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.

Read more about this topic: Pauli Equation

Famous quotes containing the words special and/or cases:

“Jack: A politician, huh?
Editor: Oh, county treasurer or something like that.
Jack: What’s so special about him?
Editor: They say he’s an honest man.”
—Robert Rossen (1908–1966)

“The world men inhabit ... is rather bleak. It is a world full of doubt and confusion, where vulnerability must be hidden, not shared; where competition, not co-operation, is the order of the day; where men sacrifice the possibility of knowing their own children and sharing in their upbringing, for the sake of a job they may have chosen by chance, which may not suit them and which in many cases dominates their lives to the exclusion of much else.”
—Anna Ford (b. 1943)