Hydrogen energy levels and wavefunctions

Hydrogen energy spectrum

Solving the Schrödinger equation for the Coulomb potential gives hydrogen bound-state energies

E_n=-rac{13.6 eV}{n^2}

for

$n=1,2,3,ldots$

More precisely, the energy includes the reduced mass and fundamental constants, but the simple formula is extremely useful. The negative sign means the electron is bound. As $n oinfty$, $E_n o0$, corresponding to ionization.

Principal quantum number

In the ideal nonrelativistic hydrogen atom without fine structure, energy depends only on the principal quantum number $n$, not on $l$ or $m$. This creates degeneracy: many different states have the same energy.

For a given $n$, allowed values are

$l=0,1,ldots,n-1$

and for each $l$,

$m=-l,ldots,l$

The total number of spatial states for a given $n$ is

$n^2$

not including spin.

Wavefunction structure

Hydrogen wavefunctions have the separated form

$psi*{nlm}(r, heta,phi)=R*{nl}(r)Y_l^m( heta,phi)$

The radial function controls dependence on distance from the nucleus. The spherical harmonic controls angular shape.

The probability of finding the electron in a volume element is

$dP=|psi*{nlm}|^2d^3r$

with

$d^3r=r^2sin heta,dr,d heta,dphi$

Radial probability distribution

To ask for the probability of finding the electron between $r$ and $r+dr$, integrate over angles. The radial probability density is

$P(r)=r^2|R*{nl}(r)|^2$

up to angular normalization conventions. The factor $r^2$ comes from the volume of spherical shells. Even if $R(r)$ is largest near the origin, the most probable radius may be away from the origin because larger shells contain more volume.

For the ground state, the most probable radius is the Bohr radius:

a_0=rac{4piepsilon_0hbar^2}{m_ee^2}

approximately

$a_0approx0.529 ext{Å}$

Ground state

The hydrogen ground state has quantum numbers

$n=1,quad l=0,quad m=0$

It is spherically symmetric. Its wavefunction is proportional to

$psi*{100}propto e^{-r/a_0}$

The probability density is highest at the nucleus, but the radial probability distribution peaks at $r=a_0$.

Nodes

Hydrogen orbitals have radial and angular nodes. The total number of nodes is

$n-1$

The number of angular nodes is $l$, and the number of radial nodes is

$n-l-1$

Nodes are surfaces or radii where the wavefunction is zero.

Orbitals, not orbits

Hydrogen orbitals are probability distributions, not classical electron paths. The electron is not a tiny planet moving along a definite orbit in the Schrödinger picture. The orbital describes probability amplitudes for position measurements.

The big idea

Hydrogen's quantum states are labeled by $n$, $l$, and $m$. In the simplest model, energies follow $E_n=-13.6 eV/n^2$ and depend only on $n$. Wavefunctions split into radial and angular parts, giving the familiar orbital shapes and probability distributions of atomic physics.