Pauli equation ($i\hslash {\partial }_t\Psi =H_P\Psi$)

For a nonrelativistic spin-one-half electron of charge $-e$, with $e>0$, the Pauli Hamiltonian isThe wavefunction has two spin components and $B=\nabla \times A$. If $\pi =-i\hslash \nabla +eA$, then $[{\pi }_i,{\pi }_j]=-ie\hslash {ϵ}_{ijk}{B}_k$; the Pauli matrix multiplication law gives $(\sigma \cdot \pi {)}^2={\pi }^2+e\hslash \sigma \cdot B$, explaining the spin-coupling sign. Thus $i\hslash {\partial }_t\Psi =H_P\Psi$ extends the Schrödinger equation to include the electron's magnetic moment.

A two-component complex wave function describing the two spin states of a nonrelativistic spin-$1/2$ particle. Under a spatial rotation by angle $\theta$ about $\hat{\mathbf{n}}$ it transforms as $\psi \mapsto e^{-i\theta \hat{\mathbf{n}}\cdot \mathbf{\sigma }/2}\psi$, with Pauli matrices $\mathbf{\sigma }$. A Pauli spinor may describe a massive particle; two components alone do not imply the massless Weyl equation.

For $B=B(y)e_z$ with $B(y)\ge B_0>0$, take $a(y)=(e/\hslash ){\int }_0^{y}B(s)ds$. The spin-down transverse Hamiltonian at momentum label $-k$ is proportional to $M^{†}M$, where $M={\partial }_y+k+a(y)$. It is nonnegative, and $M{\chi }_k=0$ gives the displayed zero mode. Since $a^{\prime }\ge e{B}_0/\hslash$, its exponent has a negative quadratic upper bound at both infinities, proving square integrability for every $k$. The continuously varying momentum label supplies an infinitely degenerate zero-energy space.