Explain why spectral lines of the hydrogen atom are split by an external magnetic field. What determines the number and spacing of these lines?

Atoms can absorb or emit photons, which causes the electrons to transition between different energy levels. The energies of these photons correspond to the differences between these energy levels. When an atom emits photons, these energies create a unique pattern called the spectrum. Spectral lines are the visible lines that appear in an emission spectrum, representing the photons emitted by the atom at specific energy levels.

The Zeeman effect is the phenomenon in which the spectral lines of an atom split into multiple components when subjected to an external magnetic field. This occurs due to the interaction of the magnetic field with the magnetic moment of the electron, which causes its energy levels to shift. In the presence of a magnetic field, the energy levels of a hydrogen atom split into several sub-levels, causing the spectral lines to split as well.

The number of split spectral lines depends on the quantum numbers of the electron in the hydrogen atom. In particular, the quantum number that defines the angular momentum of the electron, called the orbital quantum number (\(l\)), is significant here. For a given value of \(l\), the spectral lines will split into \(2l+1\) components.

The spacing of the split spectral lines is determined by the strength of the external magnetic field and the electron's angular momentum. The difference in energy between the split levels, usually represented by \(\Delta E\), can be approximated by the formula: \(\Delta E = \mu_B g_l B \) Here, \(\mu_B\) is the Bohr magneton, a physical constant; \(g_l\) is the Landé g-factor, which depends on the atomic structure and the electron's quantum numbers; and \(B\) is the strength of the external magnetic field. The greater the product of these three factors, the larger the energy separation between the split levels and the larger the spacing between the split spectral lines. In summary, when a hydrogen atom is subjected to an external magnetic field, the interaction between the magnetic field and the electron's magnetic moment causes the splitting of the spectral lines. The number of split lines depends on the angular momentum quantum number, while the spacing between the split lines is determined by the strength of the magnetic field, the electron's angular momentum, and the Landé g-factor.