Pauli equation explained
In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-1/2 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] In its linearized form it is known as Lévy-Leblond equation.
Equation
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:
Here
\boldsymbol{\sigma}=(\sigmax,\sigmay,\sigmaz)
are the
Pauli operators collected into a vector for convenience, and
} = -i\hbar \nabla is the
momentum operator in position representation. 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):
|\psi\rangle=\psi+|d\uparrow\rangle+\psi-|d\downarrow\rangle\stackrel{ ⋅ }{=}\begin{bmatrix}\psi+\\
\psi-
\end{bmatrix}
.
The Hamiltonian operator is a 2 × 2 matrix because of the Pauli operators.
\hat{H}=
\left[\boldsymbol{\sigma} ⋅ (\hat{p
} - q \mathbf) \right]^2 + q \phi
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:
(\boldsymbol{\sigma} ⋅ a)(\boldsymbol{\sigma} ⋅ b)=a ⋅ b+i\boldsymbol{\sigma} ⋅ \left(a x b\right)
Note that unlike a vector, the differential operator
} - q\mathbf = -i \hbar \nabla - q \mathbf has non-zero cross product with itself. This can be seen by considering the cross product applied to a scalar function
:
} - q\mathbf\right) \times \left(\mathbf - q\mathbf\right)\right]\psi = -q \left[\mathbf{\hat{p}} \times \left(\mathbf{A}\psi\right) + \mathbf{A} \times \left(\mathbf{\hat{p}}\psi\right)\right] = i q \hbar \left[\nabla \times \left(\mathbf{A}\psi\right) + \mathbf{A} \times \left(\nabla\psi\right)\right] = i q \hbar \left[\psi\left(\nabla \times \mathbf{A}\right) - \mathbf{A} \times \left(\nabla\psi\right) + \mathbf{A} \times \left(\nabla\psi\right)\right] = i q \hbar \mathbf \psi
where
is the magnetic field.
For the full Pauli equation, one then obtains[2]
for which only a few analytic results are known, e.g., in the context of Landau quantization with homogenous magnetic fields or for an idealized, Coulomb-like, inhomogeneous magnetic field.[3]
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 and A is now an operator. We obtain
(\hat{p}-q\hat{A})2=|\hat{p
}|^ - q(\mathbf\times\mathbf \hat)\cdot \mathbf +\fracq^2\left(|\mathbf|^2|\mathbf|^2-|\mathbf\cdot\mathbf|^2\right) \approx \mathbf^ - q\mathbf \hat\cdot\mathbf B\,,
where is the particle angular momentum operator and we neglected terms in the magnetic field squared . Therefore, we obtain
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
+\mu\rmgJmj|B|-e\phi\right]|\psi\rangle=i\hbar
|\psi\rangle
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
where
is the
orbital quantum number related to
and
is the total orbital quantum number related to
.
From Dirac equation
The Pauli equation can be inferred from the non-relativistic limit of the Dirac equation, which is the relativistic quantum equation of motion for spin-1/2 particles.[4]
Derivation
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
, leading to the
Lévy-Leblond equation.
[5] Thus
Inserted in the upper component of Dirac equation, we find Pauli equation (general form):
From a Foldy–Wouthuysen transformation
The rigorous derivation of the Pauli equation follows from Dirac equation in an external field and performing a Foldy–Wouthuysen transformation considering terms up to order
. Similarly, higher order corrections to the Pauli equation can be determined giving rise to
spin-orbit and
Darwin interaction terms, when expanding up to order
instead.
[6] 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
\gamma\mup\mu\to\gamma\mu
A\mu+a\sigma\mu\nuF\mu\nu
where
is the
four-momentum operator,
is the
electromagnetic four-potential,
is proportional to the
anomalous magnetic dipole moment,
F\mu\nu=\partial\muA\nu-\partial\nuA\mu
is the
electromagnetic tensor, and
are the Lorentzian spin matrices and the commutator of the
gamma matrices
.
[7] [8] In the context of non-relativistic quantum mechanics, instead of working with the Schrödinger equation, Pauli coupling is equivalent to using the Pauli equation (or postulating
Zeeman energy) for an arbitrary
g-factor.
See also
References
Books
- Book: Schwabl, Franz. Quantenmechanik I . Springer . 2004 . 978-3540431060.
- Book: Schwabl, Franz. Quantenmechanik für Fortgeschrittene . Springer . 2005 . 978-3540259046.
- Book: Claude Cohen-Tannoudji . Bernard Diu . Frank Laloe . Quantum Mechanics 2. Wiley, J . 2006 . 978-0471569527.
Notes and References
- Pauli. Wolfgang. Wolfgang Pauli. 1927. Zur Quantenmechanik des magnetischen Elektrons. Zeitschrift für Physik. de. 43. 9–10. 601–623. 10.1007/BF01397326. 1927ZPhy...43..601P. 128228729. 0044-3328.
- Book: Physics of Atoms and Molecules. Bransden, BH. Joachain, CJ. 1983. Prentice Hall. 1st. 638. 0-582-44401-2.
- Sidler . Dominik . Rokaj . Vasil . Ruggenthaler . Michael . Rubio . Angel . 2022-10-26 . Class of distorted Landau levels and Hall phases in a two-dimensional electron gas subject to an inhomogeneous magnetic field . Physical Review Research . en . 4 . 4 . 043059 . 10.1103/PhysRevResearch.4.043059 . 2022PhRvR...4d3059S . 253175195 . 2643-1564. 10810/58724 . free .
- Book: Greiner, Walter. Relativistic Quantum Mechanics: Wave Equations. 2012-12-06. Springer. 978-3-642-88082-7. en.
- Book: Greiner, Walter . Quantum Mechanics: An Introduction . 2000-10-04 . Springer Science & Business Media . 978-3-540-67458-0 . en.
- Fröhlich . Jürg . Studer . Urban M. . 1993-07-01 . Gauge invariance and current algebra in nonrelativistic many-body theory . Reviews of Modern Physics . en . 65 . 3 . 733–802 . 10.1103/RevModPhys.65.733 . 1993RvMP...65..733F . 0034-6861.
- Book: Das, Ashok. Lectures on Quantum Field Theory. 2008. World Scientific. 978-981-283-287-0. en.
- 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. en. 11. 1. 67–72. 10.1007/BF00417466. 1986LMaPh..11...67B. 120901078. 0377-9017.