Asymmetric potential well and discrete energy levels

Question

Let us consider an asymmetric potential, which is piecewise defined as $V_1$ for $x<0$, $0$ when $0<x<a$ and $V_2$ for $x>a$, together with the condition $V_1 > V_2 >0$.

In the first region, the wavefunction will be oscillatory, while in the middle region, it will display an exponential decay. In the third region it will oscillate again but with decreased amplitude.

In order to find out the discrete energy levels of this setup, does $E$ need to be greater or less than the potential?

I think the former is known as an unbound state and the latter as a bound state? Can states with discrete energy levels be found for either case?

Answer 1

As with all problems of this type, begin by writing down the (time-independent) Schroedinger equation, separating the three regions.

The solutions can be split into 3 categories depending on the energy of the state relative to $V_1$ and $V_2$.

If the energy of the state is lower than both $V_1$ and $V_2$, the solution is oscillatory within the well and exponentially decaying outside. If the energy is between $V_1$ and $V_2$, the solution is oscillatory within the well and on the region with $V_1$. If the energy is above both $V_1$ and $V_2$, we get unbound states which are oscillatory (with different wavenumber) in the three regions. In all cases you need to use the continuity of $\psi$ and $\psi'$ to figure out the constants that arise in the solution to piece together the solutions.