852_2010hw9_Solutions - PHYS852 Quantum Mechanics II,...

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PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 9: Solutions Topics covered: hydrogen hyper-fine structure, Wigner-Ekert theorem, Zeeman effect 1. Relations between ~ V and ~ J : For a rotation by φ about the z-axis, we have U V z U = V z , U V x U = cos φV x - sin φV y , and U V y U = sin φV x + cos φV y , where U = e - ( i/ ~ ) φJ z . (a) Consider an infinitesimal rotation by δφ , and use these expressions to show: [ J z ,V z ] = 0 , (1) [ J z ,V x ] = i ~ V y , (2) [ J z ,V y ] = - i ~ V x . (3) Write out the six additional commutators generated by cyclic permutation of the indices. For an infinitesimal rotation, we can expand U as U 1 - i ~ φJ z , so that keeping terms up to first-order in φ gives V x + i ~ φ [ J z ,V x ] = V x - φV y (4) V y + i ~ [ J z ,V y ] = φV x + V y (5) V z + i ~ [ J , V z ] = V z (6) from which we can read off: [ J z ,V x ] = i ~ V y (7) [ J z ,V y ] = - i ~ V x (8) [ J z ,V z ] = 0 (9) Cyclic permutation of indices then gives: [ J x ,V y ] = i ~ V z [ J y ,V z ] = i ~ V x (10) [ J x ,V z ] = - i ~ V y [ J y ,V x ] = - i ~ V z (11) [ J x ,V x ] = 0 [ J y ,V y ] = 0 (12) b.) Use the results from (a) to show: [ J z ,V ± ] = ± ~ V ± (13) [ J ± ,V ± ] = 0 (14) [ J ± ,V ] = ± 2 ~ V z (15) 1
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where V ± = V x ± iV y . [ J z ,V ± ] = [ J z ,V x ] ± i [ J z ,V y ] = i ~ V y ± ~ V x = ± ~ ( V x ± iV y ) = ± ~ V ± (16) [ J ± ,V ± ] = [ J x ,V x ] ± i [ J x ,V y ] ± i [ J y ,V x ] - [ J y ,V y ] = ~ V z ± ~ V z = 0 (17) [ J ± ,V ] = [ J x ,V x ] i [ J x ,V y ] ± i [ J y ,V x ] + [ J y ,V y ] = ± ~ V z ± ~ V z = ± 2 ~ V z (18) 2. Derivation of Wigner-Ekert theorem : Verify Eqs. (108)-(127) in the Atomic Physics lecture notes. Eq. (108): [ J z ,V z ] = 0 [ J z ,V z ] | kjm i = 0 J z ( V z | kjm i ) = V z J z | kjm i J z ( V z | kjm i ) = ~ mV z | kjm i (19) Eq. (109): [ J z ,V ± ] = ± ~ V ± [ J z ,V ± ] | kjm i = ± ~ V ± | kjm i
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852_2010hw9_Solutions - PHYS852 Quantum Mechanics II,...

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