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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)
i\frac{\partial}{\partial t}\psi = H\psi
where
H = \vec{\alpha}\cdot \vec{p} + \beta m
(\psi 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 \partial/\partial t, but not with the hamiltonian H. How can this be true? If \left[\vec{\alpha},H\right] \neq 0 and H = i \left(\partial/\partial t\right) how can \left[\vec{\alpha},\partial/\partial t\right]=0 ? What am I missing?
The Dirac equation is (following Shankar)
i\frac{\partial}{\partial t}\psi = H\psi
where
H = \vec{\alpha}\cdot \vec{p} + \beta m
(\psi 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 \partial/\partial t, but not with the hamiltonian H. How can this be true? If \left[\vec{\alpha},H\right] \neq 0 and H = i \left(\partial/\partial t\right) how can \left[\vec{\alpha},\partial/\partial t\right]=0 ? What am I missing?
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