Homework 12

Problem 1:

The spin operator on a spin 1/2 particle is given by Image2827.gif (957 bytes).

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

|y>=(1/2(1/2))(|+(1)>x|-(2)>x -|-(1)>x|+(2)>x )

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

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

(d)  The rotation operator

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

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 Sz 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 Bx=-wx/g, By=-wy/g, Bz=-wz/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)[wxSx+wySy+wzSz].

Calculate the matrix of M in the {|+>, |->} eigenbasis of Sz.  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-((wx2+wy2)/w02)sin2(w0t/2). 

Give a geometrical interpretation.