Bound State Solutions of the Schrödinger Equation

Abstract

When the Hamiltonian for a system is independent of time, there is an essential simplification in that the general solution of the Schrödinger equation can be expressed as a function of spatial coordinates and a function of time. Thus, assuming the potential energy function to be independent of time, the one-dimensional time dependent Schrödinger equation [see Eq. (25) of Chapter 4] is given by

$$ i\frac{{\partial \psi }}{{\partial \mu }} = - \frac{{{^2}}}{{2\mu }}\,\frac{{{\partial ^2}\psi }}{{\partial {x^2}}} + V\left( x \right)\;\psi \left( {x,t} \right) $$

((1))

where μ represents the mass of the particle. The above equation can be solved by using the method of separation of variables

$$ \psi \left( {x,t} \right) = \psi \left( x \right)\;T\;\left( t \right) $$

((2))

Substituting in Eq. (1) and dividing by ψ(x, t), we obtain

$$ \frac{{i}}{{T\left( t \right)}}\;\frac{{dT}}{{dt}} = \frac{1}{{\psi \left( x \right)}}\left[ { - \frac{{{^2}}}{{2\mu }}\;\frac{{{d^2}\psi }}{{d{x^2}}} + V\left( x \right)\;\psi \;\left( x \right)} \right] $$

((3))

Only questions about the results of experiments have a real significance and it is only such questions that theoretical physics has to consider.

P.A.M. Dirac in The Principles of Quantum Mechunics