- #1
arivero
Gold Member
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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
<br /> (i \hbar {\partial \over \partial x_0} + {\bf \sigma} . i \hbar \nabla)<br /> (i \hbar {\partial \over \partial x_0} - {\bf \sigma} . i \hbar \nabla)<br /> \phi= (mc)^2 \phi<br />
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
<br /> \phi^R\equiv{1 \over mc} <br /> (i \hbar {\partial \over \partial x_0} - {\bf \sigma} . i \hbar \nabla) \phi<br />
and \phi^L\equiv\phi. Then we have
<br /> i \hbar ({\bf \sigma} . \nabla - {\partial \over \partial x_0} ) \phi^L= - m c \phi^R<br />
<br /> i \hbar (-{\bf \sigma} . \nabla - {\partial \over \partial x_0} ) \phi^R= - m c \phi^L<br />
Now you see the trick. These are the usual left and right handed two-component spinors; if you define
<br /> \psi=<br /> \begin{pmatrix}{\phi^R + \phi^L \cr \phi^R - \phi^L}<br /> \end{pmatrix}<br />
then the equation for the four component spinor \psi is just Dirac equation!
<br /> (i \hbar {\partial \over \partial x_0} + {\bf \sigma} . i \hbar \nabla)<br /> (i \hbar {\partial \over \partial x_0} - {\bf \sigma} . i \hbar \nabla)<br /> \phi= (mc)^2 \phi<br />
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
<br /> \phi^R\equiv{1 \over mc} <br /> (i \hbar {\partial \over \partial x_0} - {\bf \sigma} . i \hbar \nabla) \phi<br />
and \phi^L\equiv\phi. Then we have
<br /> i \hbar ({\bf \sigma} . \nabla - {\partial \over \partial x_0} ) \phi^L= - m c \phi^R<br />
<br /> i \hbar (-{\bf \sigma} . \nabla - {\partial \over \partial x_0} ) \phi^R= - m c \phi^L<br />
Now you see the trick. These are the usual left and right handed two-component spinors; if you define
<br /> \psi=<br /> \begin{pmatrix}{\phi^R + \phi^L \cr \phi^R - \phi^L}<br /> \end{pmatrix}<br />
then the equation for the four component spinor \psi is just Dirac equation!
!