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

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter7: Hamilton's Principle-lagrangian And Hamiltonian Dynamics

Section: Chapter Questions

Problem 7.36P

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