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- Homework Statement
- At time t=0, the wave function of a particle is

##\Psi(x,0)=(\frac{\alpha}{\pi})^{\frac{1}{4}}e^{ik_{0}x-ax^{2}/2}##

##\alpha## and ##k_{0}## are real constants

What is the wavefunction at time t in momentum space, ##\tilde{\Psi}(k,t)##?

- Relevant Equations
- ##\tilde{\Psi}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty }^{\infty}dxe^{ikx}\Psi(x)##

##U(t,t_{0})=e^{-\frac{i}{\hbar}\hat{H}t} = e^{-\frac{i}{\hbar}\frac{\hat{p^2}t}{2m}} ##

Since it asks for the time evolution of the wavefunction in the momentum space, I write : ##\tilde{\Psi}(k,t) = < p|U(t,t_{0})|\Psi> = < U^\dagger(t,t_{0})p|\Psi>##

Since ##U(t,t_{0})^\dagger = e^{\frac{i}{\hbar}\frac{\hat{p^2}t}{2m}}##, the above equation becomes

##\tilde{\Psi}(k,t) = e^{\frac{-i}{\hbar}\frac{p^2t}{2m}} < p|\Psi> = e^{\frac{-i}{\hbar}\frac{p^2t}{2m}} \frac{1}{\sqrt{2\pi}}\int_{-\infty }^{\infty}dxe^{ikx}\Psi(x)##

Evaluate this with ##\Psi(x)=(\frac{\alpha}{\pi})^{\frac{1}{4}}e^{ik_{0}x-ax^{2}/2}##, I end up getting :

##\tilde{\Psi}(k,t) = \frac{1}{\sqrt{a}}(\frac{\alpha}{\pi})^{1/4}e^{-\frac{(k-k_{0})^2}{2a}}e^{\frac{-i}{\hbar}\frac{p^2t}{2m}}##

Since ##p=\hbar k##,

##\tilde{\Psi}(k,t) = \frac{1}{\sqrt{a}}(\frac{\alpha}{\pi})^{1/4}e^{-\frac{(k-k_{0})^2}{2a}}e^{\frac{-i}{\hbar}\frac{\hbar^2k^2t}{2m}}##

Is this the right way to solve this problem?

Since ##U(t,t_{0})^\dagger = e^{\frac{i}{\hbar}\frac{\hat{p^2}t}{2m}}##, the above equation becomes

##\tilde{\Psi}(k,t) = e^{\frac{-i}{\hbar}\frac{p^2t}{2m}} < p|\Psi> = e^{\frac{-i}{\hbar}\frac{p^2t}{2m}} \frac{1}{\sqrt{2\pi}}\int_{-\infty }^{\infty}dxe^{ikx}\Psi(x)##

Evaluate this with ##\Psi(x)=(\frac{\alpha}{\pi})^{\frac{1}{4}}e^{ik_{0}x-ax^{2}/2}##, I end up getting :

##\tilde{\Psi}(k,t) = \frac{1}{\sqrt{a}}(\frac{\alpha}{\pi})^{1/4}e^{-\frac{(k-k_{0})^2}{2a}}e^{\frac{-i}{\hbar}\frac{p^2t}{2m}}##

Since ##p=\hbar k##,

##\tilde{\Psi}(k,t) = \frac{1}{\sqrt{a}}(\frac{\alpha}{\pi})^{1/4}e^{-\frac{(k-k_{0})^2}{2a}}e^{\frac{-i}{\hbar}\frac{\hbar^2k^2t}{2m}}##

Is this the right way to solve this problem?

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