Lévy-Leblond equation explained

In quantum mechanics, the Lévy-Leblond equation describes the dynamics of a spin-1/2 particle. It is a linearized version of the Schrödinger equation and of the Pauli equation. It was derived by French physicist Jean-Marc Lévy-Leblond in 1967.^[1]

Lévy-Leblond equation was obtained under similar heuristic derivations as the Dirac equation, but contrary to the latter, Lévy-Leblond equation is not relativistic. As both equations recover the electron gyromagnetic ratio, it is suggested that spin is not necessarily a relativistic phenomenon.

Equation

For a nonrelativistic spin-1/2 particle of mass m, a representation of the time-independent Lévy-Leblond equation reads:

\left\{\begin{matrix} E\psi + (\boldsymbol \sigma ⋅ p c)\chi =0 \\ (\boldsymbol \sigma ⋅ pc )\psi + 2 mc ^{2\chi=0\end{matrix} \right.}

where c is the speed of light, E is the nonrelativistic particle energy,

p = - i\hbar \nabla is the momentum operator, and \boldsymbol \sigma = (\sigma _{x,\sigma} _{y,\sigma} _{z)} is the vector of Pauli matrices, which is proportional to the spin operator S=\tfrac12\hbar \boldsymbol \sigma . Here \psi,\chi are two components functions (spinors) describing the wave function of the particle.

By minimal coupling, the equation can be modified to account for the presence of an electromagnetic field,

\left\{\begin{matrix} (E - q V)\psi + [\boldsymbol \sigma ⋅ ( p-q A)c]\chi=0 \\ {[\boldsymbol \sigma ⋅ ( p-q A ) c] } \psi + 2 mc ^{2\chi} = 0 \end{matrix}\right.

where q is the electric charge of the particle. V is the electric potential, and A is the magnetic vector potential. This equation is linear in its spatial derivatives.

Relation to spin

In 1928, Paul Dirac linearized the relativistic dispersion relation and obtained Dirac equation, described by a bispinor. This equation can be decoupled into two spinors in the non-relativistic limit, leading to predict the electron magnetic moment with a gyromagnetic ratio $g=2$. The success of Dirac theory has led to some textbooks to erroneously claim that spin is necessarily a relativistic phenomena.^[2]

Jean-Marc Lévy-Leblond applied the same technique to the non-relativistic energy relation showing that the same prediction of $g=2$ can be obtained. Actually to derive the Pauli equation from Dirac equation one has to pass by Lévy-Leblond equation. Spin is then a result of quantum mechanics and linearization of the equations but not necessarily a relativistic effect.^[3]

Lévy-Leblond equation is Galilean invariant. This equation demonstrates that one does not need the full Poincaré group to explain the spin 1/2. In the classical limit where $c\; \backslash to\; \backslash infty$, quantum mechanics under the Galilean transformation group are enough. Similarly, one can construct classical linear equation for any arbitrary spin.^[4] Under the same idea one can construct equations for Galilean electromagnetism.

Relation to other equations

Schrödinger's and Pauli's equation

See main article: Pauli equation. Taking the second line of Lévy-Leblond equation and inserting it back into the first line, one obtains through the algebra of the Pauli matrices, that

(1)/(2 m) (\boldsymbol \sigma ⋅ p) ^{2 \psi - E\psi = \left[ (1)/(2 m)} p ^{2-E\right]\psi=0} ,

which is the Schrödinger equation for a two-valued spinor. Note that solving for

\chi also returns another Schrödinger's equation. Pauli's expression for spin- particle in an electromagnetic field can be recovered by minimal coupling:\left\{ (1)/(2 m) [\boldsymbol \sigma ⋅ ( p-q A)] ^{2+qV\right\}\psi=E\psi} .

While Lévy-Leblond is linear in its derivatives, Pauli's and Schrödinger's equations are quadratic in the spatial derivatives.

Dirac equation

See main article: Dirac equation. Dirac equation can be written as:

\left\{\begin{matrix} ( l{E}-mc ^{2)\psi+(\boldsymbol} \sigma ⋅ p c)\chi =0 \\ (\boldsymbol \sigma ⋅ pc )\psi + ( l{E}+ mc ^{2)\chi=0\end{matrix} \right.} where $\backslash mathcal$ is the total relativistic energy. In the non-relativistic limit, $E\backslash ll\; mc^2$ and $\backslash mathcal\backslash approx\; mc^2+E+\backslash cdots$ one recovers, Lévy-Leblond equations.

Heuristic derivation

Similar to the historical derivation of Dirac equation by Paul Dirac, one can try to linearize the non-relativistic dispersion relation $E=\backslash frac$. We want two operators and linear in $\backslash mathbf\; p$ (spatial derivatives) and E, like^[5]

\left\{\begin{matrix}\Theta\Psi = [AE + B ⋅ p c +2 mc ^{2C]\Psi=0} \\ \Theta'\Psi = [A'E + B' ⋅ p c +2 mc ^{2C'} ]\Psi =0 \end{matrix} \right. for some $A,A\text{'},\; \backslash mathbf\; B=(B\_x,B\_y,B\_z),\backslash mathbf\; B\text{'}=(B\_x\text{'},B\_y\text{'},B\_z\text{'}),\; C,C\text{'}$, such that, their product recovers the classical dispersion relation, that is(1)/(2 mc ^{2}) \Theta'\Theta = E - (p ^{2})/(2 m) ,where the factor is arbitrary an it is just there for normalization. By doing carrying out the product, one find that there is not solution if $A,A\text{'},B\_i,\; B\_i\text{'},\; C,C\text{'}$ are one dimensional constants. The lowest dimension where there is a solution is 4. Then $A,A\text{'},\; \backslash mathbf\; B,\; \backslash mathbf\; B\text{'},\; C,C\text{'}$ are matrices that must satisfy the following relations:\left\{\begin{matrix} A'A =0 \\ C'C =0 \\ A'B _{i+B} _{i'A=0\\ C'B} _{i+B} _{i'C=0\\ A'C+C'A=I} _{4\\ B} _{i'B} _{j+B} _{j'B} _{i=-2\delta} _{ij} \end{matrix}\right.

these relations can be rearranged to involve the gamma matrices from Clifford algebra.^[6] $I\_N$ is the Identity matrix of dimension N. One possible representation is

A = A' = \begin{pmatrix} 0 & 0 \ I _{2} & 0 \end{pmatrix}, B _{i=-B} _{i'=\begin{pmatrix}\sigma} _{i} & 0 \ 0 & \sigma _{i\end{pmatrix},} C = C' = \begin{pmatrix} 0 & I _{2} \ 0 & 0 \end{pmatrix} ,

such that $\backslash Theta\backslash Psi=0$, with $\backslash Psi=(\backslash psi,\; \backslash chi)$, returns Lévy-Leblond equation. Other representations can be chosen leading to equivalent equations with different signs or phases.

References

1. Lévy-Leblond . Jean-Marc . 1967-12-01 . Nonrelativistic particles and wave equations . Communications in Mathematical Physics . en . 6 . 4 . 286–311 . 10.1007/BF01646020 . 1432-0916.
2. Book: Jammer, Max . The Conceptual Development of Quantum Mechanics . 1966 . McGraw-Hill . en.
3. Book: Feynman, Richard P. . Quantum Electrodynamics . 2018-02-19 . CRC Press . 978-0-429-97287-4 . en.
4. Hurley . William J. . 1971-05-15 . Nonrelativistic Quantum Mechanics for Particles with Arbitrary Spin . Physical Review D . en . 3 . 10 . 2339–2347 . 10.1103/PhysRevD.3.2339 . 0556-2821.
5. Book: Greiner, Walter . Quantum Mechanics: An Introduction . 2000-10-04 . Springer Science & Business Media . 978-3-540-67458-0 . en.
6. Wilkes . James M . 2020-05-01 . The Pauli and Lévy-Leblond equations, and the spin current density . European Journal of Physics . 41 . 3 . 035402 . 10.1088/1361-6404/ab7495 . 0143-0807. 1908.03276 .