Infinite square well with barrier in the middle

[christianpoved]

Homework Statement

Show that the energy levels of a double square well $$V_{S}(x)= \begin{cases} \infty, & \left|x\right|>b\\ 0, & a<\left|x\right|<b\\ \infty, & \left|x\right|<a \end{cases}$$ are doubly degenerate. (Done)

Now suppose that the barrier between -a and a is very high, but finite. Assume that the potential between -a and a is symmetric about the origin. There is now the possibility of tunneling from one well to the other, and this possibility has the effect of splitting the degeneracy of the double well in part (a). Let Eo be an energy level of the well in part (a), and assume that Eo is reasonably less than the barrier height. Assume that in the neighborhood of Eo the reflection amplitude of the barrier at -a is of the form $$\exp(-i\delta (E))$$ where delta is real, positive and much smaller than 1. Also assume the transmission amplitude to be of the form $$iJ (E))$$ where J is small and positive. Show that to lowest order in delta and J the well with the finite barrier has two levels E corresponding to each degenerate level of the double square well in (a), given by $$E_{\pm}=E_{0}-\frac{p_{0} \hbar}{m (b-a)}(\delta (E_{0}) \mp \frac{J(E_{0})}{2})$$

Homework Equations

I guess that the relations between the transmission an reflection amplitudes are important, also i need a relation between the energy and the amplitudes

The Attempt at a Solution

I did part (a) solving the Schrödinger equation by regiones arriving to this wavefunction

$$\psi_{1}\left(x\right)=A_{1}\sin\left(n\pi\frac{x+b}{b-a}\right)$$
between -b and -a.
$$\psi_{2}\left(x\right)=\pm A_{2}\sin\left(n\pi\frac{x-a}{b-a}\right)$$
between a and b. With an energy of $$E=\frac{n_{2}^{2}\hbar^{2}\pi^{2}}{2m\left(b-a\right)^{2}}$$ showing that every energy state is doubly degenerated. But i have no idea how to do part b

 

[Simon Bridge]

Well what happens to the wavefunctions etc if the barrier in the middle is not infinite?