Question

What do the subscripts in the wave function `psi_(nlm_l)(r,theta,phi)` indicate?

Answer 1

They specify what orbitals the wave functions describe. They are the quantum numbers.

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The wave function `psi_(nlm_l)(r,theta,phi) = R_(nl)(r)Y_(l)^(m_l)(theta,phi)` for the hydrogen atom describes the state of its atomic orbitals, and require the use of quantum numbers to describe which one we refer to.

So if we wish to specify a `2p` orbital, we pick the wave function with `n = 2`, `l = 1`, and some `m_l` in the set `{-1,0,+1}`. We then know that we refer to:

`psi_(21-1)(r,theta,phi) = R_(21)(r)Y_(1)^(-1)(theta,phi) = psi_(2px)`

`psi_(211)(r,theta,phi) = R_(21)(r)Y_(1)^(1)(theta,phi) = psi_(2py)`

`psi_(210)(r,theta,phi) = R_(21)(r)Y_(1)^(0)(theta,phi) = psi_(2pz)`

And these would be given by one radial component (specifying that we refer to `2p` orbitals):

`R_(21)(r) = 1/(2sqrt6)(Z/a_0)^(3//2)sigmae^(-sigma//2)`

where `sigma = Zr//a_0` (`a_0 = "0.529177 pm"` being the Bohr radius and `Z` being the atomic number),

and three different angular components (specifying which particular `2p` orbitals we refer to):

`Y_(1)^(-1)(theta,phi) = 1/(2sqrt2) sqrt(3/pi) sintheta e^(-iphi)`

`Y_(1)^(1)(theta,phi) = 1/(2sqrt2) sqrt(3/pi) sintheta e^(iphi)`

`Y_(1)^(0)(theta,phi) = 1/2 sqrt(3/pi)costheta`

And in fact, these three `Y_(l)^(m_l)(theta,phi)` can be used represent any hydrogen-like `p` orbital, not just the `2p`, as they do not depend on the principal quantum number `n`.