Show that the operator= sin e cos o Sx + sin 0 sin o Sy +cos 6 Sz, (1.84)which represents the spin operator for the component ofspin along a direction determined by the spherical polarangles 0 and ø, has eigenvalues ± and eigenstates ofthe formcos(0/2)I↑) = ( sin(@/2) e*eiø )(1.85)sin(8/2)=(- cos(9/2) e*(1.86)Convince yourself that these results agree with the Rie-mann sphere representation in Fig. 1.8. Show further that1 00 1(1.87)14 01-1-iFig. 1.8 The Riemann sphere represents thespin states of a spin- particle. The spinvector S lies on a unit sphere. A line from thesouch pole of the sphere to S cuts the harizon-tal equitorial planc (shaded) at y = x + iywhere the horizontal plane is considered asan Argand diagram. The numerical value ofthe complex number g is shown for six cases.namcly S parallel or antiparallci to the x, yand : axes.

The secular equation for finding out the eigenvalues of a matrix A is:A-λI=0………(1)where,λ are the…

Show that the operator Se,o = sin e cos o Sx +sin 0 sin o y+cos e Sz, (1.84) which represents the spin operator for the component of spin along a direction determined by the spherical polar angles 0 and ø, has eigenvalues +, and eigenstates of the form cos(0/2) sin(0/2) eie ) | 1) =(, 14)=( (1.85) sin(0/2) (1.86) - cos(9/2) eiø Convince yourself that these results agree with the Rie- mann sphere representation in Fig. 1.8. Show further that (6 ?). 1 1 0 (1.87) 1

Show that the operator Se,o = sin e cos o Sx +sin 0 sin o y+cos e Sz, (1.84) which represents the spin operator for the component of spin along a direction determined by the spherical polar angles 0 and ø, has eigenvalues +, and eigenstates of the form cos(0/2) sin(0/2) eie ) | 1) =(, 14)=( (1.85) sin(0/2) (1.86) - cos(9/2) eiø Convince yourself that these results agree with the Rie- mann sphere representation in Fig. 1.8. Show further that (6 ?). 1 1 0 (1.87) 1

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

Show that the operator
= sin e cos o Sx + sin 0 sin o Sy +cos 6 Sz, (1.84)
which represents the spin operator for the component of
spin along a direction determined by the spherical polar
angles 0 and ø, has eigenvalues ± and eigenstates of
the form
cos(0/2)
I↑) = ( sin(@/2) e*
eiø )
(1.85)
sin(8/2)
=(- cos(9/2) e*
(1.86)
Convince yourself that these results agree with the Rie-
mann sphere representation in Fig. 1.8. Show further that
1 0
0 1
(1.87)
14

01
-1
-i
Fig. 1.8 The Riemann sphere represents the
spin states of a spin- particle. The spin
vector S lies on a unit sphere. A line from the
souch pole of the sphere to S cuts the harizon-
tal equitorial planc (shaded) at y = x + iy
where the horizontal plane is considered as
an Argand diagram. The numerical value of
the complex number g is shown for six cases.
namcly S parallel or antiparallci to the x, y
and : axes.

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