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solution8

Course: PHYS 572, Fall 2008
School: Cornell
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572 FALL PHYSICS 2007 Solution 8 1.(a) It is easy to see that B = z/2 z/2 + z = 0 , and B = (y/2 + y/2) + (x/2 x/2) = 0 .1 x y The magnitude of the magnetic eld is B= (x zx/2)2 + (y zy/2)2 + (B0 + (2z 2 x2 y 2 )/4)2 . Assuming that B0 &gt; 0, we expand B near the origin of R3 to the lowest order B B0 + Hence B = B0 + 1 2 2 B0 2 1 (x2 + y 2 ) + z 2 + higher order terms. 2 1 (2 x2 + 2 y 2 +...

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572 FALL PHYSICS 2007 Solution 8 1.(a) It is easy to see that B = z/2 z/2 + z = 0 , and B = (y/2 + y/2) + (x/2 x/2) = 0 .1 x y The magnitude of the magnetic eld is B= (x zx/2)2 + (y zy/2)2 + (B0 + (2z 2 x2 y 2 )/4)2 . Assuming that B0 > 0, we expand B near the origin of R3 to the lowest order B B0 + Hence B = B0 + 1 2 2 B0 2 1 (x2 + y 2 ) + z 2 + higher order terms. 2 1 (2 x2 + 2 y 2 + 2B0 (2z 2 x2 y 2 )/4) . 2B0 Let 2 /B0 /2 = , then up to quadratic order in the coordinates 1 B = B0 + (x2 + y 2 + z 2 ) . 2 (b) S is unitary because SS = = = = 1 Actually, 1 2 1 2 1 2 1 2 1+bzi bz 1+bz 1+bz+i bz 1+bz bz bz 1+bz+ 1+bz 1+bz (b z) (b z) 1+bz+ 1+bz 2 1 (b z) 1+bz+ 1+bz =1. = 0 for any smooth function . 1 PHYSICS 572 FALL 2007 And since Hmag = B, SHmag = 2 Then, SHmag S = ((1 + b z) B + B (b z) + 2 B (b z) (b z) B (b z) ) 1+bz ((1 + b z) B + 2B((b z)b z) 2 (b z) (b z) + B ) 1+bz ((1 + b z) B + 2B((b z)b z) + (1 b z) B) 2 (2(b z)( B) + 2(b z)( B) 2Bz ) 2 Bz . B (b z) 1+bz B+ 1+bz . = = = = (c) We know that S 2 S = 2 + 2S( S ) + S( 2 S ), Let A = S( S ), then ( + A) ( + A) = = 2 2 + 2A + A + A A + 2S( S ) + (S( S )) + S( S ) S( S ) . By the identity S( S ) = ( S)S , we can rewrite it as + 2S( S ) = + 2S( S ) 2 = + 2S( ) S = S 2 S . 2 2 + (S( S )) ( S)S S( S ) + ( S) ( S ) + S( 2 S ) ( S) ( S ) + S( 2 S ) (d) Apply the unitary transformation S on H, combining (b) and (c), we get, SHS = 2 2M ( + A) ( + A) B z . Ignoring the vector potential (A), the Hamiltonian becomes SHS = 2 2 2M B z . 2 PHYSICS 572 FALL 2007 For a spin-down particle, SHS 2 = 2 2M 2 2 + B 1 + B0 + (x2 + y 2 + z 2 ) 2 2M It is a three-dimensional harmonic oscillator, so the ground state is M 3/8 M r2 ), (x, y, z) = ( 2 2 ) exp( 2 with the eigenvalue E0 = 2. Dene O1 = We need to prove | Oi = Oi | . For the rst operator, O1 | = = = With integral by part, O1 | = = dx d1 d2 + dx 2 dx 1 2 0 d dx d dx 3 2 d dx + B0 . M , O2 = 0 d i dx d i dx 0 0 d dx 0 dx dx dx 0 d dx d 2 dx d dx 0 1 2 1 2 1 2 d 1 dx d d 2 1 1 2 dx dx dx 1 0 1 2 = | O1 . By the denition, O1 is Hermitian. Alternatively, set = , we get | O1 = O1 | = which implies the expected value, O1 = O1 | O1 , . 3 PHYSICS 572 FALL 2007 It is straightforward to repeat this process to prove O2 is Hermitian, O2 | = = = = = dx 0 d i dx d i dx 0 1 2 d 1 dx 1 2 dx i d2 dx dx dx i i 1 2 d d 2 1 i 1 2 dx dx d2 d1 +i i dx 2 dx 1 2 0 d i dx d i dx 0 dx 1 1 2 = | O2 . However, alternatively, since O2 = U O1 U where U= ei/4 0 0 ei/4 is a unitary matrix and O1 is Hermitian, O2 is also Hermitian. 4
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