Nonrelativistic Limit of the Dirac Equation

For consistency, the Dirac equation should reduce to the Schrödinger wave equation for nonrelativistic quantum mechanics in the nonrelativistic limit. We examine the nonrelativistic limit for the case of a positive-energy Dirac particle in the presence of an electromagnetic potential.

The Dirac equation with the electromagnetic potential can be written as

$$i\hbar\frac{\partial\psi}{\partial t} = \left[ c \vec{\alpha... ...\frac{e}{c} \vec{A} \right) + eA_0 + \beta mc^2 \right] \psi ,$$ (5.27)

where $\hat{p} = -i\hbar\vec{\nabla}$ is the momentum operator in 3-space.

Consider a two-component representation of, $\psi(\vec{x},t)=\left[\begin{array}{c} \phi(\vec{x},t) \\ \chi(\vec{x},t) \end{array} \right]$, where the four-component spinor $\psi$ is decomposed into two two-component spinors $\phi$ and $\chi$. Substitution of this form into the Dirac equation gives

$$i\hbar\frac{\partial}{\partial t} \left( \begin{array}{c} \p... ...^2 \left( \begin{array}{r} \phi \\ -\chi \end{array}\right) .$$ (5.28)

If the rest energy, $mc^2$, is the largest occurring energy, our two-component solution is approximately

$$\left[ \begin{array}{c} \phi(\vec{x},t) \\ \chi(\vec{x},t) \... ... \\ \chi_0(\vec{x},t) \end{array} \right] e^{-imc^2t/\hbar},$$ (5.29)

where $\phi_0(\vec{x},t)$ and $\chi_0(\vec{x},t)$ are slowly varying functions of time. Substitution of this nonrelativistic solution into the Dirac equation now gives

$$i\hbar\frac{\partial}{\partial t} \left( \begin{array}{c} \p... ...2mc^2 \left( \begin{array}{c} 0 \\ \chi_0 \end{array}\right) .$$ (5.30)

If the kinetic energy is small compared to the rest energy, $\chi_0$ is a slowly varying function of time and

$$\left\vert i\hbar\frac{\partial\chi_0}{\partial t}\right\vert \ll \vert mc^2\chi_0\vert .$$ (5.31)

If the electrostatic potential is weak, the potential energy is small compared to the rest energy:

$$\vert eA_0\chi_0\vert\ll \vert mc^2\chi_0\vert .$$ (5.32)

With the last two approximations, the lower component in equation 5.30 becomes

$$0 \approx c\vec{\sigma}\cdot \left( \hat{p} -\frac{e}{c}\vec{A} \right)\phi_0 - 2mc^2\chi_0,$$ (5.33)

$$\chi_0 = \frac{\vec{\sigma}\cdot \left( \hat{p} -\frac{e}{c}\vec{A} \right)}{2mc}\phi_0.$$ (5.34)

The lower component, $\chi$, is often referred to as the ``small'' component of the wave function $\psi$, relative to the ``large'' component, $\phi$. The small component is approximately $v/c$ less than the large component in the nonrelativistic limit.

The upper component of equation 5.30 becomes

$$i\hbar\frac{\partial\phi_0}{\partial t} = \frac{ \vec{\sigma... ... \hat{p} -\frac{e}{c}\vec{A} \right) }{2m}\phi_0 + eA_0\phi_0.$$ (5.35)

Using the identity 2.24 and realizing that operating on a wave function $\psi$ ,

$$\vec{\nabla}\times\vec{A}\psi + \vec{A}\times(\vec{\nabla}\psi) = (\vec{\nabla}\times\vec{A})\psi$$ (5.36)

we have

$$\vec{\sigma}\cdot\left(\hat{p}-\frac{e}{c}\vec{A}\right) \vec{\sigma}\cdot\left(\hat{p}-\frac{e}{c}\vec{A}\right) = \left( \hat{p} -\frac{e}{c}\vec{A} \right)^2+ i\vec{\sigma}\cdot\... ... \right)\times \left(-i\hbar\vec{\nabla} - \frac{e}{c}\vec{A} \right) \right] ,$$

$$= \left( \hat{p} -\frac{e}{c}\vec{A} \right)^2 - \frac{e\hbar}{c} \vec{\sigma}\cdot(\vec{\nabla}\times\vec{A}) ,$$

$$= \left( \hat{p} - \frac{e}{c}\vec{A} \right)^2 - \frac{e\hbar}{c} \vec{\sigma}\cdot\vec{B} .$$ (5.37)

Therefore the Dirac equation becomes

$$i\hbar\frac{\partial\phi_0}{\partial t} = \left[ \frac{\left... ...c{e\hbar}{2mc} \vec{\sigma}\cdot\vec{B} + eA_0 \right]\phi_0 .$$ (5.38)

This is the two-component Pauli equation for the theory of spin in nonrelativistic quantum mechanics. The two components of $\phi_0$ describe the spin degrees of freedom.

The Pauli equation, and thus the Dirac equation, yields the correct gyromagnetic factor of $g=2$ for a free electron. To see this, we turn on a weak, homogeneous magnetic field

$$\vec{A}=\frac{1}{2}\vec{B}\times\vec{x}.$$ (5.39)

Substitution of this weak field into the operator in the Pauli equation gives

$$\left( \hat{p} -\frac{e}{c}\vec{A} \right)^2 = \left( \hat{p} - \frac{e}{2c} \vec{B}\times\vec{x} \right)^2 \approx \hat{p}^2 -\frac{e}{c}(\vec{B}\times\vec{x})\cdot\hat{p} ,$$

$$= \hat{p}^2 -\frac{e}{c}\vec{B}\cdot\hat{L} ,$$ (5.40)

where $\hat{L}=\vec{x}\times\hat{p}$ is the operator of orbital angular momentum. We have neglected the term quadratic in $\vec{A}$.

Defining $\hat{s}=\frac{1}{2}\hbar\vec{\sigma}$ as the spin operator, the Pauli equation now becomes

$$i\hbar\frac{\partial\phi_0}{\partial t} = \left[ \frac{\hat{p}^2}{2m} -\frac{e}{2mc} \hat{L}\cdot\vec{B} -\frac{e\hbar}{2mc} \vec{\sigma}\cdot\vec{B} + eA_0 \right] \phi_0 ,$$

$$= \left[ \frac{\hat{p}^2}{2m} -\frac{e}{2mc} \hat{L}\cdot\vec{B} -\frac{e}{mc} \hat{s}\cdot\vec{B} + eA_0 \right] \phi_0 ,$$

$$= \left[ \frac{\hat{p}^2}{2m} -\frac{e}{2mc} (\hat{L}+2\hat{s})\cdot\vec{B} + eA_0 \right] \phi_0.$$ (5.41)

Generally, the intrinsic magnetic moment $\vec{\mu}$ is related to the spin vector $\vec{s}$ by

$$\vec{\mu} = g \mu_B \vec{s} ,$$ (5.42)

where $\mu_B = e\hbar/2mc$ is the Bohr magneton and $g$ is called the Lande $g$-factor. $g\mu_B = \mu/s$ is the gyromagnetic ratio, i.e. the ratio of the magnetic to mechanical moment. Thus the Dirac theory predicts for particles of spin-1/2 that

$$g = 2 .$$ (5.43)

Predicting the correct gyromagnetic ratio of the electron is one of the great triumphs of the Dirac equation.

Experimental values for the $g$-factors for real spin-1/2 particles have been measured:

$$\textrm{electron} \quad g_{\mathrm{exp}} = 2\left( 1 + \frac{\alpha}{2\pi} + \cdots \right) ,$$ (5.44)

$$\textrm{proton} \quad g_{\mathrm{exp}} = 2(1 + 1.79) ,$$ (5.45)

$$\textrm{neutron} \quad g_{\mathrm{exp}} = 2(0 -1.91) .$$ (5.46)

In the case of the proton and neutron, a microscopic view of these systems reveals that they are far from being simple point-like spin-1/2 structures. In the case of the electron, there exists no substructure. As far as we know the electron really is a point particle. However, the electron can fragment into an e-$\gamma$ system which yields a modification to the $g$-factor, but only at order $\alpha$. From experiment, $g=2.00232$ and the difference from $g=2$ can be account for by higher-order contributions. This is a great triumph for quantum electrodynamics and is one of the reasons its the ``best'' theory we have.

Since the Pauli equation is a nonrelativistic wave equation for a spin-1/2 particle, the Dirac equation describes a particle with spin-1/2 both at low and high velocities. The spin comes into the Dirac theory when the second-order differential Klein-Gordon equation is linearized.