# Sagging Bottom Quantum Box and Pertubation

1. Mar 10, 2009

### TFM

1. The problem statement, all variables and given/known data

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 Schrodinger equation. What is the perturbation potential $$\Delta$$V (x)?

c)

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

2. Relevant equations

3. 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.

Many Thanks

TFM

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2. Mar 11, 2009

### 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