A hydrogen atom is prepared in the normalised state | = 1 3 |2,0,0 + 1 3 |2,1,0 + 1 3…
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics
The Structure of Quantum Mechanics, Ch The postulates, measurement and collapse
Problem
A hydrogen atom is prepared in the normalised state ∣ψ⟩=31∣2,0,0⟩+31∣2,1,0⟩+31∣2,1,1⟩, where ∣n,l,m⟩ denotes an energy eigenstate of hydrogen with principal quantum number n, orbital quantum number l and magnetic quantum number m, and the three kets are mutually orthonormal. The energy of the n=2 level is E2=−3.4eV.
((a)) Write the projector P^E2 onto the whole n=2 eigenspace (spin ignored), using all four kets ∣2,l,m⟩ with n=2, and say which of its terms act on ∣ψ⟩.
((b)) Use P(E2)=⟨ψ∣P^E2∣ψ⟩ to find the probability that a measurement of energy on this atom returns E2.
((c)) State, without recalculating, what the probability would be if the three coefficients were instead 1/2, 1/2, 1/2 (still a normalised state built only from n=2 kets).