- #1
arivero
Gold Member
- 3,496
- 173
Sakurai credits B. L. van der Waerden 1932 a pretty derivation of Dirac equation from two-component wave functions. First decompose E^2-p^2=m^2 as
[tex]
(i \hbar {\partial \over \partial x_0} + {\bf \sigma} . i \hbar \nabla)
(i \hbar {\partial \over \partial x_0} - {\bf \sigma} . i \hbar \nabla)
\phi= (mc)^2 \phi
[/tex]
This phi has two components, but it is a second order equation, so another two components are needed (say, the first derivative of phi) to fully specify a solution. Instead, we define
[tex]
\phi^R\equiv{1 \over mc}
(i \hbar {\partial \over \partial x_0} - {\bf \sigma} . i \hbar \nabla) \phi
[/tex]
and [tex]\phi^L\equiv\phi[/tex]. Then we have
[tex]
i \hbar ({\bf \sigma} . \nabla - {\partial \over \partial x_0} ) \phi^L= - m c \phi^R
[/tex]
[tex]
i \hbar (-{\bf \sigma} . \nabla - {\partial \over \partial x_0} ) \phi^R= - m c \phi^L
[/tex]
Now you see the trick. These are the usual left and right handed two-component spinors; if you define
[tex]
\psi=
\begin{pmatrix}{\phi^R + \phi^L \cr \phi^R - \phi^L}
\end{pmatrix}
[/tex]
then the equation for the four component spinor [tex]\psi[/tex] is just Dirac equation!
[tex]
(i \hbar {\partial \over \partial x_0} + {\bf \sigma} . i \hbar \nabla)
(i \hbar {\partial \over \partial x_0} - {\bf \sigma} . i \hbar \nabla)
\phi= (mc)^2 \phi
[/tex]
This phi has two components, but it is a second order equation, so another two components are needed (say, the first derivative of phi) to fully specify a solution. Instead, we define
[tex]
\phi^R\equiv{1 \over mc}
(i \hbar {\partial \over \partial x_0} - {\bf \sigma} . i \hbar \nabla) \phi
[/tex]
and [tex]\phi^L\equiv\phi[/tex]. Then we have
[tex]
i \hbar ({\bf \sigma} . \nabla - {\partial \over \partial x_0} ) \phi^L= - m c \phi^R
[/tex]
[tex]
i \hbar (-{\bf \sigma} . \nabla - {\partial \over \partial x_0} ) \phi^R= - m c \phi^L
[/tex]
Now you see the trick. These are the usual left and right handed two-component spinors; if you define
[tex]
\psi=
\begin{pmatrix}{\phi^R + \phi^L \cr \phi^R - \phi^L}
\end{pmatrix}
[/tex]
then the equation for the four component spinor [tex]\psi[/tex] is just Dirac equation!
!