The spin operator on a spin 1/2 particle is given by .
|y>=(1/2^{½})(|+(1)>_{x}|-(2)>_{x} -|-(1)>_{x}|+(2)>_{x} )
Show that |y> is an eigenstate of S_{x}, where S=S(1)+S(2).
(c) Show that |y> is also an eigenstate of S_{z}, as well as S^{2} with eigenvalue zero.
(d) The rotation operator
R(f)=cos(f/2)I+isin(f/2)s_{z},
where I is the identity operator, generates rotations around the z-axis of angle f. Apply R(f) on the state |y> to show that |y> is invariant under rotations around the z-axis.
Note: R(f)(|+(1)>|-(2)>)= (R(f)|+(1)>R(f)|-(2)>)
Consider a spin 1/2 particle with magnetic moment m=gS in a magnetic field
B=(B_{1}coswt, B_{1}sinwt, B_{0}).
At time t the state of the particle is |y(t)>=a_{+}(t)|+>+a_{-}(t)|->. At t=0 a_{+}(0)=1 and a_{-}(0)=0.
(a) Describe the geometry of the magnetic field.
(b) Find a_{+}(t)
and a_{-}(t).
(c) Find the probability of measuring
S_{z} and
obtaining the eigenvalue -h/2 at time t.
Under what conditions can this probability become equal to 1?
(d) Now assume the two states {|+>,|->} correspond to two
sublevels of an excited level; n atom are being excited per unit time, all
to the state |+>. An atom decays by spontaneous emission of
radiation with a probability per unit time 1/t, which is the same for
the two sublevels |+> and |->. After a time much
longer than t, what is the number N of atoms, which decay per
unit time from the state |->? Plot N versus B_{0}
for a fixed w.
Consider a spin ½, of magnetic moment M=gS, placed in a magnetic field B_{0} of components B_{x}=-w_{x}/g, B_{y}=-w_{y}/g, B_{z}=-w_{z}/g. Set w_{0}=-g|B_{0}|
(a) Show that the evolution operator of this spin is U(t,0)=exp(-iMt), where M is the operator M=(1/h)[w_{x}S_{x}+w_{y}S_{y}+w_{z}S_{z}].
Calculate the matrix of M in the {|+>, |->} eigenbasis of S_{z}. Show that M^{2}=(w_{0}/2)^{2}.
(b) Put the evolution operator into the form U(t,0)=cos(w_{0}t/2)-(2i/w_{}0)Msin(w_{0}t/2).
(c) Consider a spin which at time t=0 is in the state |y(0)>=|+>. Show that the probability of finding it in the state |+> at time t is
P_{++}(t)=1-((w_{x}^{2}+w_{y}^{2})/w_{0}^{2})sin^{2}(w_{0}t/2).
Give a geometrical interpretation.