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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
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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