Probability current and it's meaning

[nateHI]

I understand how to derive the probability current from the continuity equation and probability density. I was able to follow the proof at the beginning of http://www.youtube.com/watch?v=NSi27LC6plE" lecture perfectly.
However, now I'm wondering, what if we were to prove the probability density from the continuity equation and the equation for probability current? To do so we would need to derive the probability current in a similar way the wave function, and probability density, for a particles position was derived.

My question is, does it make sense to start with deriving the probability current and then use the continuity equation to derive the probability density? I thought about this and tried to figure out where I would start but drew a blank.

 

[tom.stoer]

I have never seen this in a book or lecture but in principle what one can do is the following:

1) write down a Lagrangian density L for quantum mechanics; L contains the wave function and the potential (or some other interaction term like electromagnetic potentials, spin couplings etc.); but for the basic idea a simple Lagrangian will do

2) vary the Lagrangian density w.r.t. the fave function in order to derive the Schrödinger equation; this is a consistency check

3) identify a global symmetry of the Lagrangian density; this will be the global U(1) rotation of the wave function, i.e. the global phase

4) derive the Noether 4-current for this symmetry: 4-current = (probability density, probability 3-current)

This construction demonstrates that qm is nothing else but a field theory with a global symmetry, a conserved 4-current (continuity equation) and a conserved charge; the conserved charge is nothing else but the normalization constant of the wave function; density and current follow simultaniously from the Noether theorem.

I think this is a derivation rather closed to "from first principles"