λ = h/p and momentum eigenfunctions

Attach a phase φ = k⋅x − ωt to a free particle and ask what relativity permits. The phase at an event counts how many crests have swept past it, and a count of discrete events is something all observers agree on: φ must be a Lorentz scalar. Since φ is the contraction of x^μ = (ct, x) with the coefficients (ω/c, k) — up to the overall sign the metric signature fixes — those coefficients must themselves form a four-vector. A free particle carries exactly one four-vector, p^μ = (E/c, p), so k^μ can only be proportional to it, and the Planck–Einstein relation E = ħω — whose constant h the stopping-potential slope of topic 2.3 measured — fixes the constant: k^μ = p^μ/ħ. The space part is p = ħk, i.e. λ = h/p. The same algebra prices the wave's two speeds: the phase velocity ω/k = E/p exceeds c for a massive particle, but the group velocity dω/dk = dE/dp = v is the particle's own — the crests race ahead carrying nothing while the packet travels with the electron.

For electrons the bench form is λ = h/√(2mₑ eV) = 1.226 nm/√V. At the 54 V of Davisson and Germer, λ = 1.226/√54 = 0.167 nm — the same few-tenths-of-a-nanometre scale as a crystal lattice, which is why a crystal is the natural diffraction grating and a ruled one is hopeless. At 10 kV, λ = 12.26 pm non-relativistically; the correction of the last formula card pulls it to 12.20 pm. Scale the object up and the relation is just as obedient and far more brutal: a 1 μg dust grain drifting at 1 mm s⁻¹ has p = 10⁻¹² kg m s⁻¹ and λ = 6.6×10⁻²² m, seven orders of magnitude below a nuclear radius of a few femtometres, with no slit structure anywhere to match — nothing diffracts it, and the classical world sits at the small-λ face of the same formula. The practical skill: read the supply voltage, produce λ, and know before the experiment which spacings can Bragg-scatter the beam.

Now translate the wave statement into the working language. Define p̂ = −iħ d/dx and act on e^{ikx}: the derivative brings down ik, the prefactor −iħ turns that into ħk, and the function returns unchanged — p̂ e^{ikx} = ħk e^{ikx}. That is an eigenvalue equation, and it is why λ = h/p is more than an analogy: measured momenta are eigenvalues of p̂, and the plane wave of wavelength 2π/k is the eigenfunction belonging to ħk. The operator is not guessed. Applying e^{(−ip̂a/ħ)} to ψ(x) reproduces the Taylor series of ψ(x − a), so p̂ is the generator of translations, exactly the role p plays in classical mechanics through the Poisson bracket. Hermiticity comes from integration by parts: ⟨φ|p̂ψ⟩ − ⟨p̂φ|ψ⟩ equals a boundary term −iħ[φ*ψ] that vanishes for normalisable states — a first sight of the domain questions Unit 3 makes precise. One convention to fix: with the time factor e^(−iωt), positive k means motion towards +x.

Try to admit e^{ikx} as a state and it fails at the first axiom: |e^{ikx}|² = 1 at every x, so ⟨ψ|ψ⟩ = ∫|ψ|² dx diverges, and scaling by an amplitude A only rescales the divergence by |A|². The momentum operator has a perfectly real spectrum yet not one eigenvector in L²(R). The repair is a change of normalisation, not of function. Write ψₚ(x) = e^(ipx/ħ)/√(2πħ) and compute the overlap of two of them: ⟨p|p′⟩ = (1/2πħ)∫e^{(i(p′−p)x/ħ)} dx = δ(p − p′), the Fourier representation of the Dirac δ. Orthogonality survives — distinct momenta have zero overlap — but the 'length' of each basis element is a delta spike rather than 1, and the units say so: ψₚ carries (J s)^(−1/2), so ⟨p|p′⟩ carries 1/[p], as a density in p must. These improper kets live one rung outside L², in a rigged Hilbert space, and are kept because they make the completeness relation ∫dp |p⟩⟨p| = 1 an exact operator identity.

Insert that identity in front of any state: |ψ⟩ = ∫dp φ(p)|p⟩ with φ(p) = ⟨p|ψ⟩ — the momentum representation, concretely the Fourier transform of ψ(x). Born's rule then reads |φ(p)|² as a probability density: P(p₁ < p < p₂) = ∫|φ(p)|² dp, and the probability of any exact value is an integral over a single point, which is zero. Sharp momentum is never occupied. A beam quoted as 'momentum p₀' is shorthand for a packet whose φ(p) is sharply peaked: a Gaussian of width σₓ = 1 nm around a 54 eV electron has σₚ/p₀ = ħ/(2σₓ p₀) ≈ 1.3% — monochromatic by any experimental standard and still honestly in L². The narrower φ(p), the wider the packet in x; squeeze Δp to zero and the state stretches into the infinite plane wave you were forbidden to occupy. So read λ = h/p as a statement about where φ(p) is centred, not about a wavelength the particle 'has'.

Numerics never meets a delta function, and the reason is instructive. Put the particle on a ring of circumference L — periodic boundary conditions, the natural home of numpy.fft. The allowed wavenumbers are kₙ = 2πn/L, and e^{ikₙ x}/√L is now genuinely normalised: ⟨kₘ|kₙ⟩ = δₘₙ, Kronecker rather than Dirac. On an N-point grid the momentum operator is a matrix the discrete Fourier transform diagonalises exactly, so numpy.fft.fft(ψ) is literally the change to p̂'s eigenbasis, with eigenvalues ħkₙ up to the Nyquist momentum πħN/L. As L grows, the ladder of allowed momenta packs together with density L/2πħ states per unit momentum, sums over n become (L/2πħ)∫dp, and box normalisation goes over smoothly to delta normalisation. That limit is the honest content of ⟨p|p′⟩ = δ(p − p′): it is what Kronecker orthonormality becomes when the box outgrows every measurement, and every simulation in this course tacitly runs at finite L on the safe side of it.