QM: free particle

1. Nov 29, 2017

MementoMori96

1. The problem statement, all variables and given/known data
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 $$\hbar \vec k$$ ?

2. Relevant equations

3. 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 ?

2. Nov 30, 2017

eys_physics

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

3. Nov 30, 2017

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 ... "

4. Nov 30, 2017

5. Dec 2, 2017

Orodruin

Staff Emeritus
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.