Questions: Hydrogen Atom Wavefunctions and Atomic Orbitals

Question 1

The 2p wavefunction ψ_{210} has regions where its value is negative (below the nodal plane). What is the probability of finding the electron in one of those negative-ψ regions?

• A. Negative, because probability tracks the sign of ψ
• B. Zero, because negative ψ means the electron is excluded
• C. Positive and nonzero, because probability density is |ψ|², which is always ≥ 0
• D. Equal to the probability in the positive-ψ lobe, because |ψ|² is symmetric

Answer: (C)

The wavefunction ψ itself can be negative — it is a mathematical amplitude, not a probability. Probability density is |ψ|², which is always non-negative. A negative lobe of ψ has the same |ψ|² as a positive lobe of equal magnitude, so the electron is found there with equal probability. The sign of ψ matters only when wavefunctions interfere (e.g., in bonding/antibonding combinations), not for single-orbital probabilities.

Question 2

In the hydrogen atom, the 2s and 2p orbitals have the same energy (they are degenerate) because energy depends only on the principal quantum number n.

• T. True
• F. False

Answer: True

For the pure hydrogen atom (one electron, no electron-electron repulsion), the Coulomb potential has a special symmetry that makes all subshells with the same n exactly degenerate: E_n = −13.6/n² eV regardless of l. This accidental degeneracy is unique to hydrogen. In multi-electron atoms, electron-electron repulsion breaks this, making 2s lower in energy than 2p.

Question 3

Why does the radial probability distribution P(r) = r²|R_{nl}(r)|² include an r² factor, rather than simply plotting |R_{nl}(r)|²?

Model answer: The r² factor accounts for the increasing volume of thin spherical shells as radius grows. A shell of thickness dr at radius r has volume 4πr² dr. Even if |R_{nl}|² decreases with r, the shell volume grows, so the actual probability of finding the electron in that shell is proportional to r²|R_{nl}|². Plotting just |R_{nl}|² would give the amplitude at a single point, not the probability in a shell — which is the physically meaningful quantity for locating the electron.

This distinction is critical: the 1s wavefunction has its maximum amplitude at r = 0, but the most probable radius (peak of P(r)) is at the Bohr radius a₀ because the r² factor overwhelms the exponential decay near the nucleus. Students who skip the r² factor incorrectly conclude that s-orbital electrons are most likely found at the nucleus.