Introduction to Quantum Mechanics · Problem 2.3
Time-Independent Schrödinger Equation — David J. Griffiths
Physics — Quantum Mechanics · infinite_square_well
Problem
Show that there is no acceptable solution to the (time-independent) Schrödinger equation for the infinite square well with or . (This is a special case of the general theorem in Problem 2.2, but this time do it by explicitly solving the Schrödinger equation, and showing that you cannot satisfy the boundary conditions.)Intuitive explanation
The key is recognizing that the Schrödinger equation's form changes dramatically depending on whether the energy is positive or negative, which directly determines whether solutions can oscillate or only decay. For , the spatial equation produces exponential solutions rather than sines and cosines—and these exponentials cannot simultaneously vanish at both walls of the infinite square well. By solving explicitly and applying the boundary conditions, you'll see that only positive energy allows the oscillatory behavior needed to satisfy the hard walls.
Intuitive explanation
The key is recognizing that the Schrödinger equation's form changes dramatically depending on whether the energy is positive or negative, which directly determines whether solutions can oscillate or only decay. For , the spatial equation produces exponential solutions rather than sines and cosines—and these exponentials cannot simultaneously vanish at both walls of the infinite square well. By solving explicitly and applying the boundary conditions, you'll see that only positive energy allows the oscillatory behavior needed to satisfy the hard walls.