<Dd> (E − e φ) ψ + − c σ ⋅ (p − e c A) ψ − = m c 2 ψ + − (E − e φ) ψ − + c σ ⋅ (p − e c A) ψ + = m c 2 ψ − (\ displaystyle (\ begin (aligned) (E-e \ phi) \ psi _ (+) - c (\ boldsymbol (\ sigma)) \ cdot \ left (\ mathbf (p) - (\ frac (e) (c)) \ mathbf (A) \ right) \ psi _ (-) & = mc ^ (2) \ psi _ (+) \ \ - (E-e \ phi) \ psi _ (-) + c (\ boldsymbol (\ sigma)) \ cdot \ left (\ mathbf (p) - (\ frac (e) (c)) \ mathbf (A) \ right) \ psi _ (+) & = mc ^ (2) \ psi _ (-) \ end (aligned))) </Dd> <P> Assuming the field is weak and the motion of the electron non-relativistic, we have the total energy of the electron approximately equal to its rest energy, and the momentum reducing to the classical value, </P> <Dl> <Dd> E − e φ ≈ m c 2 p ≈ m v (\ displaystyle (\ begin (aligned) E-e \ phi & \ approx mc ^ (2) \ \ p& \ approx mv \ end (aligned))) </Dd> </Dl> <Dd> E − e φ ≈ m c 2 p ≈ m v (\ displaystyle (\ begin (aligned) E-e \ phi & \ approx mc ^ (2) \ \ p& \ approx mv \ end (aligned))) </Dd>

Energy of a spin in a magnetic field