Sagging Bottom Quantum Box and Pertubation

[TFM]

Homework Statement

Consider a particle in a one-dimensional “box” with sagging bottom

v(x) = -V_0sin(\pi x/L) for 0 \leq x \leq L

infinity outside of thius (x > L, x < 0)

a)

Sketch the potential as a function of x.

b)

For small V_0 this potential can be considered as a small perturbation of a “box” with a straight bottom, for which we have already solved the Schrödinger equation. What is the perturbation potential \DeltaV (x)?

c)

Calculate the energy shift due to the sagging for the particle in the nth stationary state to first order in the perturbation.

Homework Equations

The Attempt at a Solution

I have completed the first section with a graph as attached. I am not sure on the second part. I know for a inharmonic oscillator,

v(x) = V_0(x) + \lambda x^4

where \lambda x^4 is the \Delta v

But I am not sure what to do in this question.

Can anyone offer any advice?

Many Thanks

TFM

 

[TFM]

Okay, I feel that this may too be very useful for this problem

V(x) = V_0(x) + \Delta V(x)

rouble is, this is a sum, where as the V(x) for this question appears to be the product, snce it is -V_0 * sin(\pi x /L)

Also, may be useful, the stationary Schrödinger Equation,

E_n\phi_n(x) = -\frac{\hbar^2}{2m} + (v_0(x) + \Delta V(x))\phi_n(x)

Is this useful?

Any suggestions?

The Ferry Man