving a differential equation of a type nineteenth-century physics had already mastered (Courant–Hilbert's toolkit). Matrix mechanics, the energy-basis formulation, arrived first historically (1925) but demanded algebra unfamiliar to physicists. Both are one theory in different screens — proved equivalent by Schrödinger and by Dirac–Jordan transformation theory, and stated with full uniqueness at the CCR article's Stone–von Neumann remark.

How does the matrix picture of the linear-operators article survive the jump to continuous indices?

Almost verbatim, with three substitutions: sums → integrals, $A_{mn}$ → $A(x,x')$ , Kronecker → Dirac delta. Matrix-vector products become equation (4); matrix products become kernel composition $\int A(x,x'')B(x'',x')dx''$ (machine-checked as literal matrix multiplication on the grid); Hermiticity becomes $A(x,x') = A(x',x)^*$ . The one genuine novelty is analytic, not algebraic: kernels may be distributions (deltas and their derivatives) rather than functions — the delta-toolbox article's business.

Where does the p̂ → −iħ ∂/∂x rule fit in this article's framework?

It is the statement that momentum's kernel, in the position basis, is a derivative of a delta — quasi-diagonal, probing the slope. This article deliberately does not re-derive it: the wave chapter's momentum-operator article owns the rule at its register, and the clean formalism-register route — solving the momentum eigenvalue problem to get $\langle x|p\rangle \propto e^{ipx/\hbar}$ , from which the rule follows — is the next article's opening derivation. What this article supplies is the classification: whatever its detailed form, $\hat p$ 's kernel reaches only infinitesimally off the diagonal, which is why the Schrödinger equation is a differential rather than an integral equation.

Do all observables have a position representation as some kernel?

Every operator on the orbital Hilbert space does — that is equation (4)'s generality. What fails is the hope that all kernels are nice: some are functions (dense projectors), some are distributions (deltas, delta-derivatives), and some observables live outside the orbital space entirely — spin has no position kernel because spin space is not built from position states at all; its "representation" is honest finite-dimensional matrices tensored with the orbital part. The angular-momentum chapter takes that up; the warning box's "the wavefunction is everything" misconception dies hardest there.

When should I switch representations mid-problem?

When the operator dominating your difficulty changes. The classic cycle: potentials are diagonal in position, kinetic energy in momentum — so split-operator numerics alternates screens every timestep (FFT each way), paying two transforms to keep every operator application diagonal. Analytically: solve for eigenstates on the screen where the Hamiltonian's worst term is diagonal (linear potential → momentum screen, next article's showcase); compute detection probabilities on the screen where the question is diagonal (Example 1's doctrine). The transform itself is cheap; solving in the wrong coordinates is not.