Dennis Partanen: Solving the Schrödinger equation by example

Abstract

In this bachelor diploma, the time-independent Schrödinger equation is derived and studied under the assumption of separable solutions and a real, time-independent potential function. We show that the resulting eigenvalues E are real and bounded below by the minimum of the potential. Furthermore, we examine how solutions to the time-independent equation relate to the full time-dependent Schrödinger equation.

Two examples are treated: the harmonic oscillator, where explicit solutions and eigenvalues are obtained using the ladder operator method, and the finite potential well, where a transcendental equation is derived. Using monotonicity and the Intermediate Value Theorem, we show that this equation has a finite number of solutions depending on the parameter values, and that at least one solution always exists. Finally, the infinite potential well is discussed as a limiting case.