All Problems Wave Properties of particles. Schrodinger Equation.
Problem 6.76
Find a particular solution of the time-dependent Schrödinger equation for a freely moving particle of mass \(m\).
\(6.76 .\) The solution of the Schrödinger equation should be sought in the form \(\Psi=\psi(x) \cdot f(t) .\) The substitution of this function into the initial equation with subsequent separation of the variables \(x\) and \(t\) results in two equations. Their solutions are \(\psi(x) \sim \mathrm{e}^{i k x}\), where \(k=\sqrt{2 m E} / \hbar, E\), is the energy of the particle, and \(f(t) \sim\) \(\sim \mathrm{e}^{-i \omega t},\) where \(\omega=E / \hbar .\) Finally, \(\Psi=a \mathrm{e}^{i(k x-\omega t)},\) where \(a\) is a certain constant.