# 3-dimensional Fourier transform of an isotropic wavefunction (depending only on $r$)

I met a question asking 3-dimensional Fourier transformation of some wave functions in my QM text.

When I saw the question first, I felt it's simple question that I could solve plugging the wavefunction in the formula. But, it made me hard that 3 Cartesian coordinates are coupled in integral.

The target wavefunction is

$$\psi(\vec{r}) = \frac{e^{- \mu r}}{r}$$

where $\mu$ is a positive constant, and $r=\sqrt{x^{2}+y^{2}+z^{2}}$.

I tried to get the Fourier-transformed function $\phi(\vec{k})$ in $\vec{k}$-space by the formula as follows:

$$\phi(\vec{k}) = \frac{1}{(2 \pi)^{3/2}}\int d^{3}r e^{-i \vec{k} \cdot \vec{r} } \frac{e^{- \mu r}}{r}$$

But, the integration is not simple for me because of the inner product. It seems that it's hard to calculate this integration in the spherical coordinate.

How can I solve it?

• You can choose a reference frame with the z axis along the $\vec{k}$ direction, and write $\vec{k}\cdot\vec{r}=kr\cos \theta$. Now the integral is easy in spherical coordinates.
– GCLL
Jan 5, 2017 at 0:07

The given wavefunction is spatially isotropic, so is the transformed wavefucntion in $\vec{k}$-space. WLOG, let z axis is along the $\vec{k}$, that is, $\vec{k} \cdot \vec{r} = kr \cos \theta$.