Pauli Equation - Special Cases

Special Cases

Both spinor components satisfy the Schrödinger equation. For a particle in an externally applied B field, the Pauli equation reads:

Pauli equation (B-field)

\underbrace{i \hbar \frac{\partial}{\partial t} |\psi_\pm\rangle = \left( \frac{( \mathbf{p} -q \bold A)^2}{2 m} + q \phi \right) \hat 1 \bold |\psi\rangle }_\mathrm{Schr\ddot{o}dinger~equation} - \underbrace{\frac{q \hbar}{2m}\boldsymbol{\sigma} \cdot \bold B \bold |\psi\rangle }_\mathrm{Stern \, Gerlach \, term}

where

\hat 1 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}

is the 2 × 2 identity matrix, which acts as an identity operator.

The Stern–Gerlach term can obtain the spin orientation of atoms with one valence electron, e.g. silver atoms which flow through an inhomogeneous magnetic field.

Analogously, the term is responsible for the splitting of spectral lines (corresponding to energy levels) in a magnetic field as can be viewed in the anomalous Zeeman effect.

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