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Time Evolution for particle with potential suddenly removed

  1. Mar 19, 2015 #1
    1. The problem statement, all variables and given/known data

    This is a problem from my Statistical Mechanics book by Pathria.

    At ##t=0##, the ground state wavefunction of a one-dimensional quantum harmonic oscillator with potential ##V(x)=\frac{1}{2}\omega_0^2 x^2## is given by,

    [tex]\psi(x,0)=\frac{1}{\pi^{1/4} \sqrt{a}}e^{-\frac{x^2}{2a^2}}[/tex]

    where ##a=\sqrt{\frac{\hbar}{m \omega_0}}##. At ##t=0##, the harmonic potential is abruptly removed. Use the momentum representation of the wavefunction at ##t=0## and the time-dependent Schrodinger equation to determine the spatial wavefunction and density at time ##t>0##.
    2. Relevant equations
    [tex]i\hbar \frac{\partial}{\partial t}\psi=\hat{H} \psi[/tex]

    3. The attempt at a solution

    So first I took the Fourier transform of the wave function given at ##t=0## to get,

    [tex]\Phi(k,0)=-\frac{\sqrt{a}}{\pi^{1/4}}e^{\frac{1}{2}i a ^2 k^2}[/tex]

    So now that at ##t>0## the potential is removed, the new Hamiltonian is that of a free particle. I want to see how the original wave function evolves with time with this new potential. So I will multiply it by the time evolution operator.

    [tex]\Phi(k,t)=\Phi(k,0)e^{-i \frac{\hat{H}t}{\hbar}}=\Phi(k,0)e^{-i \frac{\hat{p}^2 t}{2m\hbar}}[/tex]

    Generally, we can write this as,

    [tex]|\psi(t)>=e^{-i \frac{\hat{p}^2 t}{2m\hbar}}|\psi_0>[/tex]

    The whole goal is to find the spatial wave function so lets project it onto ##x##.

    [tex]<x|\psi(t)>=<x|e^{-i \frac{\hat{p}^2 t}{2m\hbar}}|\psi_0>[/tex]

    What is given in the beginning of the problem is ##<x|\psi_0>## so I do the following.

    [tex]<x|\psi(t)>=\int <x|e^{-i \frac{\hat{p}^2 t}{2m\hbar}}|x'><x'|\psi_0>dx'[/tex]

    Now, the Hamiltonian in the exponential purely depends on the momentum operator, so it would be easiest to deal with momentum eigenkets. So I will insert two more identities in there.

    [tex]<x|\psi(t)>=\int <x|p'><p'|e^{-i \frac{\hat{p}^2 t}{2m\hbar}}|p><p|x'><x'|\psi_0>dx'dpdp'[/tex]

    [tex]<x|\psi(t)>=\int <x|p'>e^{-i \frac{p^2 t}{2m\hbar}}\delta(p'-p)<p|x'><x'|\psi_0>dx'dpdp'[/tex]

    [tex]<x|\psi(t)>=\int <x|p>e^{-i \frac{p^2 t}{2m\hbar}}<p|x'><x'|\psi_0>dx'dp[/tex]

    My problem is, I do not know what ##<x|p>## and ##<p|x'>## are. I remember back when I took quantum mechanics dealing with these types of terms but for the life of me I cannot remember what their value is or why! If anyone can help I would greatly appreciate it!

    EDIT:
    A ha! I just realized that these terms are simply the plane wave solution to the free particle! I will go though it now to make sure I have no other issues.
     
    Last edited: Mar 19, 2015
  2. jcsd
  3. Mar 20, 2015 #2
    Hi. I think you're taking a detour here...
    You obtained the full wave function in momentum space:
    Φ(k,t ) = Φ(k,0)⋅exp(–ik2t/2m)
    (remember in momentum space p-hat is just p, not a differential operator)
    So to get the spatial wave function, just take the inverse Fourier transform.
     
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