Pauli Equation

Prerequisites: Schrödinger Equation, Spin. Full path (19 pages)

Intuitive
Concrete
Abstract
Why is it interesting?
Definitions

$\left( \frac{1}{2m}(\vec \sigma \cdot ( \vec p - q\vec A))^2 + q\phi \right) \Psi = i \hbar \partial_t \Psi$

Intuitive

The Pauli equation describes how the state of a quantum system with half-integer spin changes in time.

In contrast, the Schrödinger equation describes the time evolution of systems without spin.

Concrete

The Pauli equation is the non-relativistic limit of the Dirac equation.

The wave function $\Psi$ has two complex components, one for spin up and one for spin down along a chosen axis, and the Pauli matrices act on these components. We write $\vec\pi = \vec p - q\vec A$ with $\vec p = -i\hbar\nabla$ for the kinetic momentum.

The spin term. For ordinary vectors $\vec a,\vec b$ the Pauli matrices satisfy $(\vec\sigma\cdot\vec a)(\vec\sigma\cdot\vec b) = \vec a\cdot\vec b + i\vec\sigma\cdot(\vec a\times\vec b)$ . The components of $\vec\pi$ are operators that do not commute: $[\pi_i,\pi_j] = i\hbar q(\partial_i A_j - \partial_j A_i)$ , so $\vec\pi\times\vec\pi = i\hbar q\vec B$ with $\vec B = \nabla\times\vec A$ . Therefore

$(\vec\sigma\cdot\vec\pi)^2 = \vec\pi^2 - \hbar q\,\vec\sigma\cdot\vec B ,$

and the Pauli equation takes the form

$i\hbar\,\partial_t\Psi = \left(\frac{(\vec p - q\vec A)^2}{2m} + q\phi - \frac{q\hbar}{2m}\,\vec\sigma\cdot\vec B\right)\Psi .$

The first two terms are the Schrödinger Hamiltonian of a charged particle, acting on each component separately. The last term is the energy $-\vec\mu\cdot\vec B$ of a magnetic moment $\vec\mu = \frac{q\hbar}{2m}\vec\sigma = g\frac{q}{2m}\vec S$ in the magnetic field, with spin $\vec S = \frac{\hbar}{2}\vec\sigma$ and gyromagnetic ratio $g = 2$ . Without a magnetic field the two components decouple and each obeys the Schrödinger equation.

Non-relativistic limit of the Dirac equation. In the Dirac representation, the Dirac equation with electromagnetic potentials reads $i\hbar\partial_t\Psi_D = \left(c\vec\alpha\cdot\vec\pi + \beta mc^2 + q\phi\right)\Psi_D$ with a four-component spinor. We split off the rest energy, $\Psi_D = e^{-imc^2t/\hbar}\begin{pmatrix}\psi\\ \chi\end{pmatrix}$ with two-component $\psi,\chi$ , and get

$\begin{aligned} i\hbar\,\partial_t\psi &= c\,\vec\sigma\cdot\vec\pi\,\chi + q\phi\,\psi , \\ i\hbar\,\partial_t\chi &= c\,\vec\sigma\cdot\vec\pi\,\psi + q\phi\,\chi - 2mc^2\chi . \end{aligned}$

If all kinetic and potential energies are small compared to $mc^2$ , the term $-2mc^2\chi$ dominates the second line, so $\chi \approx \frac{\vec\sigma\cdot\vec\pi}{2mc}\psi$ . Inserting this in the first line gives

$i\hbar\,\partial_t\psi = \left(\frac{(\vec\sigma\cdot\vec\pi)^2}{2m} + q\phi\right)\psi ,$

which is the Pauli equation for $\psi$ . The value $g = 2$ comes out without being put in. The next order in $1/c^2$ gives the relativistic kinetic correction, the spin-orbit coupling and the Darwin term, which are responsible for the fine structure of hydrogen.

Abstract

The state space of a non-relativistic spin $1/2$ particle is $\mathcal H = L^2(\mathbb R^3)\otimes\mathbb C^2$ . On it, the Pauli Hamiltonian is

$\hat H = \frac{1}{2m}\left(\vec\sigma\cdot(\vec p - q\vec A)\right)^2 + q\phi .$

The first term is the square of the first-order operator $\frac{1}{\sqrt{2m}}\,\vec\sigma\cdot(\vec p - q\vec A)$ , a three-dimensional Dirac-type operator, so for reasonable potentials it is a non-negative self-adjoint operator.

The equation is covariant under gauge transformations. With a real function $\lambda(\vec x,t)$ , the replacements $\vec A\to\vec A + \nabla\lambda$ , $\phi\to\phi - \partial_t\lambda$ and $\Psi\to e^{iq\lambda/\hbar}\Psi$ map solutions to solutions. Geometrically, $\Psi$ is a section of the tensor product of a $U(1)$ line bundle with connection $q\vec A$ , $q\phi$ and the trivial spinor bundle $\mathbb R^3\times\mathbb C^2$ .

The symmetry group of non-relativistic physics is the Galilei group. Its action on wave functions is projective, and the unitary representations relevant for quantum mechanics are representations of a central extension, in which the mass appears as the central charge (Bargmann). The spin $1/2$ representations act on $L^2(\mathbb R^3)\otimes\mathbb C^2$ with rotations acting on $\mathbb C^2$ through $SU(2)$ , the double cover of $SO(3)$ . The free Pauli equation is the equation of motion for these representations.

The value $g = 2$ is often presented as a prediction of relativity, because it follows from the Dirac equation. Lévy-Leblond showed that it also follows in a purely Galilean setting: if one asks for a first-order wave equation that is invariant under the Galilei group, in analogy with Dirac's construction, and then couples it minimally to the electromagnetic field, the resulting equation for the two physical components is the Pauli equation with $g = 2$ . The value is thus a consequence of minimal coupling in a first-order formulation, and the measured deviation $g - 2$ of the electron comes from quantum field theory corrections.

Nonrelativistic particles and wave equations by Jean-Marc Lévy-Leblond

Why is it interesting?

The Pauli equation is the correct non-relativistic equation to describe spin $1/2$ particles.

Definitions

$\Psi$ is the wave function,
$m$ the mass of the particle,
$q$ the charge of the particle,
$\vec{\sigma}$ the Pauli matrices,
$\vec{p}$ the momentum,
$\vec A$ the vector potential,
$\phi$ the electric scalar potential and
$\hbar$ the reduced Planck constant.

Take note that $\vec \sigma$ , a "vector of matrices" is only used as a convenient short-hand notation for the sums that appear in the equation. For example, $\vec \sigma \vec p = \sigma_1 p_1 + \sigma_2 p_2 + \sigma_3 p_3.$