# Local Conservation Of Charge And Quantum Mechanics

1. Jul 12, 2013

### Dr_Pill

I understand it from a classical viewpoint, like the flow of a fluid.
But shouldn't an electron obey the rules of QM?

How is teleporting from one place to another forbidden in QM, wat about tunneling.Where is your continuous flow now?

I don't get it.This simple local conservation of charge principle is not compatible with QM in my opinion.

Thanks if somebody can explain this.

2. Jul 12, 2013

### tom.stoer

In order to understand this you should start with the canonical formalism in classical mechanics. First use Noether's theorem to find a conserved quantity Q related to a symmetry (in the canonical formalism Q is translated into an operator which acts as a generator of this symmetry). Charge conservation means {Q,H} = 0, which is the Hamiltonian equation of motion, which is translated into [Q,H] = 0 in quantum mechanics. So the statements a) that a Hamiltonian H has a symmetry generated by Q, and b) that Q is conserved under time evolution generated by H are equivalent.

Note that the equation [Q,H] = 0 is an equation on the level of operators!

Now a local version of this charge conservation is the continuity equation for currents. Again this is an operator equation. As an example, the conservation of the electric current

$\partial_\mu j^\mu = 0$

holds in QED as an operator equation expressed in terms of field operators of the electron field

$j^\mu = \bar{\psi}\gamma^\mu\psi$

Now back to your states, e.g. for tunneling. The above mentioned operator equations are valid in the so-called Heisenberg picture, where operators a time-dependent (and only by special operators like conserved charges are time independent, which is a non-trivial statement), and where all states a time-independent - regardless which system and which state we want to describe.

So all states a time-independent, and the above mentioned operator equations hold for each state in the Hilbert space.

Suppose there is an eigenstate

$Q|q\rangle = q|q\rangle$

Then

$[H,Q]|q\rangle = 0$

and even

$\partial_\mu j^\mu |q\rangle = 0$

This holds for arbitrary states, but especially for eigenstates of Q, which are a complete set and span the total Hilbert space.

3. Jul 13, 2013

### Bill_K

You seem to be under the impression that quantum mechanics allows things to jump willy-nilly from point A to point B. Totally not true. Particles in QM do not teleport!

To begin with, for the Schrodinger equation one can derive a continuity equation ∂ρ/∂t + ∇·J = 0, where ρ is the probability density and J the probability current. Like all continuity equations, this says that the quantity within a certain volume can change only by virtue of current flowing into and out of the volume across the boundary.

Likewise as quoted in your reference, quantum electrodynamics has a conservation law relating the charge density and electric current. There is also one for energy-momentum conservation.