Pretty simple problem I thought.

[land]

Homework Statement

The potential energy of a particle is defined by the piecewise function:

V(x) = infinity if x<0
V(x) = -V0 if 0<x<b
V(x) = 0 if x>b

So it's like a square well with one side being infinite. I need to find the condition on V0 and b so that no bound stationary states exist, then for there to be exactly three stationary states.

Homework Equations

Uh.. not sure. For a regular square well, I have that if V0 > 0 there is at least one stationary state. So for there to be none, V0 has to be less than zero? But this isn't quite the same as a square well.

The Attempt at a Solution

See above. I also know that each stationary state must have a node at x=0. I just don't know how to put all this together..

 

[land]

OK... could really use some help on this one. I thought if I started with the individual wave equations, I could figure it out. So for what I'm calling Region I, 0<x<b, I came up with $$\Psi=C_I^+cos[\sqrt{2m(V_0-\epsilon)}(\frac{x}{\hbar}})]+C_I^-sin[\sqrt{2m(V_0-\epsilon)}(\frac{x}{\hbar})]$$, and for region II, x>b, $$\Psi=C_{II}e^{-\sqrt{2m\epsilon}(\frac{x}{\hbar})$$, where epsilon is the binding energy. I'm doing this based on the assumption that it's like a regular square well, so I have no idea if that's right or not. Now before I can impose the condition that there is no node at x=0, it seems like I should figure out the constants. To do that I tried to impose that the wavefunctions and their derivative are continuous at x=b.. but that still leaves you with 2 equations and 3 unknowns, so it can't be solved yet.

A nudge in the right direction (or even an acknowledgment that I'm totally wrong) would be much appreciated.

 

[land]

Hm.. could it be that I'm not explaining the problem clearly enough? Let me know if so. I tried to draw something to post here but it's sort of confusing. It's just a semi-infinite square well, as nearly as I can tell.. but I really think I'm doing this wrong.

 

[land]

let me rephrase.. how can it be that there are no stationary states in a potential well? i think if i can understand that, the rest of it will be easy.

 

[Dick]

Your wavefunction should vanish at x=0. There may be fewer constants than you think. Nudge.