What is the correct separable Schrödinger equation in spherical coordinates?

Question

What is the correct separable Schrödinger equation in spherical coordinates?

Some articles use this formula: $$ \Psi(r,\theta,\phi) = R(r)\cdot\Theta(\theta)\cdot \Phi(\phi), $$ and some of them use: $$ \Psi(r,\theta,\phi) =\frac{R(r)\cdot \Theta(\theta)\cdot \Phi(\phi) }{r}. $$

So which is true?

Answer 1

Both formulas are correct. And it is essentially a matter of taste which one you prefer:

• If you use the separation approach $$\Psi(r,\theta,\phi) = R(r) \cdot \Theta(\theta) \cdot \Phi(\phi)$$ then you get the radial differential equation $${\frac {1}{r^{2}}}{\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)-{\frac {\ell (\ell +1)}{r^{2}}R+{\frac {2m}{\hbar ^{2}}}\left[E-V(r)\right]R=0}$$ (like in Wikipedia - Particle in a spherically symmetric potential - Derivation of the radial equation) with the normalization condition $$\int_0^\infty dr\ r^2 |R(r)|^2=1.$$
• If you use the separation approach $$\Psi(r,\theta,\phi) = \frac{u(r)}{r} \cdot \Theta(\theta) \cdot \Phi(\phi)$$ then you get the radial differential equation $$\frac {d^2u}{dr^2}-{\frac {\ell (\ell +1)}{r^{2}}u+{\frac {2m}{\hbar ^{2}}}\left[E-V(r)\right]u=0}$$ with the normalization condition $$\int_0^\infty dr\ |u(r)|^2=1.$$

At the end you get the same solutions $\Psi(r,\theta,\phi)$ in both cases because $R(r)=\frac{u(r)}{r}$. However, the approach with $u(r)$ leads to simpler math and has more similarity with a 1-dimensional Schrödinger equation.

Answer 2

If I am not wrong, then the second one is used for (I guess) when we solve the radial part. We have to put $R(r)=u(r)/r$, so maybe in second one they have written $R(r)/r$ instead of $R(r)$ is because they put directly $R(r)/r$. I am not sure though.