Consider a qubit used as a clock. The Hamiltonian for the system is H = | , where is some constant frequency. (a) What are the energy Eigen states and their energies? (b) Write down the time...

Consider a qubit used as a clock. The Hamiltonian for the system is H = |
, where

is some constant frequency.

(a) What are the energy Eigen states and their energies?

(b) Write down the time evolution operator U(t) for this system.

(c) We prepare the clock in an initial state

What is the state |ψ(t)at a later time?

(d) What is E for this system?

(e) Let t denote the time it takes for the system to evolve from |ψ(0)to some other distinguishable state. Calculate E t and compare your result to the time–energy uncertainty relation in Eq. 5.33.