QM: Solving Free Particle with 3D Wave Function Using Fourier Transform

[MementoMori96]

Homework Statement

Hi, i have this problem:
In a 3D space, a free particle is described by :
$$ \Psi = Ne^{-ar} $$ with $$ r=| \vec r | $$
at the time t=0 .
How can we write the wave function whit $$ \hbar \vec k $$ ?

Homework Equations

The Attempt at a Solution

I know how to resolve this exercise in 1D but here i have to calculate a Fourier Transform in a 3D space, how can i do ?

 

[eys_physics]

In a 3D space, a free particle is described by :

Ψ=Ne−arΨ=Ne−ar​

\Psi = Ne^{-ar} with

From where did you get this formula? It is not the wave function of a free particle, but for a $$1/r$$ potential.

 

[MementoMori96]

Hi, the problem gives this formula as a wave function to study and, in the last point of exercise, it says: "suppose that it describes a free particle ... "

 

[TSny]

Here's an example of the Fourier transform of a similar function of r.
http://blog.cupcakephysics.com/elec...urier-transform-of-the-coulomb-potential.html

 

[Orodruin]

From where did you get this formula? It is not the wave function of a free particle, but for a $$1/r$$ potential.

It is not inconceivable that it describes a free particle at a single time. For example, you could imagine the particle being in the ground state of a $1/r$ potential for times $t \leq 0$ and the potential is turned off at $t = 0$. This would correspond to a non-adiabatic transition and the state would have to be projected onto the free particle eigenstates in order to determine its time evolution. As alluded to by TSny, this can be done through Fourier transform.