Hydrogen's 2s state (n=2, ℓ=0) and 2p states (n=2, ℓ=1) have the same energy. What accounts for this degeneracy?
ABoth states have ℓ = 0, so their angular momenta are identical
BBoth states have the same magnetic quantum number mℓ = 0
CEnergy depends only on n in the Coulomb potential, so all n=2 states are degenerate regardless of ℓ
DThe spherical harmonics Y_ℓ^m are energy eigenstates with the same eigenvalue for all ℓ
In the hydrogen atom, energy depends only on the principal quantum number n: E_n = −13.6 eV/n². States with different ℓ but the same n share the same energy — this is the n²-fold degeneracy. The 2s (ℓ=0) and all three 2p (ℓ=1, mℓ = −1, 0, +1) states all have energy −3.4 eV. This degeneracy is a special property of the 1/r Coulomb potential, reflecting a hidden SO(4) symmetry. It is lifted by perturbations such as spin-orbit coupling.
Question 2 Multiple Choice
How many distinct quantum states (ignoring spin) share the energy E_3 = −13.6/9 eV?
A3, because mℓ can take values −1, 0, +1
B5, because the largest ℓ is 2 and mℓ has 5 values for ℓ=2
C9, because for n=3, summing 2ℓ+1 over ℓ = 0, 1, 2 gives 1+3+5 = 9
D6, because there are 3 possible ℓ values each with 2 mℓ values
For n=3, ℓ can be 0, 1, or 2. For ℓ=0: mℓ=0 only → 1 state. For ℓ=1: mℓ = −1, 0, +1 → 3 states. For ℓ=2: mℓ = −2, −1, 0, +1, +2 → 5 states. Total: 1+3+5 = 9 = n² = 9. The general formula is n² distinct states per energy level (ignoring spin). This is the n²-fold degeneracy specific to the Coulomb potential.
Question 3 True / False
In the quantum mechanical hydrogen atom, the electron follows a definite circular orbit whose radius is given by a₀n², just as the Bohr model describes.
TTrue
FFalse
Answer: False
This is the Bohr model's picture, which quantum mechanics replaces. In quantum mechanics, the electron has no definite position or trajectory; it exists in a state described by a wavefunction ψ(r,θ,φ), and |ψ|² gives a probability density for finding the electron at each point. Orbitals are probability clouds, not paths. The most probable radius for the 1s state equals the Bohr radius a₀ — a coincidence of the most probable value — but the electron can be found at any radius.
Question 4 True / False
The three quantum numbers n, ℓ, and mℓ emerge from three separate equations when the hydrogen Schrödinger equation is solved by separating variables in spherical coordinates.
TTrue
FFalse
Answer: True
Separation of variables splits the hydrogen Schrödinger equation into three independent equations: a radial equation (yielding n and constraining ℓ ≤ n−1), a polar angle equation (yielding ℓ), and an azimuthal equation (yielding mℓ, constrained to |mℓ| ≤ ℓ). Each quantum number is defined by the boundary conditions on its equation. They are not independent in the sense that n constrains ℓ, which constrains mℓ — but each number arises from its own separated equation.
Question 5 Short Answer
The Bohr model correctly predicts hydrogen's energy levels but is said to give 'the right answer for the wrong reason.' What did Bohr assume that quantum mechanics corrects, and what does quantum mechanics actually say about where the electron is?
Think about your answer, then reveal below.
Model answer: Bohr assumed electrons travel on well-defined circular orbits at specific radii (a₀n²), with angular momentum quantized as nℏ. Quantum mechanics shows there are no definite orbits: the electron has a wavefunction ψ(r,θ,φ) whose square gives a probability density. The electron has no trajectory — its position is fundamentally indeterminate between measurements. The most probable radius for the ground state coincidentally equals the Bohr radius a₀, but the electron can be found at any radius. Quantum mechanics also introduces ℓ and mℓ absent from the Bohr model, and gives the 1s state zero angular momentum (ℓ=0), contradicting Bohr's requirement of one unit.
The Bohr model's success was partly accidental — the Coulomb potential and orbit quantization happen to give correct energies. But the model fails for multi-electron atoms, cannot predict spectral intensities or selection rules, and gives incorrect angular momenta. Quantum mechanics provides the correct framework by replacing deterministic trajectories with probabilistic wavefunctions and grounding the energy formula in a rigorous eigenvalue equation.