Dirac equation

How to construct

Starting from the relativistic relation between energy and momentum:

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 E^2=\vec p \; ^{2}c^2+m^2c^4}

or $E=c{\sqrt {p^{2}+m^{2}c^{2}}}$

From this equation we can not directly replace $E,{\vec {p}}$ by the corresponding operators since we don't have the definition for the square root of an operator. Therefore, first we need to linearize this equation as follows:

$E=c{\sqrt {p^{2}+m^{2}c^{2}}}=c{\sqrt {(p_{x}^{2}+p_{y}^{2}+p_{z}^{2})+m^{2}c^{2}}}=c(\alpha _{x}p_{x}+\alpha _{y}p_{y}+\alpha _{z}p_{z})+\beta mc^{2}$

where ${\mathbf {\alpha }}_{x},\alpha _{y},\alpha _{z}$ and ${\mathbf {\beta }}$ are some operators independent of ${\vec {p}}$.

From this it follows that:

$c^{2}(p_{x}^{2}+p_{y}^{2}+p_{z}^{2}+m^{2}c^{2})=[c(\alpha _{x}p_{x}+\alpha _{y}p_{y}+\alpha _{z}p_{z})+\beta mc^{2}].[c(\alpha _{x}p_{x}+\alpha _{y}p_{y}+\alpha _{z}p_{z})+\beta mc^{2}]$

Expanding the right hand side and comparing it with the left hand side, we obtain the following conditions for ${\mathbf {\alpha }}_{x},\alpha _{y},\alpha _{z}$ and ${\mathbf {\beta }}$ :

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 _{i}^2=\beta ^2=1}

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 \bold \alpha_ {i}\alpha_ {j}+\alpha_ {j}\alpha_ {i}=\{\alpha_ {i},\alpha_ {j}\}=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 \bold \alpha_ {i} \beta+\alpha_ {j} \beta=\{\alpha_ {i},\beta\}=0}

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 i=1,2,3} corresponds to 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 x, y, z}

In order to describe both particle (positive energy state) and antiparticle (negative energy state); spin-up state and spin-down state, the wave function must have 4 components and all operators acting on such states correspond to 4 by 4 matrices. Therefore, 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 \bold \alpha _{x},\alpha _{y},\alpha _{z}} 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 \bold \beta} are 4 by 4 matrices. It is convention that these matrices are given as follows (in the form of block matrices for short):

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_{x}=\left(\begin{array}{cc}0& \sigma_{x}\\ \sigma_{x}&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 \qquad \alpha_{y}=\left(\begin{array}{cc}0& \sigma_{y}\\ \sigma_{y}&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 \qquad \alpha_{z}=\left(\begin{array}{cc}0& \sigma_{z}\\ \sigma_{z}&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 \qquad \beta= \left(\begin{array}{cc}1&0\\0&-1\end{array}\right)}

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 \sigma_{x}, \;\sigma_{y}, \;\sigma_{z}} are 2 by 2 Pauli matrices.

Let's define:

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 \vec \alpha=\alpha _{x} \hat x+\alpha _{y} \hat y+\alpha _{z} \hat z}

Then we can write:

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 E=c \vec \alpha \vec p +\beta mc^2}

Substituting all quantities by their corresponding operators, we obtain Dirac equation:

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 \frac {\partial \psi}{\partial t}=H \psi}

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 H=c \vec \alpha \vec p + \beta mc^2}

Dirac equation can also be written explicitly as follows:

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 \frac {\partial \psi_{1}}{\partial t}=c(p_{x}-ip_{y}) \psi _{4}+cp_{z} \psi _{3} + mc^2 \psi _{1}}

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 \frac {\partial \psi_{2}}{\partial t}=c(p_{x}+ip_{y}) \psi _{3}-cp_{z} \psi _{4} + mc^2 \psi _{2}}

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 \frac {\partial \psi_{3}}{\partial t}=c(p_{x}-ip_{y}) \psi _{2}+cp_{z} \psi _{1} - mc^2 \psi _{3}}

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 \frac {\partial \psi_{4}}{\partial t}=c(p_{x}+ip_{y}) \psi _{1}-cp_{z} \psi _{2} - mc^2 \psi _{4}}

In the present of electromagnetic field, Dirac equation becomes:

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 \frac {\partial }{\partial t} -e \phi) \psi = \left [ c \vec \alpha (\frac {\hbar}{i} \vec \nabla - \frac {e}{c} \bold A)+\beta mc^2 \right ]\psi}

Continuity equation

Dirac equation and its adjoint equation:

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 \frac {\partial \psi}{\partial t}=(-i \hbar c \vec \alpha \vec \nabla + mc^2 \beta) \psi}

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 \frac {\partial \psi ^{\dagger}}{\partial t}=(i \hbar c \vec \nabla \psi ^{\dagger} \vec \alpha + mc^2 \psi ^{\dagger} \beta )}

Multiplying Dirac equation by 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 \psi ^{\dagger}} from the left and the adjoint equation by 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 \bold \psi} from the right, we get:

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 \psi ^{\dagger} \frac {\partial \psi}{\partial t}=-i \hbar c \psi ^{\dagger} \vec \alpha \vec \nabla \psi+ mc^2 \psi ^{\dagger} \beta \psi}

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 \frac {\partial \psi ^{\dagger}}{\partial t} \psi=i \hbar c \vec \nabla \psi ^{\dagger} \vec \alpha \psi+ mc^2 \psi ^{\dagger} \beta \psi}

Subtracting one from the other, we get:

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 \left ( \psi ^{\dagger} \frac {\partial \psi}{\partial t} + \frac {\partial \psi}{\partial t} \psi ^{\dagger} \right )=-i \hbar c \left [ \psi ^{\dagger} \vec \alpha \vec \nabla \psi + \vec \nabla \psi ^{\dagger} \vec \alpha \psi \right ] = -i \hbar c \vec \nabla \left ( \psi ^{\dagger} \vec \alpha \psi \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 \Rightarrow \frac {\partial}{\partial t} \left ( \psi ^{\dagger} \psi \right )+ \vec \nabla \left ( c \psi ^{\dagger} \vec \alpha \psi \right ) = 0}

Therefore, we can define:

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 \rho = \psi ^{\dagger} \psi } as probability density

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 \vec j = c \left ( \psi ^{\dagger} \vec \alpha \psi \right )} as probability current density