Covariant notation and the Klein-Gordon equation

Units are SI with ħ and c kept, and the metric is g_μν = diag(1, −1, −1, −1), for the rest of the course. Coordinates form x^μ = (ct, x), so x⁰ = ct is a length. Lowering an index flips the spatial sign, x_μ = (ct, −x), and a contraction reads a_μb^μ = a⁰b⁰ − a⋅b. A derivative with respect to an upper-index coordinate carries a lower index, ∂_μ = ∂/∂x^μ = (∂/∂(ct), ∇), and raising it gives ∂^μ = (∂/∂(ct), −∇). The four-momentum is p^μ = (E/c, p), and Quantum Mechanics I's rules E = iħ∂/∂t and p = −iħ∇ assemble into one statement, p^μ = iħ∂^μ: the time component is iħ(1/c)∂/∂t = E/c, and the spatial components are iħ∂^k = −iħ∂/∂x^k. Test it on Quantum Mechanics I's plane wave, ⟨x|p⟩ ∝ e^(ip⋅x/ħ) times e^(−iEt/ħ), which is e^{(−ip_μxμ/ħ)} with p_μx^μ = Et − p⋅x: iħ∂^μ returns p^μ times the wave. Lorentz matrices Λ^μ_ν and the invariant E²/c² − p² = m²c² come from special relativity and are used, not re-derived.

The obvious relativistic Hamiltonian is H = √(p²c² + m²c⁴). With p = −iħ∇ it becomes √(m²c⁴ − ħ²c²∇²), whose expansion mc² − ħ²∇²/2m − ħ⁴∇⁴/(8m³c²) − … contains every power of ∇: an operator that ties φ at one point to φ up to about ħ/mc away, and treats time, one derivative, and space, all orders, so unequally that nothing about it is manifestly covariant. Use the square instead. Acting on φ, p_μp^μ = (iħ)²∂_μ∂^μ = −ħ²□, so p_μp^μφ = m²c²φ becomes (□ + (mc/ħ)²)φ = 0 with □ = (1/c²)∂²/∂t² − ∇², derived. Dimensions: ∇² is m⁻², and mc/ħ is kg m s⁻¹ over J s, which is m⁻¹; for the electron (mc/ħ)² = 6.71 × 10²⁴ m⁻². Limit m → 0: d'Alembert's □φ = 0, which each Lorenz-gauge potential obeys in vacuum, quoted from electromagnetism. Klein, Gordon, Fock and Schrödinger all published the equation in 1926, Schrödinger having first written it down late in 1925, quoted.

A Lorentz transformation x′^μ = Λ^μ_ν x^ν preserves the metric, so any contraction of an upper index with a lower one has the same value in every frame. ∂_μ transforms with the inverse matrix, so □ = ∂_μ∂^μ is a scalar operator, and if φ is a scalar field, φ′(x′) = φ(x), then (□′ + (mc/ħ)²)φ′ = 0 holds in the new frame whenever (□ + (mc/ħ)²)φ = 0 held in the old: derived. Check it by hand for a boost at βc along x. From ct′ = γ(ct − βx) and x′ = γ(x − βct), the chain rule gives ∂/∂(ct) = γ(∂′₀ − β∂′ₓ) and ∂/∂x = γ(∂′ₓ − β∂′₀), so ∂₀² − ∂ₓ² = γ²(1 − β²)(∂′₀² − ∂′ₓ²), and γ²(1 − β²) = 1. Schrödinger's iħ∂ψ/∂t = −(ħ²/2m)∇²ψ fails the same test, one time derivative against two space derivatives, and survives only Galilean boosts, with a phase. The plane-wave phase p_μx^μ = Et − p⋅x is itself a contraction, so a solution in one frame is a solution in any other, with E and p transformed as a four-vector.

Substitute φ = e^{(−i(Et − p⋅x)/ħ)}: ∂²/∂t² brings −E²/ħ², ∇² brings −p²/ħ², and the equation becomes E²/c² − p² − m²c² = 0, so E = ±Eₚ with Eₚ = √(p²c² + m²c⁴), derived. The branches are mirror hyperbolas 2mc² apart at p = 0, 1.022 MeV for the electron and 279.1 MeV for the charged pion, and a squared equation prefers neither. Being second order in time, the equation has two amplitudes per momentum in its general solution, a(p) on e^{(−iEₚt/ħ)} and b(p) on e^(+iEₚt/ħ), fixed by two initial data, φ and ∂φ/∂t at t = 0. A pulse with ∂φ/∂t = 0 at t = 0 gets a = b: half of it is negative-energy wave. Because |E| exceeds |p|c on the mass shell, free evolution and boosts never change the sign of E, so for a free particle the split is consistent. But keeping only b = 0 means imposing the square-root equation on the data, and a perturbation with ħω ≥ 2mc², ω above 1.55 × 10²¹ s⁻¹ for the electron, couples the two branches. Topic 3.02 shows what the lower branch does to the density.

The equation contains a single length, λ_C = ħ/mc, the reduced Compton wavelength: 386.16 fm for the electron, 1.868 fm for the muon, 1.414 fm for the charged pion and 0.2103 fm for the proton, from measured masses. The wave number in units of it, kλ_C = p/mc, says which regime a wave is in. For p ≪ mc the upper branch is mc² + p²/2m − p⁴/(8m³c²) + …, rest energy, Schrödinger's kinetic energy and Quantum Mechanics II's correction, which topic 3.03 recovers from the equation itself. At p = mc the exact kinetic energy is (√2 − 1)mc² = 0.414mc², and the non-relativistic 0.5mc² is 21% high. The group velocity, dE/dp = pc²/E from the spectrum, stays below c on both branches while the phase velocity E/p exceeds c in size, their product being c². A packet squeezed below λ_C contains momenta above mc and energies where the second branch is no small correction. The length is measured directly: Compton's X-ray shift at 90° is h/(mₑc) = 2πλ_C = 2.4263 pm, quoted.

For a time-independent φ the equation reads ∇²φ = φ/λ_C². Away from a point source, with spherical symmetry, (1/r)d²(rφ)/dr² = φ/λ_C², so rφ = A e^(−r/λ_C) + B e^(r/λ_C), and decay at large r keeps φ = A e^(−r/λ_C)/r, derived: Coulomb's 1/r cut off over one Compton wavelength, returning exactly to 1/r in the massless limit m → 0. Yukawa in 1935 read the short range of the nuclear force, about 2 fm, as the λ_C of a new particle and predicted a mass near 200 mₑ, quoted. The charged pion, found in 1947, has mc² = 139.57 MeV, which is 273 mₑ, and λ_C = 1.414 fm, quoted. Reading the static solution as a force carried by particles is field theory and waits for Unit 9. As a fact about the equation alone it already says that a massive scalar field cannot reach much further than ħ/mc from its source.