Orbitals

$(n,l,m_l,m_s)$

• $n\in \mathbb{N}$ 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)
  • $l\ge 0$ 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 $-l$ to $l$ 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