B = B,j cos(wt)i – B,j sin(wt)j + Bok. (1) (a) Construct the 2 x 2 Hamiltonian matrix with H = -B S (Eq. 4.158 of the textbook), where S is the spin operator. (a(t)\ b(t), (b) If x(t) = () is the spin state at time t, show that à = (Neb+wa); b= (Ne-a –wob), (2) where 2 = 7B, is related to the strength of the rf field. (c) Solve for a(t) and b(t).
B = B,j cos(wt)i – B,j sin(wt)j + Bok. (1) (a) Construct the 2 x 2 Hamiltonian matrix with H = -B S (Eq. 4.158 of the textbook), where S is the spin operator. (a(t)\ b(t), (b) If x(t) = () is the spin state at time t, show that à = (Neb+wa); b= (Ne-a –wob), (2) where 2 = 7B, is related to the strength of the rf field. (c) Solve for a(t) and b(t).
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![Please give a clear solution of question c
Suppose a spin 1/2 particle with gyromagnetic ratio 7, at rest in a static
magnetic field Bok, precesses at the Larmor frequency wo = Bo (see example
4.3 of the textbook). Now we turn on a small transverse radiofrequency (rf)
field, Brf[cos(wt)i – sin(wt)j], so that the total field is
B = B,j cos(wt)i - B,j sin(wt)j+ Bok.
(1)
(a) Construct the 2 x 2 Hamiltonian matrix with H = -B S (Eq. 4.158 of
the textbook), where S is the spin operator.
(b) If x(t) = (0)
is the spin state at time t, show that
à = (Netwb+ woa); 6 = ;
(Ne-tut a – wob),
(2)
where 2=yBrg is related to the strength of the rf field.
(c) Solve for a(t) and b(t).
(d) If the particle starts out with spin up (i.e. a(0) = 1, b(0) = 0), show the
probability of a transition to spin down, as a function of time, is
P(t) =
sin? wt
(3)
(w - wo)? + N²](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f1f57dc-12f7-4fd5-92f6-e85ad11548d9%2F160e9025-4743-42cc-b5a5-2bd6f7024e8d%2F3e2jqgh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Please give a clear solution of question c
Suppose a spin 1/2 particle with gyromagnetic ratio 7, at rest in a static
magnetic field Bok, precesses at the Larmor frequency wo = Bo (see example
4.3 of the textbook). Now we turn on a small transverse radiofrequency (rf)
field, Brf[cos(wt)i – sin(wt)j], so that the total field is
B = B,j cos(wt)i - B,j sin(wt)j+ Bok.
(1)
(a) Construct the 2 x 2 Hamiltonian matrix with H = -B S (Eq. 4.158 of
the textbook), where S is the spin operator.
(b) If x(t) = (0)
is the spin state at time t, show that
à = (Netwb+ woa); 6 = ;
(Ne-tut a – wob),
(2)
where 2=yBrg is related to the strength of the rf field.
(c) Solve for a(t) and b(t).
(d) If the particle starts out with spin up (i.e. a(0) = 1, b(0) = 0), show the
probability of a transition to spin down, as a function of time, is
P(t) =
sin? wt
(3)
(w - wo)? + N²
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