Finding Normalization Constants for a Set of Energy Eigenstates

[jaurandt]

I do not know what I'm doing wrong but I'm working on the problem of finding the normalization constants for the energy eigenstate equation for a 1D plane wave that is traveling from the left into a potential barrier where E < V at the barrier. This is from Allan Adams' Lecture 12 of his 2013 Quantum Physics 1 lectures.

The system of equations left at x = 0 is

A + B = D
ikA - ikB = -aD (for the derivatives)

And he wrote that

D = (2k)/(k + ia)

and

B = (k - ia)/(k + ia)

He said we can "invert" the original equations to get those. After many attempts, I can't figure it out. Can someone please guide me along before I pull all of my hair out?

 

[tnich]

A + B = D
ikA - ikB = -aD (for the derivatives)

I think "invert" probably refers to matrix inversion.
You have two equations and three unknowns here. That means that the system of equations has an infinite number of solutions. You could find a solution by considering A to be a parameter and solving for B and D in terms of A (treating A like a constant). Then you can set A to whatever constant value you choose (1 for example) to get corresponding values for B and D.

 

[hilbert2]

First of all, the normalization constants on both sides of the potential step have to be such that the wavefunction doesn't have a jump discontinuity at the beginning of the barrier. But that only determines the relative magnitudes of the two constants.

Unbound plane wave states are not normalized in the same way as bound states where the normalization makes the total probability of finding the particle to be 1. Maybe the correct way to set the normalization is one where the pre-factor before the barrier (when $V=0$) is the $1/\sqrt{2\pi \hbar}$, as in the link below.

https://quantummechanics.ucsd.edu/ph130a/130_notes/node138.html