— 1.4. Normalization —
The Scroedinger equation is a linear partial differential equation. As such, if is a solution to it, then (where is a complex constant) also is a solution.
Does this mean that a physical problem has an infinite number of solutions in Quantum Mechanics? It doesn’t! The thing is that besides the The Scroedinger equation one also has condition 11 to take into account. Stating 11 for the wave function:
The previous equations states the quite obvious fact that the particle under study has to be in some place at a given instant.
Since was a complex constant the normalization condition fixes in absolute value but can’t tell us nothing regarding its phase. Apparently once again one is haunted with the perspective of having an infinite number of solutions to any given physical problem. The things is that this time the phase doesn’t carry any physical significance (a fact that will be demonstrated later) and thus we actually have just one physical solution.
In the previous discussion one is obviously assuming that the wave function is normalizable. That is to say that the function doesn’t blow up and vanishes quickly enough at infinity so that the integral being computed makes sense.
At this level it is customary to say that these wave functions don’t represent physical states but that isn’t exactly true. A wave function that isn’t normalizable because integral is infinite might represent a beam of particles in a scattering experiment. The fact that the integral diverges to infinity can then be said to represent the fact that beam is composed by an infinite amount of particles.
While the identically null wave function represents the absence of particles.
A question that now arises has to do with the consistency of our normalization and this is a very sensible question. The point is that we normalize the Schroedinger equation for a given time instant, so how does one know that the normalization holds for other times?
Let us look into the time evolution of our normalization condition 15.
Calculating the derivative under the integral for the right hand side of the previous equation
The complex conjugate of the Schroedinger equation is
Hence for the derivative under the integral
Getting back to 16
Since we’re assuming that our wave function is normalizable the wave function (and its complex conjugate) must vanish for and .
In conclusion one can say that if one normalizes the wave equation for a given time interval it stays normalized for all time intervals.