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 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\sqrt{p^2+m^2c^2}}

From this equation we can not directly replace 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, \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:

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

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 \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 some operators independent of 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 p} .

From this it follows 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 c^2(p_{x}^2+p_{y}^2+p_{z}^2+m^2c^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 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} :

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 \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \ (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}\}=2\delta_{ij} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \ \ \ \ \ (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 \bold \alpha_ {i} \beta+\beta \alpha_ {i}=\{\alpha_ {i},\beta\}=0 \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \ \ \ \ \ \ (3)}

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 4x4 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 4x4 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) \qquad (4)}

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_{D} \psi \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (5)}

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_{D}=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} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (6)}

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} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (7)}

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} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (8)}

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} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (9)}

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 \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \ \ \ \ \ \ (10)}

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 \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (11)}

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

Free particle solution

Substituting 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(x,y,z,t)=\psi (\vec r, t)=\psi _{0}(\vec r)e^{(-i/\hbar)Et}} into (), we get time-dependent 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 E \psi_{0}(\vec r)=(c \vec \alpha \vec p +mc^2 \beta)\psi_{0}(\vec r)}

Let's seek for the plane wave solutions with 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 \vec p} :

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_{0}(\vec r)=u e^{(i/ \hbar) \vec p \vec r}}

u satisfies the following 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 Eu=(c \vec \alpha \vec p +mc^2 \beta)u}

u can be written 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 u= \left[\begin{array}{cc}u_{1}\\u_{2}\\u_{3}\\u_{4}\end{array}\right]=\left(\begin{array}{cc}W\\W'\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 W= \left(\begin{array}{cc}u_{1}\\u_{2}\end{array}\right) \qquad W'= \left(\begin{array}{cc}u_{3}\\u_{4}\end{array}\right)}

The equation for u can be rewritten 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 E\left(\begin{array}{cc}W\\W'\end{array}\right)= \left [ \left(\begin{array}{cc}0&c \vec \sigma \vec p \\c \vec \sigma \vec p & 0\end{array}\right)+\left(\begin{array}{cc}mc^2&0 \\0 & -mc^2\end{array}\right)\right ]\left(\begin{array}{cc}W\\W'\end{array}\right)}

or

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}(E-mc^2)W-c \vec \sigma \vec p W'=0&\\-c \vec \sigma \vec p W+(E+mc^2)W'=0\end{array}\right.}

Condition for non-trivial solutions:

$\left|{\begin{array}{cc}E-mc^{2}&-c{\vec {\sigma }}{\vec {p}}\\-c{\vec {\sigma }}{\vec {p}}&E+mc^{2}\end{array}}\right|=0$

$\Rightarrow E^{2}=c^{2}{\vec {p}}\;^{2}+m^{2}c^{4}\Rightarrow E_{\pm }=\pm c{\sqrt {{\vec {p}}\;^{2}+m^{2}c^{2}}}$

So, for a given value of momentum there are two values of energy one with positive sign the other with negative sign.

Substituting the value of energy into equation for W and W' yields :

$W={\frac {c{\vec {\sigma }}{\vec {p}}}{E-mc^{2}}}W'\qquad for\qquad E=E_{-}$

$W'={\frac {c{\vec {\sigma }}{\vec {p}}}{E+mc^{2}}}W\qquad for\qquad E=E_{+}$

Choosing the momentum along z direction, the wave functions can be written as follows:

$u_{E_{-}}=\left[{\begin{array}{cc}{\frac {cp_{z}A}{E_{-}-mc^{2}}}\\-{\frac {cp_{z}B}{E_{-}-mc^{2}}}\\A\\B\end{array}}\right]\qquad where\qquad W'=\left({\begin{array}{cc}A\\B\end{array}}\right)$

$u_{E_{+}}=\left[{\begin{array}{cc}D\\F\\{\frac {cp_{z}D}{E_{+}+mc^{2}}}\\-{\frac {cp_{z}F}{E_{+}+mc^{2}}}\end{array}}\right]\qquad where\qquad W=\left({\begin{array}{cc}D\\F\end{array}}\right)$

General solution for free particles is as follows:

$\psi ({\vec {r}},t)=\int u_{E_{-}}(A,B)e^{i(-Et+{\vec {p}}{\vec {r}})\hbar }d{\vec {p}}+\int u_{E_{+}}(D,F)e^{i(Et+{\vec {p}}{\vec {r}})\hbar }d{\vec {p}}$