Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.3.
We’ve seen that the energy of a system must always be greater than the minimum of the potential function. As a specific example of this we can look at the Schrödinger equation for the square well, between and :
If , . Integrating gives Attempting to satisfy the boundary conditions, we get giving . Then the condition gives , thus and cannot be normalized.
If , we solve the equation
with . Since , is real. The general solution is . Applying boundary conditions, we get , so . At , we have
where . Since is strictly positive (being an exponential) the only way we can get is for , implying . However, neither nor is zero here, so , so and again. Thus if , the wave function cannot be normalized and satisfy the boundary conditions.