Problem 3 The time evolution of a spin- j particle
is governed by the Hamiltonian
hat(H)(t)=omega _(0)hat(J)_(3)+omega _(1)[hat(J)_(1)cos(omega t)+hat(J)_(2)sin(omega t)].
(a) Perform a suitable SU(2) transformation in
spin- j representation to enter a rotating frame that
supports a time-independent Hamiltonian.
(b) What is the spin precession time in the
rotating frame? What is the time required for the
state ket to return to itself in the rotating frame?
(c) Find the instantaneous eigenvalues and
eigenstates of hat(H)(t) (i.e., diagonalize hat(H)(t) ).
(d) What is the time evolution operator U(t;t_(0))
in the lab frame?
Problem 3 The time evolution of a spin-i particle is governed by the Hamiltonian
H(t)=woJ3+w1[Ji cos(wt)+ J2 sin(wt)].
(5)
(a) Perform a suitable SU(2) transformation in spin-i representation to enter a rotating frame that supports a time-independent Hamiltonian.
(b) What is the spin precession time in the rotating frame? What is the time required for the state ket to return to itself in the rotating frame?
(c) Findthe instantaneous eigenvalues and eigenstates of H(t) (i.e., diagonalize H(t)).
(d) What is the time evolution operator U(t;to) in the lab frame?