Two operators A and B satisfy the commutation relations [H,  A] = -ℏωB and [H, B] = ℏωA, where ω is a constant and H is the Hamiltonian of the system. The expectation value in a state  such that at time t = 0, ⟨A⟩_ψ(0) = 0 and ⟨B⟩ψ(0) = i, is

Explanation:

Let us reconsider the system of equations:

 and 

By differentiating the first equation again, 

• Substituting the second equation into this results in 
• This differential equation is a simple harmonic one, but with a key difference: there is no negative in front of ω², leading to hyperbolic solutions.
• Specifically, we find A(t) = Csinh(ωt), for some constant C.
• Given that the expectation value  at t = 0, we find 
• Thus, in general, B(t) has to be in the form of cosh(ωt), to meet the commutation relations. Finally, given that  we need to multiply cosh(ωt) by i.
• So the time-evolved expectation value is