Kramers degeneracy derived

Units are SI, with small splittings also quoted as frequencies ΔE/h; the lesson is non-relativistic, and the metric (+, −, −, −) is not needed until Unit 3. Two results are quoted, not re-derived: an antiunitary Θ obeys ⟨Θα|Θβ⟩ = ⟨β|α⟩ (topic 1.02), and Θ² = (−1)^(2j) whatever phase is chosen (topic 1.06). Now let [Θ, H] = 0 and H|n⟩ = Eₙ|n⟩. Then HΘ|n⟩ = ΘH|n⟩ = Θ(Eₙ|n⟩) = EₙΘ|n⟩: antilinearity conjugates the number, but Eₙ is real. So Θ|n⟩ is an eigenstate of the same energy and the same norm. Is it a new state? Put α = |n⟩ and β = Θ|n⟩ into the rule: ⟨Θn|Θ²n⟩ = ⟨Θn|n⟩. With Θ² = −1 the left side is −⟨Θn|n⟩, so ⟨Θn|n⟩ = 0. A shorter route lands in the same place: if Θ|n⟩ = c|n⟩, then Θ²|n⟩ = c* Θ|n⟩ = |c|²|n⟩, which can never equal −|n⟩. Derived here: the level is at least twofold.

One orthogonal partner makes a level at least twofold; the theorem claims more, that every eigenspace W of H has even dimension. Θ maps W into itself. Pick |a⟩ in W: the pair |a⟩, Θ|a⟩ spans a plane that Θ also maps into itself, since Θ(Θ|a⟩) = −|a⟩. The rest of W, the states |c⟩ orthogonal to both, is Θ-invariant too: the rule gives ⟨Θc|Θa⟩ = ⟨a|c⟩ = 0 and ⟨Θc|a⟩ = −⟨Θc|Θ²a⟩ = −⟨Θa|c⟩ = 0. Repeat inside that remainder until W is used up: it is a sum of orthogonal pairs, so its dimension is even. For a free-ion multiplet of half-integer J, a crystal field can therefore make at most (2J + 1)/2 levels: 5 from Nd³⁺ ⁴I₉/₂ and 8 from Er³⁺ ⁴I₁₅/₂. Optical spectra agree: in Nd:YAG the ⁴I₉/₂ ground multiplet shows five crystal-field levels and the ⁴F₃/₂ upper laser level shows two, roughly 10 meV apart, never more.

The proof never used the form of H, so try the most general T-even H that a small multiplet allows. Since ΘSΘ⁻¹ = −S, a Hermitian product of k spin components carries the T-sign (−1)^k, and for S = 1 or 3/2 the T-even operators are a constant plus a rank-2 term, which in its principal axes reads D(S_z² − S(S+1)/3) + E(Sₓ² − S_y²). For S = 1 the real states |x⟩, |y⟩, |z⟩ are eigenstates at D/3 − E, D/3 + E and −2D/3: three singlets once E ≠ 0, which an even-electron ion such as Ni²⁺ is free to show. For S = 3/2, Sₓ² − S_y² = (S₊² + S₋²)/2 links m to m ± 2, so the states fall into two blocks, m = (3/2, −1/2) and m = (−3/2, 1/2), each with diagonal ±D and off-diagonal √3E. Both blocks give ε = ±√(D² + 3E²): two doublets for every D and E, Kramers' theorem checked case by case. Cr³⁺ in ruby, with 21 electrons, shows exactly two doublets, about 11.5 GHz apart.

Inside one Kramers pair a T-even perturbation can do only one thing. Let V be Hermitian with ΘVΘ⁻¹ = V. Apply the rule to α = VΘ|n⟩ and β = |n⟩: since ΘVΘ|n⟩ = VΘ²|n⟩ = −V|n⟩, it gives −⟨n|V|Θn⟩ = ⟨n|V|Θn⟩, so the off-diagonal element vanishes. Apply it to α = V|n⟩ and β = |n⟩ and it gives ⟨Θn|V|Θn⟩ = ⟨n|V|n⟩. On the pair, V is ⟨n|V|n⟩ times the identity: it moves the doublet as a block. That is the first-order statement, and nothing changes at higher order, because H + V still commutes with Θ. The T-even list is long: a crystal field V(r); the Stark term −qE⋅r, with r even and the applied E a fixed external number; spin-orbit coupling ξ L⋅S, a product of two odd factors; electron repulsion; strain. So an odd-electron ion in an electric field of any size and direction shows doublets that shift, and never a pair that splits.

Run the same algebra with a T-odd operator such as the magnetic moment, ΘμΘ⁻¹ = −μ. Now the diagonal elements come out opposite, ⟨Θn|μ_z|Θn⟩ = −⟨n|μ_z|n⟩, and the off-diagonal ones are unconstrained, so −μ⋅B on the pair is a traceless 2 × 2 Hermitian matrix with eigenvalues ±½ g_eff μ_B B. The splitting ΔE = g_eff μ_B B is linear in B, with g_eff set by the field direction through the doublet's g-tensor. With μ_B/h = 13.996 GHz T⁻¹, a doublet with g_eff = 2 opens 28.0 GHz per tesla, which is why odd-electron ions are the staple of EPR near 9.5 GHz and 0.34 T. Compare a non-degenerate level, possible only when Θ² = +1: there Θ|n⟩ = c|n⟩, the same step gives ⟨n|μ|n⟩ = −⟨n|μ|n⟩ = 0, and the Zeeman shift starts at B². At B = 10 mT an S = 1 pair held 4.0 GHz apart by E/h = 2.0 GHz widens by 0.04 GHz, while an m = ±3/2 Kramers doublet with g = 2 opens by 0.84 GHz.

The many-particle Θ is a product of one-particle factors, exp(−iπS_y/ħ) for each spin, times one K for the whole system; the factors act on different particles and commute, so Θ² = (−1)^N, derived from topic 1.06's one-particle result. “An odd number of electrons” is shorthand that drops the nuclei, and hydrogen tests it. With the proton's spin ignored, 1s is a Kramers pair, mₛ = ±½. Keep the proton and N = 2, Θ² = +1, so nothing stops the hyperfine coupling from leaving a single F = 0 level below the F = 1 triplet, measured 1420.405 751 768 MHz, or 5.87 μeV, apart. Deuterium adds a neutron, N = 3, and its ground levels are F = 1/2 and F = 3/2, two and four states, both even, 327.384 MHz apart. Unit 7's Dirac hydrogen, quoted here, obeys the theorem: its energies depend on n and j only, so every level holds 2j + 1 or 2(2j + 1) states, always even, and the measured 1057.8 MHz Lamb shift parts 2s₁/₂ from 2p₁/₂ while leaving each a pair.