Relativistic Landau levels of a spinless particle

Units are SI throughout and the metric is diag(1, −1, −1, −1), with p^μ = iħ∂^μ and, from topic 3.04, minimal coupling p^μ → p^μ − qA^μ, A^μ = (φ/c, A). For a field B along z take φ = 0 and the Landau gauge A = (0, Bx, 0), so ∇ × A = B ẑ, and the equation reads −ħ²∂²ψ/∂t² = [c²(−iħ∇ − qA)² + m²c⁴]ψ. Nothing depends on t, y or z, so try ψ = f(x) e^{(i(k_y y + k_z z − Et/ħ))}. With p_z = ħk_z and ħk_y − qBx = −qB(x − x₀), x₀ = ħk_y/(qB), dividing by 2mc² leaves [pₓ²/2m + ½mω_c²(x − x₀)²] f = [(E² − m²c⁴ − p_z²c²)/2mc²] f, with ω_c = |q|B/m. That is QM-II's displaced oscillator line for line: the same rest mass m, the same ω_c, the same centre x₀ and the same Hermite functions of (x − x₀)/ℓ_B, with ℓ_B = √(ħ/|q|B). Exactly one thing has changed. The number standing where the Schrödinger energy stood is now (E² − m²c⁴ − p_z²c²)/2mc², still an energy, as the dimensions require.

QM-II's eigenvalue (n + ½)ħω_c, quoted, now fixes E² rather than E: E² = m²c⁴ + p_z²c² + (2n + 1)ħ|q|Bc², exactly. Dimensions: ħ|q|B is (J s)(C)(kg s⁻¹ C⁻¹) = kg J, and times c² it is J², like m²c⁴. Read it as E² = Mₙ²c⁴ + p_z²c² with Mₙ = m√(1 + (2n + 1)B/B_c) and B_c = m²c²/(ħ|q|). Each level moves along B as a free particle whose rest energy includes its gyration, so its inertia along B grows with n, which no non-relativistic level shows. B_c is 4.414 × 10⁹ T for the electron's mass and charge, the field that shrinks ℓ_B to the reduced Compton wavelength ħ/mc = 386 fm. Free-field limit: let B → 0 with (2n + 1)ħ|q|B = p_⊥² held fixed, and E² = m²c⁴ + p²c² returns. Both signs of E solve the equation, so a ladder reaching down from −√(m²c⁴ + ħ|q|Bc²) mirrors the positive one. It carries topic 3.02's negative density and waits for Unit 9 to be read as antiparticles.

Put the particle in an Lₓ × L_y rectangle, periodic in y, so k_y = 2πj/L_y with j an integer. The energy never involves k_y. It only places the orbit's centre at x₀ = ħk_y/(qB), which must lie inside the sample, a range Lₓ. The number of allowed j is therefore the k_y range Lₓ|q|B/ħ divided by the spacing 2π/L_y, which is |q|BLₓL_y/h. Each level holds |q|B/h states per unit area, derived: 2.418 × 10¹⁴ m⁻² per tesla for |q| = e, one state for each flux quantum h/|q| = 4.136 × 10⁻¹⁵ Wb threading the sample. Neither the x equation nor the range of x₀ knows E, so relativity moves every level and leaves each exactly as full as QM-II found it. The count is measured. A filled level of eB/h electrons per unit area carries Hall conductance e²/h, and quantum Hall plateaus sit at h/(νe²) for ν filled levels, with h/e² = 25 812.807 Ω, the same in every device to better than a part in 10⁹. Spin-split electron levels stand in there for our spinless ones.

Write ε = (n + ½)ħω_c + p_z²/2m, QM-II's Landau energy. The exact result is E = mc²√(1 + 2ε/mc²), and the binomial series gives E = mc² + ε − ε²/(2mc²) + ε³/(2m²c⁴) − …, derived: rest energy, then QM-II, then the first relativistic correction. Topic 3.03 found that correction as −p⁴/(8m³c²), and topic 3.04's minimal coupling makes it −π⁴/(8m³c²), with π = p − qA. Here π²/2m is exactly the non-relativistic Hamiltonian H₀ and ψ is its eigenstate, so ⟨−π⁴/(8m³c²)⟩ = −⟨(2mH₀)²⟩/(8m³c²) = −ε²/(2mc²). First-order perturbation theory returns the exact ε² term, and since the perturbation is a function of H₀ the k_y degeneracy causes no trouble. The cross term in ε², −(n + ½)(B/B_c)p_z²/2m, is the first sign of Mₙ. The series converges only while 2ε is below mc², the radius of convergence of √(1 + u). In a Penning trap at 5.36 T it converges fast: ħω_c/mc² = 1.21 × 10⁻⁹, and the n → n + 1 line sits (n + 1) × 182 Hz below ν_c = 150.0 GHz, the size of shift the Harvard g-factor measurements correct for.

Subtract neighbouring levels exactly: Eₙ₊₁² − Eₙ² = 2ħ|q|Bc², so Eₙ₊₁ − Eₙ = 2ħ|q|Bc²/(Eₙ + Eₙ₊₁) = ħ|q|B/(γ̄m), with γ̄ the mean of the two levels' Lorentz factors γ = E/mc². That is derived, for every n. Classically, dp/dt = qv × B with p = γmv, and γ is constant because B does no work, so the charge circles at ω = |q|B/(γm), with γ counting the motion along B too. At large n neighbouring γs agree to order 1/n, and the quantum gap becomes ħω, as Bohr's correspondence demands: the ladder's squeeze is the slowing of a heavier orbit. The orbit matches too. Since p_⊥² = (2n + 1)ħ|q|B, the radius p_⊥/(|q|B) is √(2n + 1) ℓ_B, enclosing (n + ½)h/|q| of flux. A 7 TeV LHC proton in 8.33 T has γ = 7461 and n ≈ 5.0 × 10²². Its radius comes out at 2.80 km, the bending radius the dipoles are built for, and its level spacing is 7.0 × 10⁻¹¹ eV, ħ × 2π × 17.0 kHz, 7461 times below the non-relativistic ħω_c.

The Klein-Gordon levels depend on B only through (2n + 1)ħ|q|B. Nothing tells one orientation from another, no level splits, and the zero-point ½ħω_c survives to all orders, so at p_z = 0 the lowest level sits at mc²√(1 + B/B_c), above mc² in any field. Unit 6 derives, and we quote, the Dirac result E² = m²c⁴ + p_z²c² + (2n + 1 ± 1)ħ|q|Bc², the lower sign for a spin moment along B. The spinless rungs sit exactly midway, in E², between Dirac rungs; the lowest Dirac level stays at √(m²c⁴ + p_z²c²) in every field; and at low energy the extra ±½ħω_c is a spin energy with g = 2. Even the correction changes, from ((n + ½)ħω_c)²/(2mc²) to ((n + ½ + mₛ)ħω_c)²/(2mc²) in Penning-trap analyses. A measured contrast: graphene's electrons obey a massless Dirac equation with v_F ≈ 1 × 10⁶ m s⁻¹ in place of c, and spectroscopy finds levels at ±v_F√(2nħeB), one at zero energy and the next at 36 meV in 1 T. A massless Klein-Gordon ladder, v_F√((2n + 1)ħeB), would start at 26 meV, with no zero level.