Charge In Schroedinger's Equation

[Chronothread]

This is a simple question which I'm sure has a simple explanation. While mass of the particle is explicitly included in the Schroedinger Equation, the charge is not. Why isn't it? Is it included along with the potential energy or...?

 

[nanobug]

While mass of the particle is explicitly included in the Schroedinger Equation, the charge is not. Is it included along with the potential energy?

Yes.

 

[Chronothread]

Thank you.

 

[f95toli]

Also, you can write down a SE for systems that do NOT involve mass. A good example would be tunneling from one state to another in a superconducting component called Josephson junctions. If you write down the SE for such a junction you will find that what would be "mass" in the equation for a e.g. a traveling electron is now actually the effective capacitance $m=(\hbar/(2e) C$ of the junction (or, equivalently, m is the effective mass of a "phase particle" in phase space).

Hence, Schroedinger equations that involve a "real mass" are just special cases of a more general formalism.

 

[Rade]

...Hence, Schroedinger equations that involve a "real mass" are just special cases of a more general formalism...

I have a question...what would this "more general formalism" look like, mathematically ?

 

[samalkhaiat]

This is a simple question which I'm sure has a simple explanation. While mass of the particle is explicitly included in the Schroedinger Equation, the charge is not. Why isn't it?

Mass and charge need not be present in the same equation! They are quantities of different origin. Mass is related to a spacetime symmetry called Poincare symmetry.
All type of Charges show up in the equations of physics as a result of making those equations invariant with respect to some internal symmetries . Some of these (local) internal symmetries give rise to interactions. The strength of such interaction is determined by the corresponding charge.
For example; if you want the electric charge to appear in Schrodinger equation, you need to make this equation invariant under the local U(1) transformation:

$$\Psi(x) \rightarrow \exp \left(i q \Lambda (x) \right) \Psi(x)$$

regards

sam

 

[f95toli]

Maybe I was a bit unclear. I only meant that it is possible to write down a SE for systems where there is no real "mass" (i.e. nothing that has the unit of kg unless it is multipled by some constant), in this case the SE is for a "phase particle" with an effective "mass" given by the capacitance of the junction which is moving in phase space with the "coordinates" of the particle given by the electric phase across the junction.

Of course nothing is really moving in real space, the SE is merely describing the state of the junction (if the current through the junction is constant problem is just a simple "particle in a well" type of problem).

Hence, my point was simply that the SE is in no way limited to describing e.g. electrons and other "real" particles with properties like mass, charge etc as one could be lead to believe by the fact that there is a "m" in the equation. As the OP was asking about including charge in the SE I assumed that he/she might not know this.

 

[Riogho]

Unfortunately with all those nasty Quantum equations charge is hard to work around, because if you think of an electron, it is surrounded by virtual photons mediating the electromagnetic force around it that are continutally undergoing pair prodution causing positrons to come nearer the electron. Theoretically there are infinite positrons there, making the charge of an electron infinite.

As physicists we have no problem making infinity - infinity = a finite sum. Those weird mathematician types tend to snub their noses at us though.