4. Consider a neutron under the influence of rotating magnetic field, B(t) = B4 cos wt i – Bg sin wt j + Bok (4) where Bt and Bo are constant, and w = 2nf with f the driving frequency. The Hamiltonian describing interaction energy between the neutron and magnetic field is H(t) = -9 () B(t) · Ŝ (5) where uy = 5.051 × 10-27 J/T is called the nuclear magneton and g is the g-factor of particle, and Š is the spin operator of neutron. (a) Let the particle be in the eigenstate of lower energy at t = 0. Using the Rabi model, find the probability that the particle is found to have the higher energy for t > 0. Under what condition is the probability maximized? (b) Suppose the probability is maximized for Bo = 10, 640 gauss and Bt = 0.01 gauss at f = 31.1 MHz. Find the g-factor of neutron. (1 gauss = 10-4 T) (c) What is the role of the rotating compolent Bf of magnetic field in this system? In other words, if the rotation were stopped at arbitrary time to in the middle of experiment, what would happen to the probability after then? (d) Likewise, what is the role of the stationary compolent of magnetic field B, in this system? What would happen to the system if there were only the rotating component?
4. Consider a neutron under the influence of rotating magnetic field, B(t) = B4 cos wt i – Bg sin wt j + Bok (4) where Bt and Bo are constant, and w = 2nf with f the driving frequency. The Hamiltonian describing interaction energy between the neutron and magnetic field is H(t) = -9 () B(t) · Ŝ (5) where uy = 5.051 × 10-27 J/T is called the nuclear magneton and g is the g-factor of particle, and Š is the spin operator of neutron. (a) Let the particle be in the eigenstate of lower energy at t = 0. Using the Rabi model, find the probability that the particle is found to have the higher energy for t > 0. Under what condition is the probability maximized? (b) Suppose the probability is maximized for Bo = 10, 640 gauss and Bt = 0.01 gauss at f = 31.1 MHz. Find the g-factor of neutron. (1 gauss = 10-4 T) (c) What is the role of the rotating compolent Bf of magnetic field in this system? In other words, if the rotation were stopped at arbitrary time to in the middle of experiment, what would happen to the probability after then? (d) Likewise, what is the role of the stationary compolent of magnetic field B, in this system? What would happen to the system if there were only the rotating component?
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Transcribed Image Text:4. Consider a neutron under the influence of rotating magnetic field,
B(t) = Bf Cos wt i – Bif sin wt j + Bok
(4)
where Bit and Bo are constant, and w = 27f with f the driving frequency. The
Hamiltonian describing interaction energy between the neutron and magnetic field is
H(t) =
-9 ( B(t) - §
(5)
where uN = 5.051 × 10–27 J/T is called the nuclear magneton and g is the g-factor of
particle, and S is the spin operator of neutron.
(a) Let the particle be in the eigenstate of lower energy at t = 0. Using the Rabi
model, find the probability that the particle is found to have the higher energy for
t > 0. Under what condition is the probability maximized?
(b) Suppose the probability is maximized for Bo = 10, 640 gauss and Bf = 0.01 gauss
at f = 31.1 MHz. Find the g-factor of neutron. (1 gauss = 10-4 T)
(c) What is the role of the rotating compolent Bt of magnetic field in this system?
In other words, if the rotation were stopped at arbitrary time to in the middle of
experiment, what would happen to the probability after then?
(d) Likewise, what is the role of the stationary compolent of magnetic field B, in
this system? What would happen to the system if there were only the rotating
component?
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