Wave function phase relationships

[DmplnJeff]

The wave function is complex. I was taught that its square (probability) was actually psi times it's conjugate. Does this relationship always hold or was this only for bound and free particles?

In other words is it possible for psi and psi* to change phases during orbital state transitions?

 

[Chopin]

That's a universal property of complex numbers--the magnitude squared of the number is always equal to the number times its conjugate. You can see this by looking at the definition of a complex number:

$$Z = x + iy$$
$$Z^* = x - iy$$
$$ZZ^* = (x + iy)(x - iy) = x^2 -i^2y^2 = x^2 + y^2$$

So in polar coordinates, where $$r = \sqrt{x^2 + y^2}$$, the number times its conjugate will always be $$r^2$$.

 

[Istvan Mezo]

Let us suppose that there are two, spacelike separated electron in coordinate eigenstates. In the next moment, they begin to propagate as two spherical waves. (Better to say: the probability densities do propagate.)

What happens when these waves overlap? There are two possibilities (as I think):

1) The electrons entangle and there will be only one two-electron instead of two electrons,
2) They overlap without any interaction.

 

[dextercioby]

The wave function is complex. I was taught that its square (probability) was actually psi times it's conjugate. Does this relationship always hold or was this only for bound and free particles?

Its square modulus is always psi*psi. The easiness with which we ascribe a probabilistic interpretation to this modulus is dictated on whether psi is an element of a Hilbert space, thus is finite norm, or its modulus is finite, so it can be rescaled to unity.

In other words is it possible for psi and psi* to change phases during orbital state transitions?

The orbital state transitions are determined by an external intervention in an initially closed atomic system. The dynamics is then described by the Schroedinger equation whose solution cannot be really found analytically. The phase of the wavefunction during transition cannot be therefore determined exactly, but only proved to different than the one pertaining to a wavefunction of an unperturbed atomic system.

 

[DmplnJeff]

If psi^2 represents the probability distribution of the location, could psi and psi* represent two locations for the same object. For example a point mass could conceivably have a slightly separate location for its momentum moment and its gravitational moment.

Basically I'm trying to understand what solutions are available for Schroedinger's equation during the transition (if Schroedinger's equation applies?). So far what I've read amounts to a transition being some sort of miracle. It just happens.

Possible solutions depend on whether psi^2 is one number squared or two numbers.

 

[DmplnJeff]

Let me try rephrasing the question. It's my understanding the Hermitian nature of the phase space arises from the conservation of energy through the Noether theorem.

That theorem applies to non-dissipative spaces. Yet during the absorption (or emission) of a photon energy is not being conserved (locally). Why doesn't that make a difference?