Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.2.
We’ve seen that the time-independent Schrödinger equation can be separated into two ordinary differential equations, one in space and one in time. We’ve also seen that the solution of the spatial equation can be taken to be real. Starting with the spatial Schrödinger equation we can also derive a condition on the energy .
If , the minimum value of the potential, then for all . This means that and have the same sign everywhere. If has a maximum, from elementary calculus we must have so at the point of the maximum itself must be negative. Similarly, any minima of must occur where is positive. Therefore, cannot tend to 0 as so it can’t be normalized. Thus any physically acceptable solution must have . This condition is the analog of the classical condition that the energy (kinetic plus potential) of a particle can’t be less than the minimum of the potential. That is, a particle at rest at the bottom of a potential well has the lowest possible energy.