Q concerning Schrödinger’s equation

[JDługosz]

I was reading Deep Down Things: The Breathtaking Beauty of Particle Physics by Bruce A. Schumm. He explains the Time-independent Schrödinger equation as a statement of the conservation of energy. You put in the function V(x) to describe the potential energy of the system, and then you can solve for ψ(x). I follow all that.

Later, he discusses local gauge invariance, and added a term A(x) which allows you to exactly undo any arbitrary phase changes you made in ψ; then goes on to identify A with an electric field and points out that the phase doesn't matter, so introducing local gauge invariance rather than being an ad-hoc fix inspired a way to handle all the bosons in the Lie group (1 in the case of em-field).

So, the function A represents a force field.
If you put the things influencing the particle in the A function (nearby electric charges), then what is the V function used for? Wasn't that being used to represent the potential energy of being near other charged objects? What am I missing?

TIA,
—John

 

[Mentz114]

The A function is a gauge potential required to enforce local phase-invariance. It is not a physical potential like V(x) in the Hamiltonian.

 

[JDługosz]

Then how is it "identified with an electric field"?

 

[Mentz114]

Hmm. Let me modify my remark, because A does appear in the Hamiltonian. For instance in QED the interaction term between electrons and the EM field is $$j_{\mu}A^{\mu}$$. However, the vector potential A is not ascribed physical reality, but allows a very good shorthand way of writing Maxwells equations.

I'm not sure if if I understand your question. Naively I'd say that V(x) is used to denote potential that is not from the EM field.

I'm handicapped by not having the book - but I wish I did because it sounds good.

 

[JDługosz]

I'm not sure if if I understand your question. ...
I'm handicapped by not having the book - but I wish I did because it sounds good.

Amazon Readerlink supports the ability to search and browse this book. Search for "local gauge" Start there and it will let you back up a few pages and go forward a few pages.

On page 224 it states,

Take any electromagnetic force field ... write down a function, and call it A(x) ... let this function A(x) be the one we put into the wave equation...

On page 226 it mentions "...electromagnetic fields that happen to be around." so it is talking about the actual context of the particle.

As for your remark about the Hamiltonian, you lost me.

—John

 

[Mentz114]

Hi John, although I've made several purchase there in the past, I'm not eligible for Amazons readerlink.

I searched for 'Hamiltonian' in the book, and there were no hits ! But "He explains the Time-independent Schrödinger equation as a statement of the conservation of energy" amounts to the same thing as saying the Hamiltonian is time invariant.

Anyhow, I can't go any further. Maybe someone whose got the book will see this.

M

 

[Mentz114]

Hi John,
I've read the pages in question and it seems to me the author is explaining verbally what is shown here http://quantumrelativity.calsci.com/Physics/EandM2.html. Scroll down the article to "Hermann Weyl and Gauge Invariance".

Note that everything hinges on what happens when the global phase $$e^{i\chi}$$ is replaced with a local phase $$e^{i\chi(x)}$$.

Also, I notice that he mentions 'the energy function', which I would call the Hamiltonian.

M