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

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

This preview shows page 1 - 2 out of 3 pages.

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 ( ) ( ) ( ) 1 2 1 2 D D D = R R R R One easy way to make this equation work out is to define ( ) D = R R 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 θ , we see that they give ( ) ( ) ( ) 1 0 0 1 0 0 ˆ , 0 cos sin 0 1 0 sin cos 0 1 cos 0 sin 1 0 ˆ , 0 1 0 0 1 0 sin 0 cos 0 1 cos sin 0 1 0 ˆ , sin cos 0 1 0 0 0 1 0 0 1 θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ = = = i j k R R R We now match this to the formula (9.6), which says to linear order ( ) ( ) ˆ ˆ ˆ ˆ , 1 so , 1 i i θ θ θ = = r r S r S r R R = = We can now read off the three spin matrices pretty easily. 0 0 0 0 0 0 0 0 0
Image of page 1
Image of page 2

You've reached the end of your free preview.

Want to read all 3 pages?

  • Spring '04
  • Unknow
  • Physics, Work, Commutator, Lz, sz, Anticommutativity

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes