# About the size of the orbitals $s$, $p$, $d$, etc, in $H$ atom

Can we define a size for the H atom orbitals which are not spherically symmetric, e.g. $$p, d$$ etc? For example, is it meaningful to say that the $$(n+1)p$$ orbital is larger in its extent than $$np$$ orbital ($$n$$ is the principal quantum number)?

• The orbitals are spherically symmetric: en.wikipedia.org/wiki/Spherical_harmonics#/media/… – safesphere May 26 at 7:44
• Only s orbitals are spherically symmetric. – Poutnik May 26 at 8:18
• Why should make less sense to speak about the size of p,d,,f orbitals than about the size of s orbitals? – Poutnik May 26 at 8:24
• @Poutnik as you said, only s orbitals are spherically symmetric. so what do you mean by size? – mithusengupta123 May 26 at 8:33
• @mithusenguota123 It depends what we mean by orbital 1/particular solution of the wave equation 2/ quantum state of electron 3/ geometrical shape described by the surface of the same arbitrary value of $\Psi \cdot \bar \Psi$. But the size of the 3/ can be taken either as the average radius as in one of the answers, either as the maximal radius with the given orbital surface probability limit. – Poutnik May 26 at 9:47

It makes perfect sense to define the average radius of any hydrogen orbital $$\psi_{nlm}(\vec{r})$$, regardless of whether it is spherically symmetric ($$l=0$$) or not:
$$\langle r\rangle = \int r|\psi_{nlm}(\vec{r})|^2 d^3r$$
Evaluating this integral for the hydrogen orbitals $$\psi_{nlm}(\vec{r})$$ you get (see FAMU-FSU Col­lege of En­gi­neer­ing - Expectation powers of $$r$$ for hydrogen) $$\langle r\rangle = a_0 \frac{3n^2 - l(l+1)}{2}$$ where $$a_0$$ is the Bohr ra­dius, about 0.53 Å.
• It's interesting that it decreases with $l$ for fixed $n$. The opposite is true for the wavefunctions in nuclear physics. I guess the difference is because the $1/r$ potential is "soft," so by putting some of the KE into radial motion, you can move the classical turning point farther out. In nuclear physics, the potential is more like a hard wall. – Ben Crowell May 26 at 13:46
You can define $$\langle r\rangle$$ or $$\langle r^2 \rangle$$ for different orbitals and compare them. You can check that the average distance of electron form the nucleus is larger for higher orbitals.