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Sagging Bottom Quantum Box and Pertubation

  1. Mar 10, 2009 #1

    TFM

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

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

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

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

    a)

    Sketch the potential as a function of x.

    b)

    For small [tex]V_0[/tex] 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 [tex]\Delta[/tex]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,

    [tex] v(x) = V_0(x) + \lambda x^4 [/tex]

    where [tex]\lambda x^4 [/tex] is the [tex] \Delta v [/tex]

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

    Can anyone offer any advice?

    Many Thanks

    TFM
     

    Attached Files:

  2. jcsd
  3. Mar 11, 2009 #2

    TFM

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

    [tex] V(x) = V_0(x) + \Delta V(x) [/tex]

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

    Also, may be useful, the stationary Schrödinger Equation,

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

    Is this useful?

    Any suggestions?

    The Ferry Man
     
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