Problem 1:

The spin operator on a spin 1/2 particle is given by .

• (a)  Find the eigenstates |+>_x and |->_x of S_x and express them in terms of the eigenstates |+> and |-> of S_z.
• (b)  Consider two spin 1/2 particles, labeled 1 and 2 respectively, in the anti-symmetric state

  |y>=(1/2^½)(|+(1)>_x|-(2)>_x -|-(1)>_x|+(2)>_x )

  Show that |y> is an eigenstate of S_x, where S=S(1)+S(2).

• (c)  Show that |y> is also an eigenstate of S_z, as well as S² with eigenvalue zero.

• (d)  The rotation operator

  R(f)=cos(f/2)I+isin(f/2)s_z,

  where I is the identity operator, generates rotations around the z-axis of angle f.  Apply R(f) on the state |y> to show that |y> is invariant under rotations around the z-axis.

  Note: R(f)(|+(1)>|-(2)>)= (R(f)|+(1)>R(f)|-(2)>)

Problem 2:

Consider a spin 1/2 particle with magnetic moment m=gS in a magnetic field 

B=(B₁coswt, B₁sinwt, B₀).  

At time t the state of the particle is |y(t)>=a₊(t)|+>+a_-(t)|->. At t=0 a₊(0)=1 and a_-(0)=0.

(a)  Describe the geometry of the magnetic field.
(b)  Find a₊(t) and a_-(t).
(c)  Find the probability of measuring S_z and obtaining the eigenvalue -h/2 at time t.  Under what conditions can this probability become equal to 1?
(d)  Now assume the two states {|+>,|->} correspond to two sublevels of an excited level; n atom are being excited per unit time, all to the state |+>.  An atom decays by spontaneous emission of radiation with a probability per unit time 1/t, which is the same for the two sublevels |+> and |->.  After a time much longer than t, what is the number N of atoms, which decay per unit time from the state |->?  Plot N versus B₀ for a fixed w.

Problem 3:

Consider a spin ½, of magnetic moment M=gS, placed in a magnetic field B₀ of components B_x=-w_x/g, B_y=-w_y/g, B_z=-w_z/g.  Set w₀=-g|B₀|

(a)  Show that the evolution operator of this spin is U(t,0)=exp(-iMt), where M is the operator M=(1/h)[w_xS_x+w_yS_y+w_zS_z].

Calculate the matrix of M in the {|+>, |->} eigenbasis of S_z.  Show that M²=(w₀/2)².

(b)  Put the evolution operator into the form U(t,0)=cos(w₀t/2)-(2i/w0)Msin(w₀t/2).

(c)  Consider a spin which at time t=0 is in the state |y(0)>=|+>.  Show that the probability of finding it in the state |+> at time t is

P₊₊(t)=1-((w_x²+w_y²)/w₀²)sin²(w₀t/2). 

Give a geometrical interpretation.