Energy eigenstates in non-symmetric potential

[dyn]

Homework Statement

A potential has V(x)=2V₀ for x<0 , V(x)=0 for 0<x<a and V(x)=V₀ for x.>a with V₀>0. The next 3 questions apply to this potential.

1 - there are 2 energy eigenstates for each energy E>V₀ but smaller than 2V₀ True or false ?
2 - there is one normalizable state for each E>0 but smaller than V₀ T or F ?
3 - there are no energy eigenstates for E<0 T or F ?

The recommended time to spend on these 3 questions is 5 minutes in total !

Homework Equations

schrodingers time independent equation

The Attempt at a Solution

I can look at the 3 regions and solve for the wavefunction. For Q1 I get an exponential decay for x<0 and oscillating functions for the other 2 regions. I could then apply continuity of the wavefunction and its derivative at the boundaries but I only have 5mins for all 3 questions. there must be a quick way. any ideas ? Thanks

 

[vela]

For Q1, I'd guess it's false. If there were two energy eigenstates, what would be the difference between them? For a free particle, you have the freedom to choose the direction of the momentum. You don't seem to have that here.