Forming Hydrogen wave functions with real spherical harmonics

[dipole]

Hi, I'm a little confused about how to apply the real spherical harmonics when building a hydrogen wave function.

I'm doing a computational project, so I want to work with a wave function which is strictly real, and I'm hoping I can do so by building the orbitals using the real spherical harmonics, $p_x, p_y, p_z,...$ etc. Can someone give me a link or reference as to how that works?

Thanks.

 

[DrClaude]

I'm doing a computational project, so I want to work with a wave function which is strictly real, and I'm hoping I can do so by building the orbitals using the real spherical harmonics, $p_x, p_y, p_z,...$ etc. Can someone give me a link or reference as to how that works?

Not sure exactly what you want, but you construct the angular part of the p orbitals by transforming from the usual basis of spherical harmonics $Y_{1,1}$, $Y_{1,0}$, $Y_{1,-1}$ to
$$
p_x = \frac{1}{\sqrt{2}} \left( Y_{1,-1} - Y_{1,1} \right) \\
p_y = \frac{i}{\sqrt{2}} \left( Y_{1,-1} + Y_{1,1} \right) \\
p_z = Y_{1,0}
$$

 

[dipole]

Hi, so my question is, does the wave function,

$$\psi_{nlm}(\vec{x}) = R_{n\ell}(r)p_{i}(\theta,\phi)$$

where $p_{i}(\theta,\phi)$ are the real spherical harmonics, satisfy the same Schrödinger equation as the wave function

$$\psi_{nlm}(\vec{x}) = R_{n\ell}(r)Y_{\ell}^m(\theta,\phi)$$ ?

Thanks.

 

[tom.stoer]

Just act with H, especially with the part containing the phi-derivative, on the new functions and check what happens with m

 

[DrDu]

Hi, so my question is, does the wave function,

$$\psi_{nlm}(\vec{x}) = R_{n\ell}(r)p_{i}(\theta,\phi)$$

where $p_{i}(\theta,\phi)$ are the real spherical harmonics, satisfy the same Schrödinger equation as the wave function

$$\psi_{nlm}(\vec{x}) = R_{n\ell}(r)Y_{\ell}^m(\theta,\phi)$$ ?

Thanks.

Yes, the states with different m are degenerate in the hydrogen atom. Hence any linear combination (notably the real ones) of these states is again an eigenstate.

 

[DrClaude]

To add to what Tom and DrDu wrote, just note that while the real spherical harmonics are eigenfunctions of the Hamiltonian of the hydrogen atom, they are not simultaneously eigenfunctions of $\hat{l}_z$, as the complex $Y_{l,m}$ are.