The three-dimensional Schrödinger equation has stationary-state solutions for the hydrogen atom potential energy only if three conditions are satisfied:

• Energy $E_n=-13.60\text{eV/}{n}^{2}n=1\text{,}2\text{,}3\text{,}\dots$

• Angular momentum $L=\sqrt{l(l+1)}\text{ℏ}l=0\text{,}1\text{,}2\text{,}3\text{,}\dots \text{,}n-1$

• $z$-component of angular momentum

  $L_z=m\text{ℏ}m=-l\text{,}-l+1\text{,}\dots \text{,}0\text{,}\dots \text{,}l-1\text{,}l$

Each state is characterized by quantum numbers $\left(n\text{,}l\text{,}m\right)$, but the energy depends only on $n$.

The probability of finding the electron within a small distance interval $\delta r$ at distance $r$ is

where $P_r\left(r\right)=4\pi r^2\left|R_{nl}\left(r\right)\right|{}^2$ is the radial probability density.

Graphs of $P_r\left(r\right)$ suggest that the electrons are arranged in shells.

The potential energy is electron-nucleus plus electronelectron. In the independent particle approximation, each electron is described by the same quantum numbers $\left(n\text{,}l\text{,}m\text{,}{m}_s\right)$ used for the hydrogen atom. The energy of a state depends on $n$ and $l$. For each $n$, energy increases as $l$ increases.

• High-$l$ states correspond to circular orbits. These stay outside the core.

• Low-$l$ states correspond to elliptical orbits. These penetrate the core to interact more strongly with the nucleus. This interaction lowers their energy.

  

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The electron has an inherent angular momentum $\vec{S}$ and magnetic moment $\vec{\mu }$ as if it were spinning. The spin angular momentum has a fixed magnitude $S=\sqrt{s(s+1)}\text{ℏ}$, where $s=\frac{1}{2}$. The $z$-component is $S_z=m_s\text{ℏ}$, where $m_s=\pm \frac{1}{2}$. These two states are called spin-up and spin-down. Each atomic state is fully characterized by the four quantum numbers $\left(n\text{,}l\text{,}m\text{,}{m}_s\right)$.

The Pauli exclusion principle says that no more than one electron can occupy each quantum state. The periodic table of the elements is based on the fact that the ground state is the lowest-energy electron configuration compatible with the Pauli principle.

The excited-state population decreases exponentially as

where $\tau =1\text{/}r$ is the lifetime and $r$ is the decay rate. It’s not possible to predict when a particular atom will decay, but the probability is

Stimulated emission of an excited state can be caused by a photon with $E_{\text{photon}}=E_2-E_1$. Laser action can occur if $N_2>N_1$, a condition called a population inversion.