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

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Physics 741 – Graduate Quantum Mechanics 1 Solution Set Q 1. [10] In chapter nine, section A, we were searching for matrices () D R which satisfy ( ) ( ) ( ) 12 1 2 DD D = RR R R One easy way to make this equation work out is to define ( ) D = Our goal in this problem is to identify the spin. (a) [4] Using the equations (7.26) and the definition of the spin matrices (9.6), work out the three spin matrices S. Equations (7.26) give the rotation matrices around each of the three axes for arbitrary angle θ . If we write these to order , we see that they give 10 0 1 0 0 ˆ ,0 c o s s i n 0 1 0s i n c o s 0 1 cos 0 sin 1 0 ˆ 1 0 0 1 0 sin 0 cos 0 1 cos sin 0 1 0 ˆ ,s i n c o s 0 1 0 00 1 0 0 1 θθ ⎛⎞ ⎜⎟ = −≈ ⎝⎠ =≈ −− i j k R R R We now match this to the formula (9.6), which says to linear order ( ) ˆˆ ˆ ˆ ,1 s o ii =− ⋅ ⋅ = rr S r S r == We can now read off the three spin matrices pretty easily. 00 0 0 0 0 0 , 0 00 , . 0 0 000 xyz Si S S i ⎛⎞⎛⎞⎛⎞ ⎜⎟⎜⎟⎜⎟ =− = = ⎝⎠⎝⎠⎝⎠ ===
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(b) [3] Find the eigenvalues of S z . This should be enough for you to conjecture
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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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solq - Physics 741 Graduate Quantum Mechanics 1 Solution...

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