Phy5646/Problem one Dim Dirac eq

Question.

Consider the Dirac equation in one dimension

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf H \psi = i\hbar \frac{\partial \psi}{\partial t}}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf H = c\alpha p_z + \beta mc^2 + V(z)} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha= \left(\begin{array}{cc} 0&\sigma_3\\ \sigma_3&0\end{array}\right)} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_3= \left(\begin{array}{cc} 1&0\\ 0&-1\end{array}\right)} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta= \left(\begin{array}{cc} I&0\\ 0&-I\end{array}\right)} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf I} being the 2 X 2 unit matrix.

(a) Show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma= \left(\begin{array}{cc} \sigma_3&0 \\ 0&\sigma_3\end{array}\right)} commutes with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\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} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf [\alpha, H] = [\alpha, c\alpha p_z + \beta mc^2 + V] = c[\sigma, \alpha]p_z + [\sigma, \beta]mc^2 = 0} .

(b) As Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf [\sigma, H] = 0} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \sigma} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf H} have common eigenfunctions. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \sigma} is a diagonal matrix. Let its eigenfunction be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{array}{cc}\psi_{1}\\\psi_{2}\\\psi_{3}\\\psi_{4}\end{array}\right)} . As

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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)} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \sigma} has eigenfunctions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{array}{cc}\psi_{1}\\0\\\psi_{3}\\0\end{array}\right)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\begin{array}{cc}0\\\psi_{2}\\0\\\psi_{4}\end{array}\right)} with eigenvalues Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1} , respectively.

Substituting these in the Dirac equation, we obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-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)} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \psi_{3} -> -\psi_{4}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \psi_{1} -> \psi_{2}} . Thus the one-dimensional Dirac equation can be written as two coupled first order differential equations.