A three-dimensional state space has an orthonormal basis that is a common eigenbasis…
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics
The Structure of Quantum Mechanics, Ch Compatible observables and uncertainty as a theorem
Problem
A three-dimensional state space has an orthonormal basis {∣1⟩,∣2⟩,∣3⟩} that is a common eigenbasis of two self-adjoint operators A^ and B^: A^∣n⟩=an∣n⟩, B^∣n⟩=bn∣n⟩, with (a1,a2,a3)=(1,1,2)ε0 and (b1,b2,b3)=(5,3,3)μ0, where ε0,μ0 are constants carrying the appropriate units. A function of an operator means f(A^)=∑af(a)P^a, as in Chapter 3.
((a)) Show that neither {A^} nor {B^} alone is a complete set of commuting observables, but {A^,B^} is.
((b)) Show that B^ cannot be written as f(A^) for any function f, and that A^ cannot be written as g(B^) for any function g.
((c)) Let C^=A^/ε0+B^/μ0. Show that {C^} alone is a complete set of commuting observables, and find polynomials f and g, each of degree at most 2, with A^=ε0f(C^) and B^=μ0g(C^).
((d)) Explain why part (c) does not contradict part (b).