Lévy-Leblond equation

Template:Short description 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.

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:

$\{\begin{array}{c}E\psi +(\mathit{\sigma }\cdot 𝐩c)\chi =0\\ (\mathit{\sigma }\cdot 𝐩c)\psi +2m{c}^2\chi =0\end{array}$

where c is the speed of light, E is the nonrelativistic particle energy, $𝐩=-i\hslash \mathrm{\nabla }$ is the momentum operator, and $\mathit{\sigma }=({\sigma }_x,{\sigma }_y,{\sigma }_z)$ is the vector of Pauli matrices, which is proportional to the spin operator $𝐒=\frac{1}{2}\hslash \mathit{\sigma }$. 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,

$\{\begin{array}{c}(E-qV)\psi +[\mathit{\sigma }\cdot (𝐩-q𝐀)c]\chi =0\\ [\mathit{\sigma }\cdot (𝐩-q𝐀)c]\psi +2m{c}^2\chi =0\end{array}$

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.

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.

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 $c\to \mathrm{\infty }$, quantum mechanics under the Galilean transformation group are enough. Similarly, one can construct a non-relativistic linear equation for any arbitrary spin. Under the same idea one can construct equations for Galilean electromagnetism.

Relation to other equations

Schrödinger's and Pauli's equation

Template:Main 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

$\frac{1}{2m}(\mathit{\sigma }\cdot 𝐩{)}^2\psi -E\psi =[\frac{1}{2m}{𝐩}^2-E]\psi =0$,

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-Template:Frac particle in an electromagnetic field can be recovered by minimal coupling:

$\{\frac{1}{2m}[\mathit{\sigma }\cdot (𝐩-q𝐀){]}^2+qV\}\psi =E\psi$.

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

Dirac equation

Template:Main Dirac equation can be written as:

$\{\begin{array}{c}(ℰ-m{c}^2)\psi +(\mathit{\sigma }\cdot 𝐩c)\chi =0\\ (\mathit{\sigma }\cdot 𝐩c)\psi +(ℰ+m{c}^2)\chi =0\end{array}$

where $ℰ$ is the total relativistic energy. In the non-relativistic limit, $E\ll m{c}^2$ and $ℰ\approx m{c}^2+E+\cdots$ 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 $E=\frac{{𝐩}^2}{2m}$. We want two operators Template:Math and Template:Math linear in $𝐩$ (spatial derivatives) and E, like

$\{\begin{array}{c}\mathrm{\Theta }\mathrm{\Psi }=[AE+𝐁\cdot 𝐩c+2m{c}^{2}C]\mathrm{\Psi }=0\\ {\mathrm{\Theta }}^{\prime }\mathrm{\Psi }=[A^{\prime }E+{𝐁}^{\prime }\cdot 𝐩c+2m{c}^2{C}^{\prime }]\mathrm{\Psi }=0\end{array}$

for some $A,A^{\prime },𝐁=(B_x,B_y,B_z),{𝐁}^{\prime }=({B_x}^{\prime },{B_y}^{\prime },{B_z}^{\prime }),C,C^{\prime }$, such that their product recovers the classical dispersion relation, that is

$\frac{1}{2m{c}^2}{\mathrm{\Theta }}^{\prime }\mathrm{\Theta }=E-\frac{{𝐩}^2}{2m}$,

where the factor Template:Math is arbitrary an it is just there for normalization. By carrying out the product, one find that there is no solution if $A,A^{\prime },B_i,{B_i}^{\prime },C,C^{\prime }$ are one dimensional constants. The lowest dimension where there is a solution is 4. Then $A,A^{\prime },𝐁,{𝐁}^{\prime },C,C^{\prime }$ are matrices that must satisfy the following relations:

$\{\begin{array}{c}{A}^{\prime }A=0\\ C^{\prime }C=0\\ A^{\prime }{B}_i+{B_i}^{\prime }A=0\\ C^{\prime }{B}_i+{B_i}^{\prime }C=0\\ A^{\prime }C+C^{\prime }A=I_4\\ {B_i}^{\prime }{B}_j+{B_j}^{\prime }{B}_i=-2{\delta }_{ij}\end{array}$

these relations can be rearranged to involve the gamma matrices from Clifford algebra. $I_N$ is the Identity matrix of dimension N. One possible representation is

$A=A^{\prime }=\left(\begin{array}{cc}0& 0\\ I_2& 0\end{array}\right),B_i=-{B_i}^{\prime }=\left(\begin{array}{cc}{\sigma }_i& 0\\ 0& {\sigma }_i\end{array}\right),C=C^{\prime }=\left(\begin{array}{cc}0& I_2\\ 0& 0\end{array}\right)$,

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