- #1
ginda770
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I was hoping someone could help me with a seeming paradox involving the Dirac equation. I have taken a non-relativistic QM course, but am new to relativistic theory.
The Dirac equation is (following Shankar)
[tex]i\frac{\partial}{\partial t}\psi = H\psi[/tex]
where
[tex]H = \vec{\alpha}\cdot \vec{p} + \beta m[/tex]
([tex]\psi[/tex] is a four component wavefunction and the alphas and beta are 4 by 4 matrices with constant entries)
It seems to me that any alpha matrix (or almost any other 4 by 4 matrix made up of constants) commutes with [tex]\partial/\partial t[/tex], but not with the hamiltonian [tex]H[/tex]. How can this be true? If [tex]\left[\vec{\alpha},H\right] \neq 0[/tex] and [tex]H = i \left(\partial/\partial t\right)[/tex] how can [tex]\left[\vec{\alpha},\partial/\partial t\right]=0[/tex] ? What am I missing?
The Dirac equation is (following Shankar)
[tex]i\frac{\partial}{\partial t}\psi = H\psi[/tex]
where
[tex]H = \vec{\alpha}\cdot \vec{p} + \beta m[/tex]
([tex]\psi[/tex] is a four component wavefunction and the alphas and beta are 4 by 4 matrices with constant entries)
It seems to me that any alpha matrix (or almost any other 4 by 4 matrix made up of constants) commutes with [tex]\partial/\partial t[/tex], but not with the hamiltonian [tex]H[/tex]. How can this be true? If [tex]\left[\vec{\alpha},H\right] \neq 0[/tex] and [tex]H = i \left(\partial/\partial t\right)[/tex] how can [tex]\left[\vec{\alpha},\partial/\partial t\right]=0[/tex] ? What am I missing?
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