Quantum mechanics

The hydrogen atom: energy levels and orbitals

The Coulomb potential gives hydrogen energy levels indexed by principal quantum number $n$ ; orbitals $|\psi_{nlm}|^2$ are probability densities, not planetary paths.

Hydrogen is the first exactly solvable Coulomb problem in quantum mechanics. Separating the Schrödinger equation in spherical coordinates yields states labeled by $n,l,m_l$ . Its simple spectrum explains spectral lines and underpins atomic structure.

E_n=-\frac{13.6\,\mathrm{eV}}{n^2},\quad n=1,2,...;\qquad l=0,...,n-1,\quad m_l=-l,...,l

Definition: Quantum numbers

The principal number $n$ sets the energy and characteristic size. Orbital number $l$ sets angular momentum $L^2=l(l+1)\hbar^2$ ; magnetic number $m_l$ gives its projection $L_z=m_l\hbar$ . For each $n$ , there are $n^2$ states excluding spin in the nonrelativistic hydrogen model.

Select 1s, 2s, or 2p to inspect a spatial model of

|\psi_{nlm}|^2

and its nodes; the cloud represents probability, not a moving path.

Spectral lines and transitions

In the nonrelativistic Coulomb model, energy depends only on $n$ , so several states are degenerate. When the atom absorbs or emits a photon, its energy equals the level difference: $h\nu=|E_f-E_i|$ . Fine structure and external effects partly lift the ideal degeneracy.

Example: Transition photon

An electron drops from $n=3$ to $n=2$ . Find the emitted photon energy.

Solution

$E_2=-13.6/4=-3.40$ eV; $E_3=-13.6/9=-1.51$ eV. Emission gives $E_\gamma=E_3-E_2=1.89$ eV.

The Coulomb potential is spherically symmetric, so the problem separates into radial and angular parts. In the nonrelativistic model, energy depends on principal quantum number $n$ but is degenerate in $l,m$ ; these label energy scale and orbital angular momentum. The Bohr radius $a_0=4\pi\epsilon_0\hbar^2/(m_e e^2)$ sets the length scale, and bound levels are $E_n=-13.6\,\text{eV}/n^2$ . Spectral lines occur when a photon satisfies $h\nu=E_i-E_f$ . Fine structure and QED corrections split levels that the basic Coulomb model treats as degenerate.

Quick check

In nonrelativistic hydrogen, which quantum number directly determines the energy?

What does an atomic orbital represent?

References

David J. Griffiths, Darrell F. Schroeter (2018). Introduction to Quantum Mechanics