# Time evolution of a particle in momentum space

• Foracle
In summary, the conversation discusses the time evolution of the wavefunction in the momentum space. The resulting equation is ##\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}}## with the use of the relevant equation, and after evaluating with a given wavefunction. There may be some sign issues, but the final result for ##\tilde{\Psi}(k, t)## seems to be correct.
Foracle
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?

Last edited:
Your work looks good to me except for some sign issues.
Foracle said:
Relevant Equations:: ##\tilde{\Psi}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty }^{\infty}dxe^{ikx}\Psi(x)##
Check to see if the factor of ##e^{ikx}## in the integrand should actually be ##e^{-ikx}##. With ##e^{-ikx}##, then I think your result for ##\tilde{\Psi}(k, t)## is correct. But, if you use your way of writing ##\tilde{\Psi}(k)##, then I believe you would get a result for ##\tilde{\Psi}(k, t)## with a factor of ##\large e^{-\frac{(k+k_0)^2}{2a}}## instead of ##\large e^{-\frac{(k-k_0)^2}{2a}}##.

(But maybe I'm the one who's getting the signs wrong. )

Foracle

## What is the significance of studying the time evolution of a particle in momentum space?

The time evolution of a particle in momentum space allows us to understand the behavior and movement of a particle over time. It also helps us to make predictions about the future behavior of the particle and to analyze its interactions with other particles.

## How is the time evolution of a particle in momentum space related to the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a particle. The time evolution of a particle in momentum space takes into account this uncertainty and shows how the particle's momentum changes over time.

## What factors affect the time evolution of a particle in momentum space?

The time evolution of a particle in momentum space is affected by various factors such as the particle's initial momentum, the forces acting on the particle, and the presence of other particles in the surrounding environment. These factors can cause the particle's momentum to change over time.

## How is the time evolution of a particle in momentum space represented mathematically?

The time evolution of a particle in momentum space is represented by a mathematical equation known as the Schrödinger equation. This equation describes the wave function of the particle, which gives information about its position and momentum at any given time.

## What are some real-world applications of studying the time evolution of a particle in momentum space?

Studying the time evolution of particles in momentum space has various applications in fields such as quantum mechanics, particle physics, and materials science. It is also used in the development of technologies such as quantum computing and in understanding the behavior of particles in complex systems.

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