(a) The z-component of B is constant, while the component of B perpendicular to the z-axis rotates ccw about the z-axis with frequency w.
(b) H(t)=-gS×B(t)=-gB_{0}S_{z}-gB_{1}(coswt S_{x}+sinwt S_{y})=w_{0}S_{z}+w_{1}(coswt S_{x}+sinwt S_{y})
Let U(R) be the rotation operator for a ccw rotation R(f) about the z-axis.
U(R)=exp(-(i/h)S_{z}f)
|y(t)>=U(t,0)|y(0)>, U(R)|y(t)>=U(R)U(t,0)|y(0)>
U(R)
can be viewed as an operator that rotates every state vector ccw through an
angle f or an
operator that changes the basis vectors and therefore rotates the coordinate
system cw through an angle f.
Let f=-wt, then U(R) rotates the coordinate system ccw with angular frequency w. The coordinate system then rotates with the magnetic field.
Consider an infinitesimal rotation.
U(R)|y(dt)> = (I+(i/h)S_{z}wdt)|y(dt)> = (I+(i/h)S_{z}wdt)(I-(i/h)Hdt)|y(0)>
U(R)|y(dt)> = (I+(i/h)S_{z}wdt)(I-(i/h)(w_{0}S_{z}+w_{1}(coswt S_{x}+sinwt S_{y}))dt)| +>
U(R)|y(dt)> = (I+(i/h)((w-w_{0})S_{z}-w_{1}S_{x}coswt-w_{1}S_{y}sinwt)dt)| +>
S is a vector observable, its components transform under rotation R as V’=R^{-1}V.
For a ccw rotation R(f)
we have V_{x}’=V_{x}cosf+V_{y}sinf.
Therefore S_{x}’=S_{x}coswt+S_{y}sinwt is the x-component of the observable S in a frame that rotates ccw with angular frequency w about the z-axis, i.e. in a frame that rotates with the magnetic field.
U(R)|y(dt)>=|y'(dt)>=(I+(i/h)((w-w_{0})S_{z}-w_{1}S_{x}’)dt)|+>
|y'(dt)>=(I+(i/h)((w-w_{0})S_{z}’-w_{1}S_{x}’)dt)|y’(0)>
|y'> denotes the state vector in a frame rotating ccw about the z-axis with angular frequency w.
In this frame the evolution operator is U(t,0)=exp(-(i/h)((w-w_{0})S_{z}’-w_{1}S_{x}’)t).
The Hamiltonian therefore is H’=(w-w_{0})S_{z}’-w_{1}S_{x}’.
The matrix of the Hamiltonian is
in the {|+>, |->} basis.
The eigenfunctions and eigenvalues are
E_{+}=+(h/2)(Dw^{2}+w_{1}^{2})^{1/2}, |y'_{+}>=cos(q/2)|+>+sin(q/2)|->,
E_{-}=-(h/2)(Dw^{2}+w_{1}^{2})^{1/2}, |y'_{-}>=-sin(q/2)|+>+cos(q/2)|->,
where tanq=-w_{1}/Dw, sinw=-w_{1}/(Dw^{2}+w_{1}^{2})^{1/2}.
Therefore:
|y’(0)> = |+> = cos(q/2)|y'_{+}>-sin(q/2)|y'_{-}>,
|y’(t)> = cos(q/2)exp(-(i/h)E_{+}t)|y'_{+}>-sin(q/2)exp((i/h)E_{+}t)|y'_{-}>
= cos(E_{+}t/h)[cos(q/2)|y'_{+}>-sin(q/2)|y'_{-}>]-i sin(E_{+}t/h)[cos(q/2)|y'_{+}>+sin(q/2)|y'_{-}>]
= cos(E_{+}t/h)|+>-i sin(E_{+}t/h)[cos(q)|+>+sin(q)|->]=b_{+}|+>+b_{-}|->
To find the eigenfunctions in the lab frame we use U^{-1}(R)|y’(t)> = |y(t)>.
exp(-(i/h)S_{z}wt)|y’(t)> = exp(-iwt/2) b_{+}|+>+exp(i/wt/2) b_{-}|-> = |y(t)>.
We now have:
a_{+}(t)= exp(-iwt/2)b_{+ }= exp(-iwt/2)(cos(E_{+}t/h)-i sin(E_{+}t/h)cos(q))
a_{-}(t)= exp(iwt/2)b_{- }= -i exp(iwt/2)sin(E_{+}t/h)sin(q)
Special case:
If
w=w_{0},
the Dw=0 and E_{+}=(h/2)w_{1},
tanq=-w_{1}/Dw=-
¥, q=-p/2, cos(q)=0,
sin(q)=-1.
Then
a_{+}(t)= exp(-iwt/2) b_{+ }= exp(-iwt/2)cos((w_{1}/2)t),
a_{-}(t)= exp(iwt/2) b_{- }= i exp(iwt/2)sin((w_{1}/2)t).
(c) P_{+-}(t)=|a_{-}(t)|^{2}= sin^{2}(E_{+}t/h)sin^{2}(q)=w_{1}^{2}/(Dw^{2}+w_{1}^{2}) sin^{2}((Dw^{2}+w_{1}^{2})^{1/2}t/2)
For the special case w=w_{0} P_{+-}(t)= sin^{2}((w_{1}/2)t). Then P_{+-}(t)= 1 if (w_{1}/2)t=np/2, n=odd.
(d) |+> and |-> are both excited states and both decay with probability 1/t per unit time. If n atoms are excited at t=0, then at time t n exp(t/t) are still in an excited state. The atoms that are in an excited state flip between |+> and |->.
If an atom is excited at t=0, then the probability of finding it in the state |-> at time t is exp(t/t) P_{+-}(t). Assume n atoms are excited per unit time. As t goes to infinity we find the total number of atoms in the |-> state by summing up the contributions from all prior time intervals, i.e. by calculating the integral .
Use .
We multiply the number of atoms thus obtained by their probability of decay 1/t to find the number N which decay from the state |-> per unit time.
N=(n/2)w_{1}^{2}/(Dw^{2}+w_{1}^{2}+(1/t)^{2})
N plotted versus Dw is a Lorentz curve with half width L=(w_{1}^{2}+(1/t)^{2})^{1/2}.
Dw=w-w_{0}=w+gB_{0}.