Question

Why do we say that `s` orbitals have zero angular momentum?

Answer 1

The angular momentum of any `s` orbital is zero, since the wave function for an `s` orbital has no angular dependence.

In other words, recall that angular momentum gives rise to irregular shapes of a given atomic orbital. Well, all `s` orbitals are spherically symmetric, so angular momentum has no influence on the shape.

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Consider some of the `s` orbital wave functions:

`psi_(1s)(r,theta,phi) = 1/sqrtpi (Z/a_0)^(3//2)e^(-Zr//a_0)`

`psi_(2s)(r,theta,phi) = 1/(4sqrt(2pi)) (Z/a_0)^(3//2)(2 - (Zr)/a_0)e^(-r//2a_0)`

`psi_(3s)(r,theta,phi) = 1/(81sqrt(3pi)) (Z/a_0)^(3//2) (27 - 18 (Zr)/a_0 + 2((Zr)/(a_0))^2)e^(-Zr//3a_0)`

If you look closely, there is no `theta` or `phi` anywhere in these wave functions. As a result, the orbitals they describe are perfect spheres, having zero angular momentum.