soln - Physics 741 Graduate Quantum Mechanics 1 Solution...

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Physics 741 – Graduate Quantum Mechanics 1 Solution Set N 1. [10] Let ( ) ,, x yz J JJ = J be three Hermitian operators that commute with the Hamiltonian H , and have angular momentum-like commutation relations x J Ji J ⎡⎤ = ⎣⎦ = y zx J J = = [ ] ,. y J J = = Define the operators 2 222 x JJJ =++ J and x y J J ± = ± Show each of the following is true: (a) [6] 2 ,0 = (this is three identities), and therefore 2 ± = We simply begin working them out: [ ] [ ] () 22 2 2 ,0, , , , , , 0, 0 , , , x yx y yx yx y zzx zxz zy y z y x x y x z z y z xz x J JJ JJ J JJJ iJ J J J J J JJ J i J ⎡⎤⎡⎤ =+ + = + + + ⎣⎦⎣⎦ =− + + = ⎤⎡ += + + + ⎦⎣ = = [] 222 ,,, 0 , , 0. z z x x z x z x y y z y xy J JJJ J JJ i J J = = + + + + + = = (b) [2] [ ] , z J ±± = ± = Again, this is most easily done by just working it out: [ ] [ ] ( ) 2 , . zz x z y y x x y x y iJJ iJ iJ J iJ J ± ± = = ± + = ± ± = ± =∓= = = = = = (c) [3] J J ± ± J = It is easiest to expand the right side and show that it is equal to the left side. ( )( ) 2 2 2 2 2 , z z z zxx yy x y z z xyz x y zxyz z z J J J J J iJ J iJ J J J iJ J iJ J J J J JJJi J J JJJJi J ± = ± ++±
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This note was uploaded on 02/15/2011 for the course PHYS 742 taught by Professor Unknow during the Spring '04 term at Harvard.

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soln - Physics 741 Graduate Quantum Mechanics 1 Solution...

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