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Give details regarding the problem. Topic: Griffith’s Time
Evolution Operator, Spin Precession. NOTE: Do not copy from other
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Provide Step By Step Solution With Formulas And Explanation Give Details Regarding The Problem Topic Griffith S Time 1 (93.54 KiB) Viewed 31 times
***Problem 9.19 An electron is at rest at the origin, in the presence of a magnetic field whose magnitude (B) is constant but whose direction rides around at constant angular velocity w on the lip of a cone of opening angle a: B(t) = Boſsin a cos(wt )+ sina sin(wt)ſ + cosæł]. [9.89) (a) Construct the 2 x 2 Hamiltonian matrix (Equation 4.158) for this system. (b) Find the exact solution to the time-dependent) Schrödinger equation, assuming the particle starts out with spin up. Hint: You can do it from scratch, or by noting that in this case the rotating wave approximation is exact, and refering to Problem 9.7. Answer: i[wi sina)/2] sin(t/2)ew/2 [9.90] where w = -eBo/m and 1 = y2 + + 2ww, cosa. [9.911 (c) Now treat the same problem by (first-order) time-dependent perturbation theory: use Equation 9.17 to calculate the approximate) probability of a transition from spin up (the initial state) to spin down, as a function of time, and compare the exact answer (from part b). State the criterion on the strength of the field that determines whether perturbation theory is applicable in this case. x() = (lcos(at/2)in ?2») etat2 ).
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