Introduction: Labeling Hydrogen States
Each hydrogen state is uniquely specified by three quantum numbers: \(n\), \(l\), and \(m\). They arise from the three parts of the separated solution.
The Three Quantum Numbers
Principal quantum number n:
- Values: \(n = 1, 2, 3, \ldots\)
- Determines energy: \(E_n = -13.6/n^2\) eV
- Labels the "shell"
Orbital angular momentum l:
- Values: \(l = 0, 1, \ldots, n-1\)
- Determines \(|\vec{L}| = \hbar\sqrt{l(l+1)}\)
- Labels: s, p, d, f... for \(l = 0, 1, 2, 3...\)
Magnetic quantum number m:
- Values: \(m = -l, \ldots, l\)
- Determines \(L_z = m\hbar\)
- Affects orientation in magnetic field
Counting States
For each \(n\): number of states = \(\sum_{l=0}^{n-1}(2l+1) = n^2\)
\(n = 1\): 1 state (1s)
\(n = 2\): 4 states (2s, 2p_x, 2p_y, 2p_z)
\(n = 3\): 9 states (3s, 3p, 3d)
The Quantum Connection
These quantum numbers directly determine atomic properties: \(n\) sets binding energy, \(l\) determines orbital shape and penetration, \(m\) determines response to magnetic fields. Adding spin (\(m_s = \pm 1/2\)) doubles the state count and leads to the Pauli exclusion principle for building up the periodic table.