Dirac equation: Difference between revisions
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where <math>H=c \vec \alpha \vec p + \beta mc^2</math> | where <math>H=c \vec \alpha \vec p + \beta mc^2</math> | ||
Dirac equation can also be written explicitly as follows: | |||
<math>i \hbar \frac {\partial \psi_{1}}{\partial t}=c(p_{x}-ip_{y}) \psi _{4}+cp_{z} \psi _{3} + mc^2 \psi _{1}</math> | |||
<math>i \hbar \frac {\partial \psi_{2}}{\partial t}=c(p_{x}+ip_{y}) \psi _{3}-cp_{z} \psi _{4} + mc^2 \psi _{2}</math> | |||
<math>i \hbar \frac {\partial \psi_{3}}{\partial t}=c(p_{x}-ip_{y}) \psi _{2}+cp_{z} \psi _{1} - mc^2 \psi _{3}</math> | |||
<math>i \hbar \frac {\partial \psi_{4}}{\partial t}=c(p_{x}+ip_{y}) \psi _{1}-cp_{z} \psi _{2} - mc^2 \psi _{4}</math> |
Revision as of 19:52, 18 April 2009
How to construct
Starting from the relativistic relation between energy and momentum:
or
From this equation we can not directly replace 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:
where and 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}
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 corresponds to
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, and are 4 by 4 matrices. It is convention that these matrices are given as follows (in the form of block matrices for short):
; ; ;
where are 2 by 2 Pauli matrices.
Let's define:
Then we can write:
Substituting all quantities by there corresponding operators, we obtain Dirac equation:
where
Dirac equation can also be written explicitly as follows: