Momentum Representation

The previous article demoted the wavefunction to a shadow: $\psi(x) = \langle x|\psi\rangle$ , one screen among many. This article builds the second screen from scratch — and then shows you why you would ever bother. The momentum basis $\{|p\rangle\}$ is constructed by solving one eigenvalue equation: demanding $\hat p|p\rangle = p|p\rangle$ in position coordinates yields, in two lines, the kernel $\langle x|p\rangle = e^{ipx/\hbar}/\sqrt{2\pi\hbar}$ — a plane wave whose normalization is fixed, not chosen, by delta-orthonormality. Components along this basis, $\phi(p) = \langle p|\psi\rangle$ , give the momentum-space wavefunction; the Born rule reads $|\phi(p)|^2$ as momentum probability density; and the kernel makes the position–momentum bridge literally a Fourier transform — the wave-mechanics fact your earlier chapter's Momentum-Space Wavefunction article taught, now derived rather than decreed. The dictionary flips symmetrically: on this screen momentum multiplies and position differentiates — $\hat x \to +i\hbar\,d/dp$ , with a plus sign that this article audits numerically, because it is the single most reliably fumbled sign in the chapter. And then the payoff, which is craft rather than formalism: some problems that are hard in position coordinates are trivial here. The attractive delta well, a matching-conditions exercise in the wells chapter, becomes three lines of algebra ending in $E = -m\alpha^2/2\hbar^2$ — the same answer, no derivative jump conditions anywhere. The linear potential (a uniform force), whose position-space treatment needs the Airy machinery of a later chapter, collapses to a first-order ODE with a closed-form phase solution. Representation choice is leverage, and this article is where you learn to pull.

Key takeaways

Momentum eigenkets in position coordinates: $\langle x|p\rangle = \dfrac{e^{ipx/\hbar}}{\sqrt{2\pi\hbar}}$ — derived by solving $\hat p|p\rangle = p|p\rangle$ ; the prefactor is forced by $\langle p|p'\rangle = \delta(p-p')$ .
$\phi(p) = \langle p|\psi\rangle$ : components along the momentum basis; $|\phi(p)|^2$ = momentum probability density; the $x\leftrightarrow p$ bridge is the Fourier transform, now a theorem.
Dictionary mirror: $\hat p \to p\,\cdot$ (diagonal) and $\hat x \to +i\hbar\,d/dp$ — plus sign, machine-audited (the minus sign belongs to $\hat p$ in the position representation).
Craft: the delta well solves algebraically in $p$ -space — $\phi(p)$ is a Lorentzian and self-consistency forces $E = -m\alpha^2/2\hbar^2$ , matching the wells chapter.
Craft: the linear potential $V = -Fx$ gives a first-order ODE in $p$ -space with exact solution $\phi_E(p) \propto e^{-i(p^3/6m - Ep)/\hbar F}$ — position space needs Airy functions.
Choose the screen where your Hamiltonian's hard term is diagonal: kinetic-dominated and momentum-coupled problems live here.
Key idea: $\phi(p)$ is the same state read in the momentum basis, reachable from $\psi(x)$ by the Fourier kernel $\langle x|p\rangle$ — and some locks only open with this key.

The core insight

$\phi(p) = \langle p|\psi\rangle$ is the Fourier transform of $\psi(x) = \langle x|\psi\rangle$ , because the change-of-basis kernel is $\langle x|p\rangle = e^{ipx/\hbar}/\sqrt{2\pi\hbar}$ ; in this basis momentum multiplies, position differentiates, and momentum-diagonal problems become easy.

Where this lives today

Momentum space is the natural representation for many modern experimental techniques. Cold-atom laboratories measure $|\phi(p)|^2$ as their primary observable — switch off the trap, wait, photograph: the time-of-flight image is the momentum distribution, magnified (the matter-wave analog of far-field diffraction), and phenomena from Bose–Einstein condensation (a spike at $p = 0$ ) to Fermi seas are seen on this screen. Angle-resolved photoemission (ARPES) maps electron removal as a function of momentum, making momentum space the natural description for band-structure measurements; crystallography has lived in the reciprocal lattice — momentum space with periodic wallpaper — for a century. On the computational side, plane-wave electronic-structure codes store wavefunctions as $\phi(\mathbf p)$ arrays and hop screens by FFT whenever a position-diagonal operator must act (Gauss's algorithm, executing Gauss's transform), and the delta-well's algebraic trick scales up into the separable-potential and $T$ -matrix methods of nuclear and scattering physics. The second screen is not an alternative formalism; it is half of the working world.

The physical picture

Return to the screens of the previous article. The position screen showed $\psi(x)$ — the state's overlap with ever-sharper location spikes. The momentum screen shows $\phi(p)$ — the state's overlap with perfectly rhythmic plane waves, one per momentum. A state narrow on one screen is broad on the other (the uncertainty article's floor made that quantitative); a state drifting in position picks up a phase twist in momentum; and questions about "how fast, which way" — scattering directions, currents, time-of-flight images — are diagonal here, answered by reading $|\phi(p)|^2$ off the screen with no further work.

Figure

The momentum screen — overlap with plane waves, one rhythm per p

Before diving in, keep the division of labor between the two representations in mind. The wave chapter's Momentum-Space Wavefunction article built $\Phi(p)$ as a Fourier transform, worked the Gaussian pair to the end, and established the operator correspondence at the wavefunction level — all of those results are assumed here and will only be recalled when needed. What that article could not yet say is why the transform kernel is $e^{ipx/\hbar}/\sqrt{2\pi\hbar}$ and not something else: the answer is an eigenvalue calculation in the abstract formalism, and it is this article's opening derivation. From there the article spends its length where the ket language pays rent — the solving craft.

The kernel, derived — and the mirror dictionary

Derivation AThe momentum eigenket in position coordinates — normalization forced, not chosen

We derive $\langle x|p\rangle = e^{ipx/\hbar}/\sqrt{2\pi\hbar}$ , the change-of-basis kernel between the two screens. The strategy: write the abstract eigenvalue equation $\hat p|p\rangle = p|p\rangle$ in position components using the position-representation rule for $\hat p$ (recalled from its wave-chapter owner), solve the resulting first-order ODE, and then — the step that fixes the prefactor — impose delta-orthonormality with the Fourier identity consumed from the continuous-spectra article, two serials on. Both the functional form and the normalization are uniquely determined by the eigenvalue equation and the orthonormality condition.

Step 1. Components of the eigenvalue equation. Let $u_p(x) \equiv \langle x|p\rangle$ . Taking $\langle x|$ -components of $\hat p|p\rangle = p|p\rangle$ and using $\hat p \to -i\hbar\,d/dx$ on the position screen:

-i\hbar\,\frac{du_p}{dx} = p\,u_p(x) \qquad (1)

Step 2. Solve: $u_p(x) = C\,e^{ipx/\hbar}$ — a pure plane wave, no other shape possible. Why the exponent's sign matters: with the physics convention $\hat p = -i\hbar\partial_x$ , positive $p$ means phase advancing toward positive $x$ — a right-mover; flip the exponent and every current in the track reverses.

Step 3. Fix $C$ by orthonormalization. Demand $\langle p|p'\rangle = \delta(p - p')$ and insert position completeness:

\langle p|p'\rangle = \int dx\, u_p^*(x)\,u_{p'}(x) = |C|^2 \int dx\, e^{i(p'-p)x/\hbar} = |C|^2\,2\pi\hbar\,\delta(p - p') \qquad (2)

using the Fourier–delta identity $\int e^{ikx}\,dx=2\pi\delta(k)$ together with the scaling property $\delta[(p-p')/\hbar]=\hbar\,\delta(p-p')$ , both established in the earlier articles Continuous Spectra and Dirac Delta Normalization. Hence $|C| = 1/\sqrt{2\pi\hbar}$ , phase conventionally zero:

\langle x|p\rangle = \frac{1}{\sqrt{2\pi\hbar}}\,e^{ipx/\hbar} \qquad (3)

Step 4. Construct the transformation between the two representations. The components of $|\psi\rangle$ in the position and momentum bases are related by inserting the completeness relation: $\phi(p) = \langle p|\psi\rangle = \int dx\,\langle p|x\rangle\langle x|\psi\rangle = \tfrac{1}{\sqrt{2\pi\hbar}}\int dx\,e^{-ipx/\hbar}\,\psi(x)$ — the Fourier transform, with the inverse following identically. The Fourier-transform pair therefore follows directly from the momentum eigenvalue equation together with completeness. In a finite box, distinct quantized momenta are orthogonal, while the same-momentum normalization approaches $L/(2\pi\hbar)$ , the finite-volume counterpart of $\delta(0)$ .

The dictionary on the new screen now reads off by symmetry. Momentum is diagonal in its own basis — $\langle p|\hat p|\psi\rangle = p\,\phi(p)$ , multiplication, by exactly the argument that made $\hat x$ multiply on the position screen. Position differentiates:

\langle p|\hat x|\psi\rangle = +i\hbar\,\frac{d\phi}{dp} \qquad (4)

where: the plus sign is the chapter's most-fumbled detail, so it earns an audit trail: the mirror derivation (differentiate the kernel with respect to $p$ instead of $x$ ) is owned by the wave-register twin and recalled here; what this article adds is a numerical sign audit — a Gaussian packet displaced to $x_0 = 3.5$ has $\phi(p) = e^{-ipx_0/\hbar}\times(\text{real Gaussian})$ , and the sandwich $\int\phi^*(+i\hbar\,\partial_p)\phi\,dp$ returns $+3.5$ while the wrong sign returns $-3.5$ (machine-checked). Mnemonic that survives exams: each screen's own operator multiplies; the conjugate operator differentiates, and the sign flips with the screen — $(-i\hbar\partial_x)$ there, $(+i\hbar\partial_p)$ here, the pair conspiring to keep $[\hat x,\hat p] = i\hbar$ representation-independent (the CCR article, next serial, makes that the headline).

Figure

Two locks, one key — what the momentum screen does to hard potentials

The solving craft: two locks this key opens

Lock one: the delta well, algebraically. The attractive $V(x) = -\alpha\,\delta(x)$ cost the wells chapter a derivative jump condition. On the momentum screen the same eigenvalue problem is algebra, because the momentum-space matrix elements of the delta potential are independent of momentum: $\langle p|V|\psi\rangle = -\tfrac{\alpha}{2\pi\hbar}\int\phi(p')\,dp'$ — every momentum couples to every other with equal strength.

Go deeper · Derivation B · 1 minDelta-well bound state in momentum space — three lines to the energy

Derivation BDelta-well bound state in momentum space — three lines to the energy

We derive the bound-state energy $E = -m\alpha^2/(2\hbar^2)$ entirely in the momentum representation, with no wavefunction matching anywhere — the article's exhibit that representation choice converts calculus into algebra. The strategy: the TISE in $p$ -components becomes an algebraic equation for $\phi(p)$ in terms of one unknown constant (an integral of $\phi$ itself); solving and demanding self-consistency fixes the energy. This is also the template for separable-potential methods used across scattering theory.

Step 1. Write the time-independent Schrödinger equation in momentum space. With $E = -B$ ( $B > 0$ for a bound state), $\langle p|\hat H|\psi\rangle = E\phi(p)$ reads

\frac{p^2}{2m}\,\phi(p) - \frac{\alpha}{2\pi\hbar}\,\underbrace{\int_{-\infty}^{\infty}\phi(p')\,dp'}_{\equiv\,K,\ \text{a number}} = -B\,\phi(p) \qquad (5)

Why the potential term collapsed to a constant: $\langle p|\delta(\hat x)|p'\rangle = u_p^*(0)u_{p'}(0) = 1/2\pi\hbar$ — flat. The delta couples all momenta equally; that democratic coupling is the whole problem.

Step 2. Solve algebraically: $\phi(p) = \dfrac{\alpha K/2\pi\hbar}{p^2/2m + B}$ — a Lorentzian in momentum, no differential equation solved.

Step 3. Self-consistency. The constant $K$ is the integral of the very $\phi$ it determines; integrating Step 2's expression and cancelling $K$ (legal since $K \ne 0$ for a nontrivial state):

1 = \frac{\alpha}{2\pi\hbar}\int_{-\infty}^{\infty}\frac{dp}{p^2/2m + B} = \frac{\alpha}{2\pi\hbar}\,\pi\sqrt{\frac{2m}{B}} \quad\Longrightarrow\quad B = \frac{m\alpha^2}{2\hbar^2} \qquad (6)

One equation, one root: exactly one bound state at $E = -m\alpha^2/2\hbar^2$ — the count and the value of the wells chapter, recovered without a single matching condition.

Step 4. Transform back to position space. Fourier-transforming the wells chapter's $\psi(x) = \sqrt\kappa\,e^{-\kappa|x|}$ gives precisely the Step-2 Lorentzian (machine-checked against quadrature), normalized by Parseval — same state, two screens, and the momentum screen never had to hear about cusps. The self-consistency integral and root are exact ( $B = m\alpha^2/2\hbar^2$ ); the Lorentzian is the Fourier transform of $\sqrt{\kappa}\,e^{-\kappa|x|}$ at fifteen $(p,\kappa,\hbar)$ spot values to $10^{-6}$ , and its normalization holds by Parseval.

Lock two: the uniform force. For $V = -Fx$ (a constant force $F$ — charged particle in a uniform field, the bouncing neutron's ramp), the position representation leads to a second-order differential equation whose solutions are Airy functions, a topic treated later. In the momentum representation, the same problem becomes considerably simpler. Using the momentum-space operator correspondence $\hat{x} = +i\hbar\,\partial/\partial p$ , $V = -F\hat x \to -i\hbar F\,d/dp$ , the time-independent Schrödinger equation becomes a first-order differential equation:

\frac{p^2}{2m}\,\phi - i\hbar F\,\frac{d\phi}{dp} = E\,\phi \quad\Longrightarrow\quad \phi_E(p) = C\,\exp\!\Bigl[-\,\frac{i}{\hbar F}\Bigl(\frac{p^3}{6m} - Ep\Bigr)\Bigr] \qquad (7)

where: the solution is a pure phase — $|\phi_E(p)| = 1$ for every $p$ (machine-checked), the momentum-space face of the fact that a falling particle's energy eigenstates contain all momenta with equal weight, organized by a cubic phase. Every first-order ODE is solvable by one integration; the entire Airy transcendence of the position picture is this phase's Fourier shadow (indeed the Airy function's standard integral representation is the transform of (7) — the approximation chapter will meet it from the other side). Solvable-by-quadrature versus special-functions: that is what a well-chosen screen buys.

Limiting cases

case	result	meaning
free particle	$\hat H$ diagonal: $\phi_E$ supported at $p = \pm\sqrt{2mE}$	the momentum screen's home game
$V =$ const	shifts $E$ ; $\phi(p)$ untouched	flat potentials are invisible in $p$ -space
$V = -\alpha\,\delta(x)$	algebraic solve; Lorentzian $\phi$ ; one bound state	equation (6)
$V = -Fx$	first-order ODE; pure-phase $\phi_E$	Airy physics, one integration
$V = \tfrac12 m\omega^2x^2$	$p$ -equation has the same form as the $x$ -equation	the oscillator's famous $x\leftrightarrow p$ symmetry — same spectrum route either way
generic $V(x)$	integral operator in $p$ -space	locality lost: the position screen wins — choose accordingly
narrow $\psi(x)$	broad $\phi(p)$ , and conversely	the reciprocity of the uncertainty article, now a kernel fact

Common trapphi of p is the Fourier transform of psi of x — so it is just math, not new physics

Half right, and the half-wrong part costs marks and understanding. Right: $\phi(p)$ contains identical information to $\psi(x)$ — one ket, two coordinate systems, invertibly connected. Wrong: "just math" — because $|\phi(p)|^2$ is directly measured physics: time-of-flight imaging in cold-atom laboratories literally photographs it (switch off the trap, let the cloud fly, positions map to initial momenta), and scattering experiments read momentum distributions before position ever enters. Equally wrong is the converse complacency about operators: on this screen it is position that becomes the derivative — with a plus sign, $+i\hbar\,\partial_p$ — and dropping or flipping that sign silently negates every $\langle\hat x\rangle$ you compute (this article's verifier exhibits the $\pm3.5$ failure explicitly). The screen is a full citizen: its density is measurable, its dictionary is mirrored, and its choice is a strategic act, not a notational one.

Worked examples

Example 1 — read the momentum screen (confidence builder). A Gaussian packet of width $\sigma = 2.0$ (natural units, $\hbar = 1$ ) drifts at $\langle x\rangle = x_0 = 3.5$ . Its momentum components are $\phi(p) = (2\sigma^2/\pi)^{1/4}e^{-\sigma^2p^2}e^{-ipx_0}$ . Extract the physics by inspection.

Step 1. Momentum distribution: $|\phi(p)|^2 = \sqrt{2\sigma^2/\pi}\,e^{-2\sigma^2p^2}$ — centered at zero (no drift in momentum), width $\sigma_p = 1/2\sigma = 0.25$ . The displacement $x_0$ is invisible in the magnitude. Normalization checks to 1 (machine-verified).

Step 2. Where did $x_0$ go? Into the phase twist $e^{-ipx_0}$ : position information lives in the phase of the momentum screen, exactly as momentum information lives in the phase of the position screen. Magnitude-only detectors on one screen are blind to the conjugate variable's mean.

Step 3. Recover it with the dictionary: $\langle\hat x\rangle = \int\phi^*\,(+i\hbar\,\partial_p)\,\phi\,dp = +3.5$ — the phase's derivative delivers the displacement, plus sign essential (the wrong sign's $-3.5$ is the verifier's cautionary exhibit).

Step 4. Uncertainty audit: $\sigma_x\sigma_p = 2.0 \times 0.25 = \hbar/2$ — the Gaussian rides the floor of the uncertainty article, visible from either screen.

Example 2 — choose your screen like a professional (exam level). For each Hamiltonian, pick the representation and name the payoff: (a) $\hat p^2/2m + V_0\cos(\hat x/a)$ ; (b) $\hat p^2/2m - F\hat x$ ; (c) $\hat p^2/2m - \alpha\,\delta(\hat x)$ ; (d) $\hat p^2/2m + \tfrac12 m\omega^2\hat x^2$ .

Step 1. (a) Position screen. The potential is an honest multiplicative function with structure ( $\cos$ ), while in $p$ -space it becomes a hopping operator connecting $\phi(p)$ to $\phi(p \pm \hbar/a)$ — a difference equation. (That hopping picture is also gold — it is the seed of band theory — but the differential equation lives in $x$ .)

Step 2. (b) Momentum screen: first-order ODE, closed-form phase (7). Position screen: Airy functions. No contest — one integration versus special functions.

Step 3. (c) Momentum screen: the delta's flat kernel makes the problem algebraic (equation 6). Position screen: solvable but needs the jump condition. Preference, not necessity — and knowing both routes is the real exam armor.

Step 4. (d) A draw, gloriously: the oscillator Hamiltonian is symmetric under $\hat x \leftrightarrow \hat p$ (up to scale), so both screens serve the same differential equation and the same Hermite spectrum. That symmetry is no accident — it is why the ladder-operator method (which never chooses a screen at all) is the oscillator's most natural language, as the algebraic-oscillator article showed. Moral: the screen is a tool; inventory your Hamiltonian's terms and put the ugliest one on the diagonal.

The calculator of Ceres and the fixed point of the transform

Origins · 1 minCarl Friedrich Gauss and how this was discovered

Carl Friedrich Gauss

1777–1855 · Göttingen

Gauss appears twice in this article: once explicitly, through Gaussian wave packets, and once implicitly, through the Fourier transform. Openly: the Gaussian is the momentum screen's favorite state — the Fourier transform's fixed point (a Gaussian transforms to a Gaussian — the unique functional form, up to scaling and phase, with that property), the unique saturator of the uncertainty floor, and Example 1's workhorse. In disguise: the transform machinery itself. In 1805, interpolating the orbits of the newly found asteroids Pallas and Juno from scattered observations, Gauss worked out — in an unpublished notebook, in Latin — an algorithmic factorization of trigonometric interpolation that anticipated ideas underlying the modern Fast Fourier Transform (FFT), complete with the divide-and-conquer structure, 160 years before Cooley and Tukey's 1965 rediscovery made it the most-executed algorithm in numerical science. (Historians of computing rank the notebook among the great near-misses; it predates even Fourier's own 1807 memoir on heat.) Every split-operator quantum simulation hopping between this chapter's two screens executes Gauss's asteroid arithmetic billions of times per run. His EM-track appearances concern flux and curvature; here he is the patron of harmonic analysis — the mathematics of reading one state on two rulers.

History

1805

— Gauss's unpublished notebook: trigonometric interpolation by what is now called the FFT, for asteroid orbits

1807

— Fourier's heat memoir proposes expanding functions in harmonics; the analytic transform follows

1926

— Born's probability reading extends to momentum space:

|\phi(p)|^2

as momentum density (Pauli's footnote)

1927

— transformation theory (Dirac, Jordan): position and momentum representations as equal citizens, connected by the kernel

1965

— Cooley and Tukey publish the FFT; Gauss's priority surfaces during the ensuing archaeology

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., §3.4 · Heideman, Johnson & Burrus, IEEE ASSP Magazine 1, 14 (1984) — Gauss and the history of the FFT. The following results have been checked numerically and symbolically: the kernel $\langle x|p\rangle=e^{ipx/\hbar}/\sqrt{2\pi\hbar}$ from the eigenvalue ODE plus delta-orthonormalization; box-regularized orthonormality of the prefactored plane waves (distinct- $p$ overlap $< 10^{-10}$ , $\delta(0) \to L/2\pi\hbar$ ); the delta-well self-consistency integral $\pi\sqrt{2m/B}$ (exact), root $B = m\alpha^2/2\hbar^2$ (exact), and the Lorentzian as the Fourier transform of $\sqrt{\kappa}\,e^{-\kappa|x|}$ (spot-checked to $10^{-6}$ , Parseval-normalized); equation (7) as the exact solution of the first-order linear-potential TISE with $|\phi_E| = 1$ ; and the $\hat x = +i\hbar\,d/dp$ sign audit on the displaced Gaussian ( $+3.5$ with the correct sign, $-3.5$ with the wrong sign).

Builds on: Hilbert Space and State Vectors · Dirac Bra-Ket Notation · Completeness Relation and Resolution of Identity · Complete Orthonormal Bases · Eigenvalues and Eigenstates

Leads to: Born Rule and Measurement Probabilities · Hermitian Operators and Observables · Linear Operators on Hilbert Space

Is φ(p) as "real" as ψ(x)? Which one does nature use?

Neither and both — they are coordinate expressions of one ket, and nature uses the ket. Operationally they are on equal footing: $|\psi(x)|^2$ is what a position detector array records, $|\phi(p)|^2$ is what a time-of-flight image or an ARPES detector records. The prejudice for $\psi(x)$ is historical (wave mechanics arrived first) and practical (interactions are position-local), not fundamental. The previous article's warning — the state is the ket, the function is a shadow — applies symmetrically here.

Why is the transform kernel a plane wave and not something more complicated?

Because it is forced: the kernel is the position-components of the momentum eigenket, and the eigenvalue equation $-i\hbar\,u_p' = p\,u_p$ admits only exponentials. Any other kernel would mean a different momentum operator. This is worth internalizing as a pattern: change-of-basis kernels between two observables' bases are always "eigenfunctions of one, written in the other" — the plane wave is to $(\hat x, \hat p)$ what the Hermite functions are to (position, oscillator-energy).

The plane waves aren't normalizable — are all these integrals even legal?

Legal with the delta-normalization license: $|p\rangle$ is an improper basis element, exactly like $|x\rangle$ , with $\langle p|p'\rangle = \delta(p-p')$ replacing unit norm and $\int dp\,|p\rangle\langle p| = \mathbb 1$ replacing the discrete sum. Physical states are packets — square-integrable smears $\int\phi(p)|p\rangle dp$ — and every measurable statement involves only those. The systematic treatment (why this works, what $\delta(0)$ means, box-regularization) is the next-but-one article, Continuous Spectra and Dirac Delta Normalization; this article consumed its two key identities with citation.

How do I remember which sign goes with which representation?

Anchor one fact and mirror it: on the position screen, $\hat p = -i\hbar\,\partial_x$ (the wave chapter's rule — right-movers $e^{+ipx/\hbar}$ must carry positive momentum). The momentum screen's $\hat x = +i\hbar\,\partial_p$ then follows from either the kernel's symmetry (the inverse transform conjugates the exponent) or, fastest, from consistency of the CCR: $[\hat x,\hat p]$ must equal $+i\hbar$ on both screens, and only the opposite-sign pairing delivers that. When in doubt, run the drifting-Gaussian audit — thirty seconds of numerics settles any sign dispute permanently.

When is the momentum representation actually worse?

Whenever the potential has local structure: a generic $V(x)$ becomes an integral (convolution) operator in $p$ -space — every momentum coupled to every other with kernel $\tilde V(p - p')$ — trading one differential equation for an integral equation. Hard walls are worst of all: the box's sharp edges make $\tilde V$ decay slowly and any truncated $p$ -space treatment converge painfully. The professional inventory of Example 2 applies: put the structured term on the diagonal. Kinetic-plus-simple-potential → momentum helps only if the potential is delta, linear, or constant; otherwise the position screen (or a clever mixed strategy like split-operator) wins.

Does the momentum representation exist for a particle in a box or on a ring?

Modified, and instructively. On a ring (periodic boundary conditions), momentum is quantized: the basis is discrete plane waves and $\phi(p)$ becomes Fourier coefficients — countable components, honest Hilbert-space vectors, no deltas needed. In a hard-wall box, the situation is subtler: the naive momentum operator is not self-adjoint on the interval (the half-line boundary-term story from the Hermiticity article), so there is no legitimate momentum observable, though one may still expand in plane waves as a computational device. The moral: "momentum representation" presupposes a good momentum operator, and boundaries are exactly where that presupposition gets audited.

What happens to the momentum representation in three dimensions?

Everything vectorizes smoothly: $\langle\mathbf r|\mathbf p\rangle = e^{i\mathbf p\cdot\mathbf r/\hbar}/(2\pi\hbar)^{3/2}$ , the transform is three-dimensional, and each Cartesian pair obeys its own dictionary and CCR. New and useful: angular structure — decomposing plane waves into spherical harmonics (the partial-wave expansion) is the workhorse of scattering theory, where the momentum representation is the natural language of beams and detectors. That machinery arrives with the scattering chapter; the kernel above is its seed.