Motivation
Why is the electron's spin Zeeman splitting twice the orbital unit?
Schrödinger theory describes charged particles coupled minimally to electromagnetic potentials; a spin-½ electron also carries an intrinsic magnetic moment. The Pauli Hamiltonian includes both the orbital response to magnetic field and a direct coupling of spin to B, with gyromagnetic factor g≈2 in the Dirac minimal-coupling limit. By controlling an orbital magnetic quantum number and a spinor preparation, we can compare orbital spacings, the exact two spin eigenenergies and intermediate expectation values in one transparent model.
Prerequisites: Minimal electromagnetic coupling, Lz and Sz eigenvalues, Pauli matrices, and two-level spinors.
Computed physics
Live observable readouts
Magnetic-field energy splitting
Spin-up and spin-down Zeeman branches
A fixed mℓ orbital shifts both spin levels by the same amount while their energy separation grows linearly with B.
In units μB B, E(mℓ,mₛ)=μB B(mℓ+2mₛ); the spin splitting is 2μB B and changing mℓ by one shifts the orbital energy by μB B.
Spin superposition
Measurement statistics and mean Zeeman energy
A spinor with a nonzero transverse component is an energy superposition, so its mean energy interpolates between the two eigenvalues.
For |ψ〉=cos(θ/2)|↑〉+sin(θ/2)|↓〉, p↑=cos²(θ/2), p↓=sin²(θ/2) and 〈HZeeman〉=μB B(mℓ+cosθ).
Derive the result
The Pauli magnetic coupling and its spin factor
For a charged spin-half particle in electromagnetic potentials, introduce kinetic momentum π=p−(q/c)A. The Pauli Hamiltonian may be written H=[σ·π]²/(2m)+qφ. Using the Pauli identity (σ·π)²=π²−(qℏ/c)σ·B gives an additional spin coupling proportional to the magnetic field.
For an electron of charge q=−e, define μB=eℏ/(2mc)>0. In a weak spatially uniform field along z, and ignoring the diamagnetic B² correction, the orbital and spin Zeeman contributions are respectively μB B Lz/ℏ and μB B σz. Thus the electron spin contributes two units of μB B per unit mₛ:
The spin eigenstates have mₛ=±½, so the two spin branches at fixed orbital state are separated by 2μB B. An arbitrary pure spin preparation has 〈σz〉=cosθ and 〈HZeeman〉=μB B(mℓ+cosθ). The displayed energy is the magnetic contribution only; any zero-field orbital energy common to the spin pair has been subtracted.
Exactly soluble limits and sanity checks
At B=0 the Zeeman splitting vanishes and both spin projections are degenerate. When θ=0 or 180°, the spinor is a stationary Zeeman eigenstate for a field aligned with z. At θ=90° the spin Zeeman expectation vanishes even though the state has equal probabilities for two energies separated by 2μB B.
What assumptions define this model?
Weak uniform magnetic field along z, a fixed orbital Lz eigenvalue mℓ, noninteracting electron with gspin=2 and neglect of B² terms, spin–orbit coupling, anomalous magnetic moment, finite-size orbital mixing and nuclear interactions. Energies are plotted in units μB times the numerical B parameter; only Zeeman contributions are included.
Self-directed physics exercises
Three independent investigations
- Find the spin splitting. Set mℓ=0 and sweep B. Read the two exact spin branches and verify their difference scales as twice the orbital magnetic unit.
- Distinguish state and eigenenergy. Choose θ=90° and compare the mean energy with both eigenenergies. Explain why an expectation value can lie between discrete measurement outcomes.
- Test the orbital shift. Change mℓ from −1 to 0 to +1 at fixed B. Verify both spin levels translate together without changing their separation.
Self-check: For an electron with mℓ=0 in a weak positive B field, what is the separation of its spin-up and spin-down Zeeman energies?
Extension: derive the Pauli magnetic term by applying (σ·π)² and using [πx,πy]=i qℏ Bz/c, then connect it to the nonrelativistic Dirac expansion.
Scientific meaning
Conclusions, caveats and further reading
The Pauli Hamiltonian explains why electron spin contributes a magnetic coupling twice the orbital Zeeman unit in the ideal Dirac limit. The distinction between energy eigenstates, general spinor preparations and operator expectations connects magnetic response to the qubit and Dirac models elsewhere in the companion.
Corresponding lecture-note sources: Pauli Hamiltonian and Dirac construction · Free Dirac spinors.