How to Find Eigenfunction and Kinetic Energy in an Infinite Square Well?

[stunner5000pt]

Homework Statement

A particle of mass m is in the ground state of the infinite square well $x \in [0,a]$
a) Find the corresponding eigenfunction of the Hamiltonian in the momentum representation.
b) Find the expectation value of the kinetic energy

Homework Equations

For the ground state of the infinite square well
$$\psi(x) = \sqrt{\frac{2}{a}}} \sin \frac{\pi x}{a}$$

The Attempt at a Solution

TO find the corresponding eigenfunction of the Hamiltonian in the momentum representation do i simply have to use this: ?

$$\Psi(p,t) = \frac{1}{2\pi\hbar} \int_{0}^{a} \Psi(x,t) \exp\left(\frac{-ipx}{\hbar}\right) dx$$

so find the eignefunction in momentum representation??

For part b, the expectation value of the kinetic energy
The kinetic energy is given by
$$\hat{T} = \frac{\hat{p}^2}{2m}$$
so to find the expectation value of the kinetic energy do i have to just do this

$$<\hat{T}> = \int_{0}^{a} \Psi(x,t)^* \frac{\hat{p}^2}{2m} \Psi(x,t) dx$$

to get the expectation value of the kinetic energy?

Thanks for your help, it is greatly appreciated!

 

[nrqed]

Homework Statement

A particle of mass m is in the ground state of the infinite square well $x \in [0,a]$
a) Find the corresponding eigenfunction of the Hamiltonian in the momentum representation.
b) Find the expectation value of the kinetic energy

Homework Equations

For the ground state of the infinite square well
$$\psi(x) = \sqrt{\frac{2}{a}}} \sin \frac{\pi x}{a}$$

The Attempt at a Solution

TO find the corresponding eigenfunction of the Hamiltonian in the momentum representation do i simply have to use this: ?

$$\Psi(p,t) = \frac{1}{2\pi\hbar} \int_{0}^{a} \Psi(x,t) \exp\left(\frac{-ipx}{\hbar}\right) dx$$

so find the eignefunction in momentum representation??

Yes (I did not check the normalization constant but that's the correct approach). However, in this simplest case you can get the answer quickly by simply writing the sin as a sum of imaginary exponentials, which gives you directly the wavefunction as a sum of two momentum eigenstates)

For part b, the expectation value of the kinetic energy
The kinetic energy is given by
$$\hat{T} = \frac{\hat{p}^2}{2m}$$
so to find the expectation value of the kinetic energy do i have to just do this

$$<\hat{T}> = \int_{0}^{a} \Psi(x,t)^* \frac{\hat{p}^2}{2m} \Psi(x,t) dx$$

to get the expectation value of the kinetic energy?

Thanks for your help, it is greatly appreciated!

That's correct.