Pauli equation

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]

EquationEdit

For a particle of mass   and electric charge  , in an electromagnetic field described by the magnetic vector potential   and the electric scalar potential  , the Pauli equation reads:

Pauli equation (general)

 

Here   are the Pauli operators collected into a vector for convenience, and   is the momentum operator. The state of the system,   (written in Dirac notation), can be considered as a two-component spinor wavefunction, or a column vector (after choice of basis):

 .

The Hamiltonian operator is a 2 × 2 matrix because of the Pauli operators.

 

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   where   is the kinetic momentum, while in the presence of an electromagnetic field it involves the minimal coupling  , where now   is the kinetic momentum and   is the canonical momentum.

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

 

Note that unlike a vector, the differential operator   has non-zero cross product with itself. This can be seen by considering the cross product applied to a scalar function  :

 

where   is the magnetic field.

For the full Pauli equation, one then obtains[2]

Pauli equation (standard form)

 

Weak magnetic fieldsEdit

For the case of where the magnetic field is constant and homogenous, one may expand   using the symmetric gauge  , where   is the position operator. We obtain

 

where   is the particle angular momentum and we neglected terms in the magnetic field squared  . Therefore we obtain

Pauli equation (weak magnetic fields)

 


where   is the spin of the particle. The factor 2 in front of the spin is known as the Dirac g-factor. The term in  , is of the form   which is the usual interaction between a magnetic moment   and a magnetic field, like in the Zeeman effect.

For an electron of charge   in an isotropic constant magnetic field, one can further reduce the equation using the total angular momentum   and Wigner-Eckart theorem. Thus we find

 

where   is the Bohr magneton and   is the magnetic quantum number related to  . The term   is known as the Landé g-factor, and is given here by

 [a]

where   is the orbital quantum number related to   and   is the total orbital quantum number related to  .

From Dirac equationEdit

The Pauli equation is the non-relativistic limit of Dirac equation, the relativistic quantum equation of motion for particles spin-½.[3]

DerivationEdit

Dirac equation can be written as:

 ,

where   and   are two-component spinor, forming a bispinor.

Using the following ansatz:

 ,

with two new spinors  ,the equation becomes

 .

In the non-relativistic limit,   and the kinetic and electrostatic energies are small with respect to the rest energy  .

Thus

 

Inserted in the upper component of Dirac equation, we find Pauli equation (general form):

 

Pauli couplingEdit

Pauli's equation is derived by requiring minimal coupling, which provides a g-factor g=2. Most elementary particles have anomalous g-factors, different from 2. In the domain of relativistic quantum field theory, one defines a non-minimal coupling, sometimes called Pauli coupling, in order to add an anomalous factor

 

where   is the four-momentum operator,   if the electromagnetic four-potential,   is the anomalous magnetic dipole moment,   is electromagnetic tensor, and   are the Lorentzian spin matrices and the commutator of the gamma matrices  .[4][5] In the context, of non-relativistic quantum mechanics, instead of working with Schrödinger equation, Pauli coupling is equivalent to use Pauli equation (or to postulate Zeeman energy) for any g-factor.

See alsoEdit

FootnotesEdit

  1. ^ The formula used here is for a particle with spin ½, with a g-factor   and orbital g-factor  .

ReferencesEdit

  1. ^ Pauli, Wolfgang (1927). "Zur Quantenmechanik des magnetischen Elektrons". Zeitschrift für Physik (in German). 43 (9–10): 601–623. Bibcode:1927ZPhy...43..601P. doi:10.1007/BF01397326. ISSN 0044-3328. S2CID 128228729.
  2. ^ Bransden, BH; Joachain, CJ (1983). Physics of Atoms and Molecules (1st ed.). Prentice Hall. p. 638–638. ISBN 0-582-44401-2.
  3. ^ Greiner, Walter (2012-12-06). Relativistic Quantum Mechanics: Wave Equations. Springer. ISBN 978-3-642-88082-7.
  4. ^ Das, Ashok (2008). Lectures on Quantum Field Theory. World Scientific. ISBN 978-981-283-287-0.
  5. ^ Barut, A. O.; McEwan, J. (January 1986). "The four states of the Massless neutrino with pauli coupling by Spin-Gauge invariance". Letters in Mathematical Physics. 11 (1): 67–72. Bibcode:1986LMaPh..11...67B. doi:10.1007/BF00417466. ISSN 0377-9017. S2CID 120901078.

BooksEdit

  • 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.