Orbitals

#Physics #Chemistry

$(n,l,m_l,m_s)$

• $n\in\mathbb{N}$ is the principle quantum number and it dictates the orbital size and energy level
• $l\in\mathbb{N}:[0,n-1]$ is the secondary (angular momentum) quantum number and it dictates the subshell type (s, p, d, or f)
  • $\displaystyle l\geq 0$ comes from the associated Legendre polynomial part of the spherical wave function
• $m_l\in\mathbb{N}:[-l,l]$ is the magnetic quantum number and it dictates the orbital orientation (x, y, z, or etc.)
  • The restriction from $\displaystyle -l$ to $\displaystyle l$ comes again from the associated Legendre polynomial
• $m_s\in{-\frac{1}{2},\frac{1}{2}}$ is the spin quantum number and it dictates the electron spin direction