Local Conservation Of Charge And Quantum Mechanics

[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?
What about Quantum Entanglement.

Continuity equations are a stronger, local form of conservation laws. For example, it is true that "the total energy in the universe is conserved". But this statement does not immediately rule out the possibility that energy could disappear from Earth while simultaneously appearing in another galaxy. A stronger statement is that energy is locally conserved: Energy can neither be created nor destroyed, nor can it "teleport" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement.

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.

 

[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.

 

[Bill_K]

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.

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 Schrödinger 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.