The hydrogen atom consists of a single proton in the nucleus orbited by a single electron. According to classical electromagnetism, this system should be stable, with the electron orbiting the nucleus indefinitely. However, classical physics fails to explain certain phenomena observed in hydrogen, such as the discrete spectrum of light emitted or absorbed by hydrogen atoms.

Quantum mechanics successfully describes the behavior of hydrogen by treating the electron's position and energy levels as quantized, meaning they can only take on certain discrete values. The problem arises when attempting to solve the Schrödinger equation, which describes the behavior of quantum mechanical systems, for the hydrogen atom.

The Schrödinger equation is a differential equation that is difficult to solve exactly for systems with more than one electron. For hydrogen, an analytical solution is possible due to its simplicity, but it still requires sophisticated mathematical techniques. The solution yields a set of allowed energy levels for the electron, known as the energy spectrum, which matches the observed spectrum of hydrogen.

However, even with the analytical solution, there are aspects of the hydrogen atom that pose challenges. One of the most notable is the fine structure of the hydrogen spectrum, which includes small energy corrections due to relativistic effects and the interaction between the electron's magnetic moment and its orbital angular momentum. These corrections are crucial for achieving agreement between theory and experiment at high precision.