- #1
arivero
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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
[tex] (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[/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} <br /> (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=<br /> \begin{pmatrix}{\phi^R + \phi^L \cr \phi^R - \phi^L}<br /> \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)<br /> (i \hbar {\partial \over \partial x_0} - {\bf \sigma} . i \hbar \nabla)<br /> \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} <br /> (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=<br /> \begin{pmatrix}{\phi^R + \phi^L \cr \phi^R - \phi^L}<br /> \end{pmatrix}[/tex]
then the equation for the four component spinor [tex]\psi[/tex] is just Dirac equation!
!