Approximate solutions of the transcendental equation for the square quantum wells by finding the real root of the cubic equation

Abstract

The energy eigenvalues E n in finite square quantum wells (SQW) cannot be found using an analytic expression. As a result, numerical methods are normally used to find the eigenvalues from a transcendental equation. In this report, it will be shown that the eigenvalue solution for a given state consists in finding the only real positive root of a depressed trinomial polynomial of third order, which is as easy to solve as a quadratic equation. The method proposed can also be applied for semi-infinite, finite, and asymmetric SQW, which are often presented in quantum mechanics textbooks at the undergraduate level. The proposed method can be applied during an exam when programmable calculators are not allowed as the real root of the trinomial polynomial can be found using the formula for the cubic equation found nearly 500 years ago.