# Probability wave function is still in ground state after imparting momentum

• xoxomae
In summary, we have a particle with an instantaneous force imparting a momentum to the ground state SHO wave function. We need to find the probability that the system is still in its ground state. To do this, we use the wave function and solve for the probability using the Fourier transform and the energy of the ground state. To solve the integral by hand, we can use integration by parts or a shortcut where the integral of the wave function squared is equal to 1.
xoxomae

## Homework Statement

An interaction occurs so that an instantaneous force acts on a particle imparting a momentum ## p_{0} = \hbar k_{0}## to the ground state SHO wave function. Find the probability that the system is still in its ground state.

## Homework Equations

##\psi _{0} =\left( \frac{mw}{\hbar\pi} \right )^\frac{1}{4} e^{-mwx^{2}/2\hbar} ##

## The Attempt at a Solution

[/B]
##\Psi(x)=\psi_{0}e^{ik_{0}x}##
This wave function gives a <p> =##\hbar k_{0}##
Im confused whether this is the correct Fourier transform to do.
##c(k)=\frac{1}{2\pi}^{0.5}\int_{-\infty}^{\infty}e^{-ikx}\psi_{0}e^{ik_{0}x}dx##
And then solving for when the wavenumber of the ground state using E0=0.5* hbar * w.
Therefore
##c(k)^2 ## = Probability

Is this correct?

You want to know the probability of a particular result of a measurement of energy.

Ah right. So would this be the probability of the system still being in the ground state?
##\left | \int_{-\infty}^{\infty} \psi_{0} \psi_{0} e^{ik_{o}x}dx\right |^2=P##

Well done - left out a star, but ##\psi_0## is real so...

Thank you. Do you by any chance know any way to solve that integral by hand? I solved it on mathematica and it gave me ##P=e^\frac{-{k_{0}}^{2}\hbar}{4mw}## but I have no idea how to solve it by hand. Thanks again.

Hint: integrate by parts. (That's normal for quantum.)
You may be able to shortcut using ##\int_\infty \psi_0^2\; dx = 1##

## 1. What is a probability wave function?

A probability wave function is a mathematical description of the quantum state of a system, which represents the probability of finding a particle in a particular location or state. It is a fundamental concept in quantum mechanics and is used to describe the behavior of subatomic particles.

## 2. What does it mean for a probability wave function to be in the ground state?

The ground state of a probability wave function is the lowest energy state that a particle can occupy. This state has the highest probability of being observed and represents the most stable configuration of the system. In other words, the particle is in its lowest possible energy state.

## 3. Can a probability wave function remain in the ground state after being imparted with momentum?

Yes, it is possible for a probability wave function to remain in the ground state even after being imparted with momentum. This is because the ground state is the lowest energy state, so the particle will always try to return to this state. However, the probability of finding the particle in a higher energy state will increase.

## 4. How does imparting momentum affect the probability wave function?

Imparting momentum to a particle changes the shape and amplitude of its probability wave function. The higher the momentum, the more spread out the wave function becomes, meaning there is a higher chance of finding the particle in a larger area. However, the overall probability of finding the particle in the ground state will decrease.

## 5. Why is the ground state important in quantum mechanics?

The ground state is important in quantum mechanics because it is the most stable and lowest energy state of a system. It serves as the starting point for understanding the behavior of particles and their interactions. The ground state also plays a crucial role in determining the properties and behavior of materials at the atomic level.

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