Pauli equation

Quantum mechanics
$\sigma_x \sigma_p \ge \frac{\hbar}{2}$ Uncertainty principle
Introduction Glossary History
Background Classical mechanics Old quantum theory Bra–ket notation Hamiltonian Interference
Fundamentals Complementarity Decoherence Entanglement Energy level Measurement Nonlocality Quantum number State Superposition Symmetry Tunnelling Uncertainty Wave function (collapse)
Experiments Afshar Bell's inequality Davisson–Germer Double-slit Elitzur–Vaidman Franck–Hertz Mach–Zehnder Popper Quantum eraser (delayed-choice) Schrödinger's cat Stern–Gerlach Wheeler's delayed-choice
Formulations Overview Heisenberg Interaction Matrix Phase-space Schrödinger Sum-over-histories (path integral)
Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger
Interpretations Overview Consistent histories Copenhagen de Broglie–Bohm Ensemble Hidden-variable Many-worlds Objective collapse Bayesian Quantum logic Relational Stochastic Scale relativity Transactional
Advanced topics Quantum chaos Quantum computing Density matrix Quantum field theory Fractional quantum mechanics Quantum information science Relativistic quantum mechanics Scattering theory Quantum statistical mechanics
Scientists Aharonov Bell Blackett Bloch Bohm Bohr Born Bose de Broglie Candlin Compton Dirac Davisson Debye Ehrenfest Einstein Everett Fock Fermi Feynman Glauber Gutzwiller Heisenberg Hilbert Jordan Kramers Pauli Lamb Landau Laue Moseley Millikan Onnes Planck Rabi Raman Rydberg Schrödinger Sommerfeld von Neumann Weyl Wien Wigner Zeeman Zeilinger

In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field. It is the non-relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927.^[1]

Equation

For a particle of mass m and charge q, in an electromagnetic field described by the vector potential A = (A_x, A_y, A_z) and scalar electric potential ϕ, the Pauli equation reads:

Pauli equation (general)

$\left[ \frac{1}{2m}(\boldsymbol{\sigma}\cdot(\mathbf{p} - q \mathbf{A}))^2 + q \phi \right] |\psi\rangle = i \hbar \frac{\partial}{\partial t} |\psi\rangle$

where σ = (σ_x, σ_y, σ_z) are the Pauli matrices collected into a vector for convenience, p = −iħ∇ is the momentum operator wherein ∇ denotes the gradient operator, and

|\psi\rangle = \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix}

is the two-component spinor wavefunction, a column vector written in Dirac notation.

The Hamiltonian operator

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

is a 2 × 2 matrix operator, because of the Pauli matrices. Substitution into the Schrödinger equation gives the Pauli equation. This Hamiltonian is similar to the classical Hamiltonian for a charged particle interacting with an electromagnetic field, see Lorentz force for details of this classical case. The kinetic energy term for a free particle in the absence of an electromagnetic field is just p²/2m where p is the kinetic momentum, while in the presence of an EM field we have the minimal coupling p = P − qA, where P is the canonical momentum.

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

(\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^[2]

Pauli equation (standard form)

$\hat{H} |\psi\rangle = \left[\frac{1}{2m}\left[\left(\mathbf{p} - q \mathbf{A}\right)^2 - q \hbar \boldsymbol{\sigma}\cdot \mathbf{B}\right] + q \phi\right]|\psi\rangle = i \hbar \frac{\partial}{\partial t} |\psi\rangle$

where B = ∇ × A is the magnetic field.

Relationship with the Schrödinger equation and the Dirac equation

The Pauli equation is non-relativistic, but it does predict spin. As such, it can be thought of as occupying the middle ground between:

The familiar Schrödinger equation (on a complex scalar wavefunction), which is non-relativistic and does not predict spin.
The Dirac equation (on a complex four-component spinor), which is fully relativistic (with respect to special relativity) and predicts spin.

Note that because of the properties of the Pauli matrices, if the magnetic vector potential A is equal to zero, 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 spinor:

\left[ \frac{\mathbf{p}^2}{2m} + q \phi \right] \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix} = i \hbar \frac{\partial}{\partial t} \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix}.

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 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.

References

↑ Wolfgang Pauli (1927) Zur Quantenmechanik des magnetischen Elektrons Zeitschrift für Physik (43) 601-623
↑ Bransden, BH; Joachain, CJ (1983). Physics of Atoms and Molecules (1st ed.). Prentice Hall. p. 638-638. ISBN 0-582-44401-2.

Schwabl, Franz (2004). Quantenmechanik I. Springer. ISBN 978-3540431060.
Schwabl, Franz (2005). Quantenmechanik für Fortgeschrittene. Springer. ISBN 978-3540259046.
Claude Cohen-Tannoudji, Bernard Diu, Frank Laloe (2006). Quantum Mechanics 2. Wiley, J. ISBN 978-0471569527.

External links

This article is issued from Wikipedia - version of the Wednesday, August 26, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.