In an orthonormal basis a Hermitian operator has the matrix A= pmatrix 2&0&0 0&1&1…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · The formalism of quantum mechanics
The Structure of Quantum Mechanics, Ch Observables, spectra and the spectral theorem
Problem
In an orthonormal basis {∣1⟩,∣2⟩,∣3⟩} a Hermitian operator has the matrix A=200011011.
((a)) Find the eigenvalues of A and identify the one that is doubly degenerate.
((b)) Build the projector P^2 onto the degenerate eigenspace from the orthonormal pair ∣1⟩ and (∣2⟩+∣3⟩)/2, and write its matrix.
((c)) Build it again from the pair (∣1⟩±(∣2⟩+∣3⟩)/2)/2 and confirm the same matrix.
((d)) Verify P^22=P^2=P^2†.