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Consider the time-dependent Schrodiner equation

  1. Apr 29, 2012 #1
    "Consider the time-dependent Schrodiner equation ..."

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

    Consider the time-dependent Schrodinger equation

    ih2ψt = [-h2/(2m)]ψxx + V(x)ψ​

    which is the underlying equation of quantum mechanics. Here V(x) is a given potential, h is the Planck's constant, and m is the mass of the particle. ψ(x,t) is the amplitude of the wave that the particle traces out. i=√(-1) is the imaginary unit.

    (a) Use the separation of variable ψ(x,t)=u(x)exp(-iEt/h) to derive the time-independent Schrodiner equation which governs u(x). Show that the resulting equation is a Sturm-Louiville eigenvalue problem with

    p(x) = h2/(2m), q(x) = V(x), r(x) = 1, λ=E.​

    The eigenvalue E represents the energy of the particle.

    (b) Solve the Sturm-Louiville eigenvalue problem in the domain -1 < x < 1 with zero potential and homogenous Dirichlet boundary conditions. Sketch the ground state (the lowest non-zero energy state). What is the energy?

    (c) Without further calculation, explain what would happen to the eigenfunctions and eigenvalues if the domain is cut in half, i.e. 0 < x < 1, with new boundary conditions u(0)=u(1)=0.

    2. Relevant equations

    A Sturm-Louville eigenvalue problem has the form

    -p(x)u''(x)-p'(x)u'(x)+q(x)=λr(x)u(x)​

    3. The attempt at a solution

    Part (a) is trivial. For part (b), doesn't "zero potential" mean V(x)=0, in which case I seem to get a trivial solution?
     
  2. jcsd
  3. Apr 29, 2012 #2

    Pengwuino

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    Re: "Consider the time-dependent Schrodiner equation ..."

    The Schrodinger's Equation with [itex]V(x,t) = 0[/itex] is the free particle situation. The solution is not trivial, though, but very easy to determine.
     
  4. Apr 29, 2012 #3
    Re: "Consider the time-dependent Schrodiner equation ..."

    How is that not trivial?
     
  5. Apr 29, 2012 #4

    Pengwuino

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    Re: "Consider the time-dependent Schrodiner equation ..."

    Well, what does the Schrodinger Equation look like with [itex]V(x,t) = 0[/itex] and the time dependence accounted for by the [itex]e^{{{-i\hbar E}\over{t}}}[/itex]?
     
  6. Apr 29, 2012 #5
    Re: "Consider the time-dependent Schrodiner equation ..."

    -h2/(2m) * u''(x) = E * u(x).

    If V(x) = 0, then which implies that u(x) = kei(√(2mE)/h)x
     
  7. Apr 29, 2012 #6
    Re: "Consider the time-dependent Schrodiner equation ..."

    Hang on a moment! I'm not sure if this is correct, but I figured out that we'll only have a ψ(x,t)≠0 if √(2mE)/h = (2n+1)/2 for some integer n≥0. Is that correct?
     
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