- #1
AxiomOfChoice
- 531
- 1
If you can answer any of the questions below, your help will be greatly appreciated.
There's an equation from E&M (I believe the definition of the antisymmetric field tensor) that reads:
<br /> F_{\mu \nu} = \frac{\partial A_\nu}{\partial x^\mu} - \frac{\partial A_\mu}{\partial x^\nu},<br />
where (of course) A is the 4-vector potential. I have no idea how to parse this equation...what's going on with taking the partial of A_\nu with respect to x^\mu? What's this notation describing, or telling me to do? I just don't see it.
Also (and this pertains to relativistic quantum mechanics), what's meant by something like
<br /> \partial_\mu [\partial_\nu \Psi(x^\mu)],<br />
where (of course) \Psi(x^\mu) is the wave function? I.e., what does it mean to operate on \partial_\nu of something with \partial_\mu? Particularly when that something I'm operating on is a function of x^\mu? (By the way: In case the notation is unfamiliar to you, \partial^\mu = \partial / \partial x_\mu.) Does that turn into something? Can I compactify that? Or do I just have to leave it as \partial_\mu [\partial_\nu \Psi(x^\mu)]? Also, would I get an equivalent expression if I interchanged the order of \partial_\mu and \partial_\nu; i.e., if I tried to evaluate \partial_\nu [\partial_\mu \Psi(x^\mu)]? Or would I get something quite different?
Does all of this boil down to "we need two distinct dummy indices, \mu and \nu, to indicate that there shouldn't be any summation going on?"
(P.S. - I would like to thank David J. Griffiths for getting me into the habit of italicizing words I want to emphasize.)
There's an equation from E&M (I believe the definition of the antisymmetric field tensor) that reads:
<br /> F_{\mu \nu} = \frac{\partial A_\nu}{\partial x^\mu} - \frac{\partial A_\mu}{\partial x^\nu},<br />
where (of course) A is the 4-vector potential. I have no idea how to parse this equation...what's going on with taking the partial of A_\nu with respect to x^\mu? What's this notation describing, or telling me to do? I just don't see it.
Also (and this pertains to relativistic quantum mechanics), what's meant by something like
<br /> \partial_\mu [\partial_\nu \Psi(x^\mu)],<br />
where (of course) \Psi(x^\mu) is the wave function? I.e., what does it mean to operate on \partial_\nu of something with \partial_\mu? Particularly when that something I'm operating on is a function of x^\mu? (By the way: In case the notation is unfamiliar to you, \partial^\mu = \partial / \partial x_\mu.) Does that turn into something? Can I compactify that? Or do I just have to leave it as \partial_\mu [\partial_\nu \Psi(x^\mu)]? Also, would I get an equivalent expression if I interchanged the order of \partial_\mu and \partial_\nu; i.e., if I tried to evaluate \partial_\nu [\partial_\mu \Psi(x^\mu)]? Or would I get something quite different?
Does all of this boil down to "we need two distinct dummy indices, \mu and \nu, to indicate that there shouldn't be any summation going on?"
(P.S. - I would like to thank David J. Griffiths for getting me into the habit of italicizing words I want to emphasize.)
