solq - 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 D R1 D R2
Physics 741 – Graduate Quantum Mechanics 1 Solution Set Q 1. [10] In chapter nine, section A, we were searching for matrices ()DRwhich satisfy ()()()1212DDD=RRR ROne easy way to make this equation work out is to define ()D=RROur 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 θ, we see that they give ()()()100100ˆ,0cossin010sincos01cos0sin10ˆ,010010sin0cos01cossin010ˆ,sincos010001001θθθθθθθθθθθθθθθθθθθθθ⎛⎞⎛⎞⎜⎟⎜⎟=−≈−⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎛⎞⎜⎟⎜⎟=≈⎜⎟⎜⎟⎜⎟⎜⎟−−⎝⎠⎝⎠−−⎛⎞⎛⎞⎜⎟⎜⎟=≈⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠ijkRRRWe now match this to the formula (9.6), which says to linear order ()()ˆˆˆˆ,1so,1iiθθθ=−⋅⋅=−⎡⎤⎣⎦rr Sr SrRR==We can now read off the three spin matrices pretty easily. 000000000