Which Potential V(x) Matches the Given Eigenfunction Ψ(x)?

[k3r0]

Homework Statement

Consider the stationary state (eigenfunction) Ψ(x) illustrated. Which of the three potentials V(x) illustrated could lead to such an eigenfunction?

Homework Equations

N/A.

The Attempt at a Solution

I think it's the second one. Where the wavefunction is negative the potential should be 0; the second potential graph drops to 0 between the two zero points of the wavefunction graph and is non-zero inside the well. Could anyone confirm if I'm right in thinking this? Been muddling with this for a while now. Thanks.

 

[hilbert2]

The first potential looks like harmonic oscillator potential and the second one is a finite square well. Are you familiar with the solutions of Schrödinger equation for those potentials?

Hint: If a particle is in a finite square potential well that is deeper than the particle's total energy, the wavefunction is sinusoidal inside the well and decays exponentially outside it.

 

[k3r0]

Thanks. I originally chose #2 but but I think I'm going to go with the last one; it's sort of sinusoidal within the well, goes towards 0 during the blip in the middle and decays outside the well.

Part (b) asks me to draw the eigenfunction for the ground state, E1, and the second energy, E2. Would they just look like these (on the left)?:

I'm having a hard time deciding what state the diagram we've been given is in. It seems to be all over the place and I'm not sure how to approach the question.

EDIT: I think the graph we've been given is E3 because it looks like the third state /should/ look, except it goes to 0 outside the well. I'm drawing E1 and E2 like the diagram above, but with the interference of the well in the middle.

 

[hilbert2]

As far as I know, if a 1D quantum system is in the N:th energy state, the wavefunction has N-1 nodes (points where it has value zero).

 

[k3r0]

Thanks for your help. :)