Consider the transition from a 3d to a 2p state of hydrogen...

Consider the transition from a 3$d$ to a 2$p$ state of hydrogen in an external magnetic field. Assume that the effects of electron spin can be ignored (which is not actually the case) so that the magnetic field interacts only with the orbital angular momentum. Identify each allowed transition by the $m_{l}$ values of the initial and final states. For each of these allowed transitions, determine the shift of the transition energy from the zero-field value and show that there are three different transition energies.

Key Concepts

Orbital Angular Momentum and Magnetic Quantum Number

In quantum mechanics, electrons in atoms are described by orbital angular momentum quantum numbers. The orbital angular momentum quantum number, l, defines the shape of the orbital (for example, l = 2 for d states and l = 1 for p states) while the magnetic quantum number, m?, specifies the orientation of the orbital’s angular momentum in space. The possible values of m? range from –l to +l in integer steps, and these values are essential for determining how energy levels split in the presence of external fields.

Zeeman Effect

The Zeeman effect describes how the energy levels of atomic states split when an external magnetic field is applied. This splitting arises because the magnetic field interacts with the magnetic dipole moment associated with the orbital angular momentum (and possibly the spin, though it is neglected in this problem). The energy shift is proportional to the magnetic quantum number m? and the strength of the magnetic field, leading to shifts of ±?_BB (with appropriate factors) for different m? values.

Selection Rules for Electric Dipole Transitions

Electric dipole transitions in atoms are governed by selection rules, which ensure conservation of angular momentum. For orbital transitions, the rules require that the orbital angular momentum quantum number change by ?l = ±1 and the magnetic quantum number change by ?m? = 0, ±1. These restrictions limit the allowed transitions between initial and final states, as only those pairs of m? values that satisfy ?m? = 0, ±1 will contribute to an observable spectral line.

Energy Shift Calculation and Distinct Transition Energies

When an external magnetic field is present, each atomic state is shifted in energy by an amount proportional to its m? value, typically expressed as ?E = ?_BB m? (ignoring electron spin contributions). In a transition between two states (such as from 3d to 2p), the overall energy shift is given by the difference ?_BB (m?_initial - m?_final). Because the allowed transitions (as determined by the selection rules) yield differences that can take on distinct values (for example, -1, 0, or +1), the resulting spectrum exhibits three distinct shifted transition energies relative to the zero-field case.

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