a) Consider a spin 1/2 particle in a magnetic field which points in an arbitrary direction....

 

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a) Consider a spin 1/2 particle in a magnetic field which points in an arbitrary direction. B = B₂ î+ By Ĵ + B₂ Ê Use the spin matrices in which S, is diagonal and write down the Hamiltonian for this system. Recall H = - B where = y Sandy is the gyromagnetic ratio. b) Now assume that the magnetic field points only in the z-direction. Find out the eigenval- ues and eigenvectors of the Hamiltonian. Write down an expression for the time dependent general state of the system. Fix the unknown expansion coefficients by assuming that ini- tially the system is in the state |(t = 0) >= c) Calculate the expectation value of the three spin components as time evolves. < S₂ (t) >=< (t)|S₁|v(t) > d) For the spin pointing in a general direction î = 1 √/2 (1) sin cos pi+sin 0 sino + cos 0 k construct the general spin matrix S. ñ. Calculate < (t)|(5.)|(t) > and check that this general expression yields the same results encountered in part c) for appropriate values of 0 and p. a) Consider a spin 1/2 particle in a magnetic field which points in an arbitrary direction. B = B₂ î+ By Ĵ + B₂ Ê Use the spin matrices in which S, is diagonal and write down the Hamiltonian for this system. Recall H = - B where = y Sandy is the gyromagnetic ratio. b) Now assume that the magnetic field points only in the z-direction. Find out the eigenval- ues and eigenvectors of the Hamiltonian. Write down an expression for the time dependent general state of the system. Fix the unknown expansion coefficients by assuming that ini- tially the system is in the state |(t = 0) >= c) Calculate the expectation value of the three spin components as time evolves. < S₂ (t) >=< (t)|S₁|v(t) > d) For the spin pointing in a general direction î = 1 √/2 (1) sin cos pi+sin 0 sino + cos 0 k construct the general spin matrix S. ñ. Calculate < (t)|(5.)|(t) > and check that this general expression yields the same results encountered in part c) for appropriate values of 0 and p.

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