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Local Gauge Transformation

If we change
$\displaystyle \mathbf{A}'=\mathbf{A}+ \nabla f,\quad \phi' = \phi -\partial_t f,$     (2.4)

the Hamiltonian in the new gauge becomes ( \bgroup\color{col1}$ H_{\rm rad}$\egroup is not changed)
$\displaystyle H'(t)$ $\displaystyle \equiv$ $\displaystyle \frac{1}{2m}\left({\bf p}-q \mathbf{A}' \right)^2 + q \phi'$ (2.5)

The time-dependent Schrödinger equations in the old and the new gauge are
$\displaystyle i\partial_t \psi = H(t)\psi,\quad i\partial_t \psi'= H'(t)\psi'.$     (2.6)

They should describe the same physics which is the case if
$\displaystyle \psi'({\bf r},t) = U \psi({\bf r},t),\quad U = e^{iqf({\bf r},t)/c}.$     (2.7)

This can be seen by
$\displaystyle i\partial_t \psi'$ $\displaystyle =$ $\displaystyle i\partial_t U \psi = (i\partial_t U) \psi+ i U \partial_t\psi
= (i\partial_t U) \psi+ i U H\psi = \left[(i\partial_t U) + U H \right] \psi$  
  $\displaystyle =$ $\displaystyle \left[(i\partial_t U) + U H \right]U^{\dagger} U \psi= \left[(i\partial_t U) + U H \right]U^{\dagger}
\psi' = H'\psi'$  
  $\displaystyle \leftrightarrow$ $\displaystyle H' = (i\partial_t U) U^{\dagger}+ U H U^{\dagger},$ (2.8)

which means
$\displaystyle H' = (i\partial_t U) U^{\dagger}+ U H U^{\dagger},\quad \psi'({\bf r},t) = U \psi({\bf r},t)$ $\displaystyle \leftrightarrow$    same physics.  

The transformation from \bgroup\color{col1}$ H$\egroup to \bgroup\color{col1}$ H'$\egroup and correspondingly \bgroup\color{col1}$ \psi$\egroup to \bgroup\color{col1}$ \psi'$\egroup is completely arbitrary and works for any Hamiltonian and transformation (operator) \bgroup\color{col1}$ U$\egroup. In the context we are discussing it here, \bgroup\color{col1}$ U$\egroup is a phase and thus an element of the group \bgroup\color{col1}$ U(1)$\egroup. The transformation \bgroup\color{col1}$ U= e^{iqf({\bf r},t)/c}$\egroup is a local gauge transformation as it involves a \bgroup\color{col1}$ {\bf r}$\egroup-dependent phase.



Subsections
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Next: Example: spatially constant electric Up: Gauge invariance in single-particle Previous: Gauge invariance in single-particle   Contents   Index
Tobias Brandes 2005-04-26