# Effect of phase change in wave function on energy

1. May 2, 2014

### jordankonisky

I understand that a local gauge transformation functions to conserve the energy of an electron as it moves through space/time. What I don’t understand is why the energy of the electron, as dictated by the momentum and potential energy terms of the Schrödinger equation changes as a function of (x) and (t). I read that this is due to the fact that the phase of the wave function (not its value) changes. Why should the total energy change as the phase angle changes?

2. May 2, 2014

### bhobba

The why of the Schroedinger equation actually boils down to, strange as it may seem, symmetry considerations, specifically the probability of outcomes is frame independent (ie it obeys the POR).

You will find this approach in Ballentine - Chapter 3:
https://www.amazon.com/Quantum-Mechanics-A-Modern-Development/dp/9810241054

The deep reason of energy's relation to time is bound up in Noethers Theorem - energy is by definition the conserved quantity related to time invariance:
http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html

Actually a change in phase alone has no effect on energy because it has no effect on the state. The deep reason for that is states are strictly speaking not elements of a vector space - they are in fact positive operators of unit trace. Pure states are those states of the form |u><u|. Such states can be mapped to a vector space but not uniquely because if c is a phase factor |cu><cu| = |u><u|.

I suggest not only reading chapter 3 of Ballentine - but the first two chapters as well where a lot of stuff glossed over in more elementary treatments is made clear.

Thanks
Bill

Last edited by a moderator: May 6, 2017