Pauli equation

In, the Pauli equation or Schrödinger–Pauli equation is the formulation of the for  particles, which takes into account the interaction of the particle's  with an external. It is the non- limit of the and can be used where particles are moving at speeds much less than the, so that relativistic effects can be neglected. It was formulated by in 1927.

Equation
For a particle of mass $$m$$ and electric charge $$q$$, in an described by the  $$\mathbf{A}$$ and the  $$\phi$$, the Pauli equation reads:

where $$\boldsymbol{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$$ are the collected into a vector for convenience, and  $$\mathbf{p} = -i\hbar \nabla$$ is the.


 * $$ |\psi\rangle = \begin{bmatrix}

\psi_+ \\ \psi_- \end{bmatrix}$$is the two-component, a  written in.

The

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

is a 2 × 2 matrix because of the. Substitution into the gives the Pauli equation. This Hamiltonian is similar to the classical Hamiltonian for a charged particle interacting with an electromagnetic field. See for details of this classical case. The term for a free particle in the absence of an electromagnetic field is just  $$\frac{\mathbf{p}^2}{2m}$$ where  is the, while in the presence of an EM field it involves the  $$\mathbf{P} = p - q\mathbf{A}$$, where $$\mathbf{P}$$ is the.

The Pauli matrices can be removed from the kinetic energy term using the :


 * $$(\boldsymbol{\sigma}\cdot \mathbf{a})(\boldsymbol{\sigma}\cdot \mathbf{b}) = \mathbf{a}\cdot\mathbf{b} + i\boldsymbol{\sigma}\cdot \left(\mathbf{a} \times \mathbf{b}\right)$$

to obtain

where $$\mathbf{B} = \nabla \times \mathbf{A}$$ is the magnetic field.

Relationship with the Schrödinger equation and the Dirac equation
The Pauli equation is non-relativistic, but it does incorporate spin. As such, it can be thought of as occupying the middle ground between:
 * The familiar Schrödinger equation (on a complex scalar ), which is non-relativistic and does not predict spin.
 * The Dirac equation (on a ), which is fully (with respect to ) and predicts spin.

Note that because of the properties of the Pauli matrices, if the magnetic vector potential $$\mathbf{A}$$ vanishes, then the equation reduces to the familiar Schrödinger equation for a particle in a purely electric potential ϕ, except that it operates on a two-component :
 * $$\left( \frac{\mathbf{p}^2}{2m} + q \phi \right) \begin{bmatrix}

\psi_+ \\ \psi_- \end{bmatrix} = i \hbar \frac{\partial}{\partial t} \begin{bmatrix} \psi_+ \\ \psi_- \end{bmatrix}.$$

Therefore, we can see that the spin of the particle only affects its motion in the presence of a magnetic field.

Relationship with Stern–Gerlach experiment
Both spinor components satisfy the Schrödinger equation. For a particle in an externally applied $$\mathbf{B}$$ field, the Pauli equation reads:

where


 * $$ \mathbb{I} = \begin{pmatrix}

1 & 0 \\ 0 & 1 \\ \end{pmatrix} $$

is the $$ 2 \times 2 $$.

The can obtain the spin orientation of atoms with one, 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.