All Problems Wave Properties of particles. Schrodinger Equation.
Problem 6.84
Using the Schrödinger equation, demonstrate that at the point where the potential energy \(U(x)\) of a particle has a finite discontinuity, the wave function remains smooth, i.e. its first derivative with respect to the coordinate is continuous.
6.84. Let us integrate the Schrödinger equation over a small interval of the coordinate \(x\) within which there is a discontinuity in \(U(x),\) for example at the point \(x=0\) : \[ \frac{\partial \psi}{\partial x}(+\delta)-\frac{\partial \psi}{\partial x}(-\delta)=\int_{-8}^{+8} \frac{2 m}{\hbar^{2}}(E-U) \psi d x \] Since the discontinuity \(U\) is finite the integral tends to zero as \(\delta \rightarrow 0 .\) What follows is obvious.