Pauli equation

Last updated

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]



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 fields

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

[lower-alpha 1]

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

From Dirac equation

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


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 .


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

From a Foldy-Wouthuysen transformation

One can also rigorously derive Pauli equation, starting from Dirac equation in an external field and performing a Foldy-Wouthuysen transformation. [3]

Pauli coupling

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 an arbitrary g-factor.

See also


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

Related Research Articles

In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's energy spectrum or its set of energy eigenvalues, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

Dirac equation Relativistic quantum mechanical wave equation

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-½ massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.

In 1851, George Gabriel Stokes derived an expression, now known as Stokes law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain combination of two Weyl spinors, specifically, a bispinor that transforms "spinorially" under the action of the Lorentz group.

In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. One GL-type superconductor is the famous YBCO, and generally all Cuprates.

Relativistic wave equations Wave equations respecting special and general relativity

In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the context of quantum field theory (QFT), the equations determine the dynamics of quantum fields. The solutions to the equations, universally denoted as ψ or Ψ, are referred to as "wave functions" in the context of RQM, and "fields" in the context of QFT. The equations themselves are called "wave equations" or "field equations", because they have the mathematical form of a wave equation or are generated from a Lagrangian density and the field-theoretic Euler–Lagrange equations.

Rabi cycle Quantum mechanical phenomenom

In physics, the Rabi cycle is the cyclic behaviour of a two-level quantum system in the presence of an oscillatory driving field. A great variety of physical processes belonging to the areas of quantum computing, condensed matter, atomic and molecular physics, and nuclear and particle physics can be conveniently studied in terms of two-level quantum mechanical systems, and exhibit Rabi flopping when coupled to an oscillatory driving field. The effect is important in quantum optics, magnetic resonance and quantum computing, and is named after Isidor Isaac Rabi.

In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely.

In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron caused by its intrinsic properties of spin and electric charge. The value of the electron magnetic moment is approximately −9.284764×10−24 J/T. The electron magnetic moment has been measured to an accuracy of 7.6 parts in 1013.

In differential geometry, the four-gradient is the four-vector analogue of the gradient from vector calculus.

Two-state quantum system Quantum system that can be measured as one of two values; sought for "quantum bits" in quantum computing

In quantum mechanics, a two-state system is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is:

Maxwell stress tensor

The Maxwell stress tensor is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impossibly difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand.

The Gross–Pitaevskii equation describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model.

In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4). According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of special relativity and relativistic spacetime.

The intent of this article is to highlight the important points of the derivation of the Navier–Stokes equations as well as its application and formulation for different families of fluids.

In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light c, and can accommodate massless particles. The theory has application in high energy physics, particle physics and accelerator physics, as well as atomic physics, chemistry and condensed matter physics. Non-relativistic quantum mechanics refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators. Relativistic quantum mechanics (RQM) is quantum mechanics applied with special relativity. Although the earlier formulations, like the Schrödinger picture and Heisenberg picture were originally formulated in a non-relativistic background, a few of them also work with special relativity.

Spin (physics) Intrinsic form of angular momentum as a property of quantum particles

In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.

Weyl equation Relativistic wave equation describing massless fermions

In physics, particularly a quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three possible types of elementary fermions, the other two being the Dirac and the Majorana fermions.

In mathematical physics, the Gordon decomposition of the Dirac current is a splitting of the charge or particle-number current into a part that arises from the motion of the center of mass of the particles and a part that arises from gradients of the spin density. It makes explicit use of the Dirac equation and so it applies only to "on-shell" solutions of the Dirac equation.


  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. 1 2 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.