s-orbital

An s-orbital is a spherical region of space around the nucleus where an electron with zero orbital angular momentum is most likely to be found. The letter ”s” is of spectroscopic origin, standing for ‘sharp’.

One-electron s-orbital wavefunctions can be expressed using the total wavefunction of a hydrogenic atom:

$\psi=R_{n,l}(r)Y_l^{\vert m_l\vert}(\theta,\phi)$

where

$R_{n,l}(r)=N_{n,l}\frac{1}{r}\biggr$\frac{r}{na_0}\biggr$^{l+1}e^{-\frac{r}{na_0}}L_{n-l-1}^{2l+1}\biggr$\frac{2r}{na_0}\biggr$$ , the radial wavefunction, is the radial component of $\psi$ .
$Y_l^{\vert m_l\vert}(\theta,\phi)=N_l^{\vert m_l\vert}P_l^{\vert m_l\vert}(cos\theta)e^{im_l\phi}$ , the spherical harmonics, is the angular component of $\psi$ .
$L_{n-l-1}^{2l+1}=\sum_{k=0}^{n-l-1}(-1)^k\frac{(n+l)!}{k!(k+2l+1)!(n-l-1-k)!}\biggr$\frac{2r}{na_0}\biggr$^k$ are the associated Laguerre polynomials.
$P_l^{\vert m_l\vert}(cos\theta)=\frac{1}{2^ll!}(1-cos^2\theta)^{\frac{\vert m_l\vert}{2}}\frac{d^{l+\vert m_l\vert}}{dcos\theta^{l+\vert m_l\vert}}(cos^2\theta-1)^l$ are the associated Legendre polynomials.
$N_{n,l}=\sqrt{\frac{(n-l-1)!}{2n(n+l)!}\frac{2^{2l+3}}{na_0}}$ is the normalisation constant of the radial wavefunction.
$N_l^{\vert m_l\vert}=\frac{1}{\sqrt{2\pi}}\biggr\[\frac{2l+1}{2}\frac{(l-\vert m_l\vert)!}{(l+\vert m_l\vert)!}\biggr\]^{\frac{1}{2}}$ is the normalisation constant of the spherical harmonics.
$n$ is the principal quantum number, where $n=1,2,\cdots$
$l$ is the orbital angular momentum quantum number (also known as azimuthal quantum number), where $l=0,1,2,\cdots$ and $l\leq n-1$ .
$m_l$ is the magnetic quantum number, where $-l\leq m_l\leq l$ .

The principal quantum number, $n$ , is also called a shell. Since the first shell is characterised by $n=1$ , the one-electron s-orbital wavefunction in the first shell, known as the 1s orbital, is defined by the set of quantum numbers $\{n,l,m_l\}=\{1,0,0\}$ . Substituting these values into the explicit formula for $R_{n,l}(r)Y_l^{\vert m_l\vert}(\theta,\phi)$ yields:

$R_{1,0}(r)Y_0^{0}(\theta,\phi)=\frac{1}{\sqrt{a_0^3\pi}}e^{-\frac{r}{a_0}}\;\;\;\;\;\;\;\;472$

Similarly, the one-electron s-orbital wavefunction in the second shell, known as the 2s orbital, is defined by the set of quantum numbers $\{n,l,m_l\}=\{2,0,0\}$ and has the formula:

$R_{2,0}(r)Y_0^{0}(\theta,\phi)=\frac{1}{4\sqrt{2\pi}}\sqrt{\biggr$\frac{1}{a_0}\biggr$^3}\biggr$2-\frac{r}{a_0}\biggr$e^{-\frac{r}{2a_0}}\;\;\;\;\;\;\;\;473$

Since $l=m_l=0$ for all s-orbitals, their wavefunctions are independent of $\theta$ and $\phi$ . This implies that the values of an s-orbital wavefunction are the same in all directions in 3D space for a given $r$ . In other words, s-orbital wavefunctions are spherically symmetrical.