A hydrogen atom in a particular orbital angular momentum state is found to have $j$ quantum numbers $\frac{7}{2}$ and $\frac{9}{2}$. If $n=5$, what is the energy difference between the $j=\frac{7}{2}$ and $j=\frac{9}{2}$ levels?

Verified step by step guidance

1. Understand the problem: The energy difference between the two levels with different total angular momentum quantum numbers (j = 7/2 and j = 9/2) in a hydrogen atom is due to the fine structure splitting. This splitting arises from relativistic corrections and spin-orbit coupling. The principal quantum number n = 5 is given, which will help determine the energy levels.
2. Recall the fine structure energy correction formula: The energy of a fine structure level is given by \( E_{n,j} = - \frac{13.6 \text{ eV}}{n^2} \left( 1 + \frac{(\alpha^2)}{n^2} \left[ \frac{n}{j + 1/2} - \frac{3}{4} \right] \right) \), where \( \alpha \) is the fine structure constant (approximately 1/137).
3. Substitute the given values for n = 5 and j = 7/2 into the formula to calculate the energy \( E_{n,j=7/2} \). Simplify the expression step by step, keeping track of the terms involving \( \alpha^2 \) and \( j \).
4. Similarly, substitute the given values for n = 5 and j = 9/2 into the formula to calculate the energy \( E_{n,j=9/2} \). Again, simplify the expression step by step.
5. Find the energy difference: Subtract \( E_{n,j=7/2} \) from \( E_{n,j=9/2} \) to determine the energy difference between the two levels. This difference will primarily depend on the term involving \( \frac{n}{j + 1/2} \), as the other terms are common for both levels.

Key Concepts

Quantum Numbers

Quantum numbers are sets of numerical values that describe the unique quantum state of an electron in an atom. They include the principal quantum number (n), which indicates the energy level, the azimuthal quantum number (l), which describes the shape of the orbital, and the magnetic quantum number (m), which specifies the orientation of the orbital. In this case, the total angular momentum quantum number j combines both orbital and spin angular momentum.

Energy Levels in Hydrogen Atom

In a hydrogen atom, the energy levels are quantized and depend primarily on the principal quantum number n. The energy of an electron in a given level can be calculated using the formula E_n = -13.6 eV/n². The energy difference between two levels can be found by calculating the energies corresponding to their respective quantum numbers and taking the absolute difference.

Fine Structure and Angular Momentum Coupling

Fine structure refers to the small energy differences between atomic states that arise from the interaction between the electron's spin and its orbital angular momentum. In this context, the j quantum number represents the total angular momentum, which is a combination of the orbital angular momentum (l) and the spin angular momentum (s). The energy levels associated with different j values can be affected by these interactions, leading to energy differences that can be calculated using quantum mechanical principles.