# Plotting the Probability Density of the Coulomb Wave Function

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1. Nov 19, 2015

### tomdodd4598

Hey there - I think I have an issue with my 3D density plots of the probability density of the Coulomb wave function. The reason I think something is going wrong is because my plots of |ψ(n=2, l=1, m=-1)|² and |ψ(2, 1, 1)|² are identical, while I would expect them to have the same shape but be rotationally symmetric along different orthogonal axes.

The above images are Mathematica's plots of |ψ(2, 1, -1)|², |ψ(2, 1, 0)|², |ψ(2, 1, 1)|², respectively. As you can see, the first and third are identical, and not the shape of 2p orbitals, while the second plot actually looks like what I would expect - one of the three 2p orbitals.

Here is my wave function - it's possible that the conversion from spherical to Cartesian coordinates is a problem, but I'm not sure:

If the above needs clarifying, do ask. Thanks for any help in advance ;)

2. Nov 19, 2015

### fzero

3. Nov 19, 2015

### Staff: Mentor

If you take suitable linear combinations of $\psi(2,1,-1)$ and $\psi(2,1,+1)$, and then find the probability densities, you do get the $p_x$ and $p_y$ orbitals.

4. Nov 20, 2015

### tomdodd4598

Ah, I've managed to make linear combinations which give the x and y-direction p orbitals, but I am still confused - does that mean some orbital wave functions aren't energy energy eigenfunctions? How/why would that be true?

5. Nov 20, 2015

### Staff: Mentor

They are still energy eigenfunctions. However, $p_x$ and $p_y$ are not eigenfunctions of $\hat{L}_z$.

6. Nov 20, 2015

### Staff: Mentor

7. Nov 20, 2015

### tomdodd4598

Right, I understand that, but I've now got more questions - what are those donut-shaped probability densities? Why aren't they valid orbitals? Also, is there a systematic way for me to combine these 'base' wave functions to create the correct electron orbitals? I managed to make an intuitive guess with the other two p orbitals, but is there a general set of linear combinations which give all of the orbitals?

Last edited: Nov 20, 2015
8. Nov 20, 2015

### Staff: Mentor

They are completely valid orbitals. But they are complex functions, which can make them more difficult to work with in certain situations. The $p_x$ and $p_y$ orbitals are completely real functions, and their orientation along the Cartesian coordinates makes them useful to understand things like chemical bonding.

See https://en.wikipedia.org/wiki/Atomic_orbital#Real_orbitals