The expectation value of an observable Ô for a system in state ψ is best described as which of the following?
AThe value of Ô at the position where |ψ|² is largest
BThe integral of ψ*Ôψ over all space
CThe eigenvalue of Ô corresponding to ψ
DThe maximum value of ψ across all space
The expectation value ⟨Ô⟩ = ∫ψ*Ôψ dτ is the quantum mechanical average — what you would measure on average over many identical experiments. Option 3 (eigenvalue) is only correct when ψ happens to be an eigenstate of Ô; in general, ψ can be a superposition of many eigenstates.
Question 2 True / False
The wavefunction ψ for an electron directly gives the probability of finding the electron at a given point in space.
TTrue
FFalse
Answer: False
ψ itself is not a probability — it can be negative, complex, or zero in ways that probability cannot. It is |ψ|², the probability density, that gives the probability of finding the electron near a given point. This distinction matters: the sign of ψ carries physical information (interference, bonding vs. antibonding character) that disappears when you square it.
Question 3 Short Answer
Why does the Schrödinger equation have exact analytical solutions only for one-electron systems, not for multi-electron atoms?
Think about your answer, then reveal below.
Model answer: Multi-electron atoms contain electron-electron repulsion terms (e²/r₁₂ for each pair) in the Hamiltonian that cannot be separated into independent one-electron equations. The many-body problem has no closed-form solution, so approximation methods like the Hartree-Fock method or perturbation theory are required.
The hydrogen atom Hamiltonian has only one electron-nucleus attraction term, which allows the equation to separate into radial and angular parts with known solutions. Adding a second electron introduces a 1/r₁₂ coupling between coordinates that prevents separation — this is the fundamental origin of why quantum chemistry beyond hydrogen requires numerical approximation.