Matching conditions for solutions to the Schrodinger equation

[AxiomOfChoice]

In standard, run-of-the-mill, one-dimensional scattering problems (e.g., finite square wells), we calculate transmission and reflection amplitudes by (in part) making sure that our wave function $\psi$ satisfies the following conditions at discontinuities of the potential:

(1) It is continuous;

(2) Its first derivative is continuous.

But why does it need to satisfy these conditions? Which of the postulates is violated if it doesn't?

 

[espen180]

It's first derivative isn't always continous. Think about the bound state of a delta potential. This needs to happen because whenever there is a discontinuity in the integral over the potential, it must be balanced by a discontinuity in the first derivative of the wave function.

The wave function has to satisfy a continuity equation, which translates to conservation of momentum. This equation in one-dimensional coordinate space is $\frac{d}{dt}|\psi (x,t)|^2 + \frac{d}{dx} j(x,t) = 0$
where $j(x,t)=\frac{\hbar}{2im}\left( \psi^* \frac{d \psi}{dx} - \psi \frac{d \psi^*}{dx}\right)$

I would think this constaint is enough to fix the amplitudes of a given scattering potential.

 

[Bill_K]

Because -ih ∂ψ/∂x = pψ and -h²/2m ∂²ψ/∂x² = (E - V)ψ

If ψ were discontinuous at a point its first derivative would be infinite, and thus the momentum would be infinite. Likewise if ∂ψ/∂x were discontinuous its second derivative would be infinite, and thus the kinetic energy (E - V) would be infinite.

 

[espen180]

Likewise if ∂ψ/∂x were discontinuous its second derivative would be infinite, and thus the kinetic energy (E - V) would be infinite.

But this need not be the case when we have an infinite discontinuity in the potential, right? I have in mind especially infinite potential barriers and delta functions, in which the integral over the potential has a discontinuity.

 

[SpectraCat]

But this need not be the case when we have an infinite discontinuity in the potential, right? I have in mind especially infinite potential barriers and delta functions, in which the integral over the potential has a discontinuity.

That is mathematically correct, and it important for solving model problems in QM. However it has no physical relevance as far as I know, since I am unaware of any infinite potential barriers or delta functions in nature.