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Integrating the time-indepdent Schrodinger equation

  1. Jun 30, 2013 #1
    1. The problem statement, all variables and given/known data

    A particle of mass m is confined to move in a one-dimensional and Diract delta-function attractive potential [itex]V(x)=-\frac{\hbar^2}{m}\alpha\delta(x)[/itex] where [itex]\alpha[/itex] is positive.

    Integrate eh time-independent Schrodinger equation between [itex]-\epsilon[/itex] and [itex]\epsilon[/itex]. Let [itex]\epsilon\to 0[/itex], show that the derivative of the eigenfunction [itex]\phi(x)[/itex] is discontinuous at x=0, and determine its jump at the point.

    2. Relevant equations
    [itex]\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2}-V(x)\psi=E\psi[/itex]

    3. The attempt at a solution

    First I want to determine if I am approaching the problem correctly.
    First we have:
    [itex]\int\limits_{-\epsilon}^{\epsilon}-\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2}+\int\limits_{-\epsilon}^{\epsilon}\frac{\hbar^2}{m}\alpha\delta(x)\psi=\int\limits_{-\epsilon}^{\epsilon}E\psi[/itex].
    Then this simplifies to
    [itex]\int\limits_{-\epsilon}^{\epsilon}\frac{\hbar^2}{m}\alpha\delta\psi=\int\limits_{-\epsilon}^{\epsilon}\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2}[/itex]
    Further simplifying
    [itex]2\alpha\int\limits_{-\epsilon}^{\epsilon}\delta(x)\psi(x)=\frac{\partial \psi}{\partial x}\bigg|_{-\epsilon}^{\epsilon}[/itex]

    The last equation shows that as $\epsilon \to 0$, the right hand side will tend to zero but the left hand side will explode. I'm wondering if my work sufficiently shows this.

    The part I am stuck on is showing what its jump is at x=0.
     
  2. jcsd
  3. Jun 30, 2013 #2

    TSny

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    Homework Helper
    Gold Member

    Your work up to here looks very good.

    Here's where you need to think some more about what you're getting. The left hand side does not explode. What does integrating over the delta function give you for any finite ##\epsilon##? Then let ##\epsilon## → 0. Similarly for the right hand side.
     
    Last edited: Jun 30, 2013
  4. Jun 30, 2013 #3
    [itex]2\alpha\int\limits_{-\epsilon}^{\epsilon}\delta(x)\psi(x)=\frac{\partial \psi}{\partial x}\bigg|_{-\epsilon}^{\epsilon}[/itex]

    This simplies to something like [itex]2\alpha\psi(0)=\Delta\left(\frac{\partial \psi}{\partial x}\right)[/itex].

    After some work I believe this simplies down to [itex]k=\alpha[/itex]

    So now our [itex]E=-\frac{\hbar^2\alpha^2}{2m}[/itex].

    I'm basically following the process that is going on on p. 71-72 of Introduction to Quantum Mechanics.
     
    Last edited: Jun 30, 2013
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