Prove that the degeneracy of the n th level in the hydrogen...

Prove that the degeneracy of the $n$ th level in the hydrogen atom is $n^{2}$; that is, verify the result $(8.77)$. (But be aware that this number gets doubled because of the electron's spin, as we describe in Chapter $9 .$ )

Key Concepts

Hydrogen Atom Model

The hydrogen atom is modeled using quantum mechanics, where the Schrödinger equation with a Coulomb potential is solved to obtain quantized energy levels. In this framework, the energy of a state depends only on the principal quantum number, which leads to multiple distinct spatial states sharing the same energy level, a phenomenon known as degeneracy.

Quantum Numbers

Quantum states in the hydrogen atom are described by a set of quantum numbers: the principal quantum number (n) determines the energy level; the orbital angular momentum quantum number (l) defines the shape of the orbital; and the magnetic quantum number (m) specifies the orientation of the orbital. The energy’s independence from l and m in the hydrogen atom results in many states having the same energy.

State Counting and Degeneracy

The degeneracy of an energy level in the hydrogen atom is determined by counting all the possible combinations of the orbital quantum numbers l and m for a fixed principal quantum number n. Since l ranges from 0 to n-1 and for each l the magnetic quantum number m runs from -l to +l, the total number of states in the nth level is the sum of (2l + 1) for l from 0 to n-1, which evaluates to n².

Electron Spin

Electron spin is an intrinsic form of angular momentum independent of orbital motion. Each spatial quantum state in the hydrogen atom can accommodate two electrons with opposite spins, effectively doubling the degeneracy. This additional two-fold multiplicity is essential for a complete understanding of the quantum state structure in atoms.

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