Phy5646/Problem one Dim Dirac eq

Question.

Consider the Dirac equation in one dimension

$\mathbf {H} \psi =i\hbar {\frac {\partial \psi }{\partial t}}$

where

$\mathbf {H} =c\alpha p_{z}+\beta mc^{2}+V(z)$,

$\alpha =\left({\begin{array}{cc}0&\sigma _{3}\\\sigma _{3}&0\end{array}}\right)$,

$\sigma _{3}=\left({\begin{array}{cc}1&0\\0&-1\end{array}}\right)$,

$\beta =\left({\begin{array}{cc}I&0\\0&-I\end{array}}\right)$,

$\mathbf {I}$ being the 2 X 2 unit matrix.

(a) Show that $\sigma =\left({\begin{array}{cc}\sigma _{3}&0\\0&\sigma _{3}\end{array}}\right)$ commutes with $\mathbf {H}$.

(b) Use the result of (a) to show that the one dimensional Dirac equation can be written as two coupled first order differential equations.

Solution.

(a) As

$[\sigma ,\alpha ]=[\left({\begin{array}{cc}\sigma _{3}&0\\0&\sigma _{3}\end{array}}\right),\left({\begin{array}{cc}0&\sigma _{3}\\\sigma _{3}&0\end{array}}\right)]=0$,

$[\sigma ,\beta ]=[\left({\begin{array}{cc}\sigma _{3}&0\\0&\sigma _{3}\end{array}}\right),\left({\begin{array}{cc}I&0\\0&-I\end{array}}\right)]=0$,

we have

$\mathbf {[} \alpha ,H]=[\alpha ,c\alpha p_{z}+\beta mc^{2}+V]=c[\sigma ,\alpha ]p_{z}+[\sigma ,\beta ]mc^{2}=0$.

(b) As $\mathbf {[} \sigma ,H]=0$, $\mathbf {\sigma }$ and $\mathbf {H}$ have common eigenfunctions. $\mathbf {\sigma }$ is a diagonal matrix. Let its eigenfunction be $\left({\begin{array}{cc}\psi _{1}\\\psi _{2}\\\psi _{3}\\\psi _{4}\end{array}}\right)$. As

$\sigma \left({\begin{array}{cc}\psi _{1}\\\psi _{2}\\\psi _{3}\\\psi _{4}\end{array}}\right)=\left({\begin{array}{cc}\psi _{1}\\-\psi _{2}\\\psi _{3}\\-\psi _{4}\end{array}}\right)=\left({\begin{array}{cc}\psi _{1}\\0\\\psi _{3}\\0\end{array}}\right)-\left({\begin{array}{cc}0\\\psi _{2}\\0\\\psi _{4}\end{array}}\right)$,

$\mathbf {\sigma }$ has eigenfunctions $\left({\begin{array}{cc}\psi _{1}\\0\\\psi _{3}\\0\end{array}}\right)$ and $\left({\begin{array}{cc}0\\\psi _{2}\\0\\\psi _{4}\end{array}}\right)$ with eigenvalues $+1$ and $-1$, respectively.

Substituting these in the Dirac equation, we obtain

$(-i\hbar c{\frac {\partial }{\partial z}}+V)\left({\begin{array}{cc}\psi _{3}\\0\\\psi _{1}\\0\end{array}}\right)+mc^{2}\left({\begin{array}{cc}\psi _{1}\\0\\-\psi _{3}\\0\end{array}}\right)=i\hbar {\frac {\partial }{\partial t}}\left({\begin{array}{cc}\psi _{1}\\0\\\psi _{3}\\0\end{array}}\right)$,

$(-i\hbar c{\frac {\partial }{\partial z}}+V)\left({\begin{array}{cc}0\\-\psi _{4}\\0\\-\psi _{2}\end{array}}\right)+mc^{2}\left({\begin{array}{cc}0\\\psi _{2}\\0\\-\psi _{4}\end{array}}\right)=i\hbar {\frac {\partial }{\partial t}}\left({\begin{array}{cc}0\\\psi _{2}\\0\\\psi _{4}\end{array}}\right)$.

Each of these represents two coupled differential equations. However, the two sets of equations become identical if we let $\mathbf {\psi } _{3}->-\psi _{4}$ and $\mathbf {\psi } _{1}->\psi _{2}$. Thus the one-dimensional Dirac equation can be written as two coupled first order differential equations.