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Xyius
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Homework Statement
This is a problem from my Statistical Mechanics book by Pathria.
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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##.
Homework Equations
[tex]i\hbar \frac{\partial}{\partial t}\psi=\hat{H} \psi[/tex]
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 let's 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.
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