Required math: algebra, calculus (partial derivatives and integration by parts), complex numbers
Required physics: Schrödinger equation, probability density
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 1.8.
The time-independent Schrödinger equation in one dimension can be separated into two equations as follows:
and the general solution is
The time component can be solved as
where is the constant of integration.
If we add a constant (in both space and time) to the potential, then the original Schrödinger equation becomes
Applying separation of variables gives us
[Since is independent of both and , we could put it in either the or the equation, but putting it in the equation eliminates it from the more complex equation, so we’ll do that.]
The solution to 8 is now
so we’ve introduced a phase factor into the overall wave function . For the time-independent Schrödinger equation, all quantities of physical interest involve multiplying the complex conjugate by some operator that depends only on , operating on . That is, we’re interested only in quantities of the form
Thus the phase factor disappears when calculating any physical quantity.