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Wave Properties of particles. Schrodinger Equation.

Problem 6.84

Using the Schrödinger equation, demonstrate that at the point where the potential energy U(x)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.

Reveal Answer
6.84. Let us integrate the Schrödinger equation over a small interval of the coordinate xx within which there is a discontinuity in U(x),U(x), for example at the point x=0x=0 : ∂ψ∂x(+δ)−∂ψ∂x(−δ)=∫−8+82mℏ2(E−U)ψdx \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 UU is finite the integral tends to zero as δ→0.\delta \rightarrow 0 . What follows is obvious.
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