|
|
|
Viewing Version
2
of
'Dirac equation'
|
[ view 'Dirac equation'
|
back to history
]
| Title of object: |
Dirac equation |
| Canonical Name: |
DiracEquation |
| Type: |
Definition |
| Created on: |
2008-03-15 10:12:57 |
| Modified on: |
2008-03-15 10:26:12 |
| Classification: |
msc:35Q40 |
| Keywords: |
relativistic, D'Alembertian |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
|
Content:
The Dirac equation is an equation derived by Paul Dirac in 1927 that describes relativistic spin $1/2$ particles (fermions). It is given by:
\section{Derivation}
Mathematically, it is interesting as one of the first uses of the spinor calculus in mathematical physics. Dirac began with the relativistic equation of total energy:
\[
E = \sqrt{p^2c^2 + m^2c^4}
\]
As Schr\"odinger had done before him, Dirac then replaced $p$ with its quantum mechanical operator equivalent, $\hat{p} \Rightarrow i\hbar \nabla$. Since he was looking for a Lorentz-invariant equation, he replaced $\nabla$ with the D'Alembertian or wave operator
\[
\Box = \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2}
\]
Note that some authors use $\Box^2$ for the D'alembertian. Dirac was now faced with the problem of how to take the square root of an expression containing a differential operator. He proceeded to factorise the d'Alembertian as follows:
\[
\nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} = (A \frac{\partial}{\partial x} + B \frac{\partial}{\partial y} + c \frac{\partial}{\partial z} + D\frac{i}{c} \frac{\partial}{\partial t})^2
\]
Multiplying this out, we find that:
\[
A^2 = B^2 = C^2 = D^2 = 1
\]
And
\[
AB + BA = BC + CB = CD + DC = 0
\]
Clearly these relations cannot be satisfied by scalars, so Dirac sought a set of four matrices which satisfy these relations. These are now known as the Dirac matrices, and are given as follows:
\[
\gamma^0 =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \end{pmatrix},
\gamma^1 \!=\! \begin{pmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & -1 & 0 & 0 \\
-1 & 0 & 0 & 0 \end{pmatrix}
\]
\[
\gamma^2 \!=\! \begin{pmatrix}
0 & 0 & 0 & -i \\
0 & 0 & i & 0 \\
0 & i & 0 & 0 \\
-i & 0 & 0 & 0 \end{pmatrix},
\gamma^3 \!=\! \begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1 \\
-1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \end{pmatrix}
\]
|
|
|
|
|
|
