1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Finite Rotations-QM

  1. Feb 14, 2004 #1
    Finite Rotations

    [tex] D^{\frac{1}{2}}[R]=exp( \frac{-i}{\hbar} \mathbf{\theta} \cdot \mathbf{J}^{\frac{1}{2}} ) = cos(\frac{\theta}{2}) I-\frac{2i}{\hbar}sin(\frac{\theta}{2}) \hat{\theta} \cdot \mathbf{J}^{\frac{1}{2}} [/tex]

    [tex] J_i^{\frac{1}{2}}=\frac{\hbar}{2} \sigma_i [/tex]
    and [itex] \sigma_i [/itex] is the appropriate pauli matrix. And I is the identity matrix.

    here is what I have so far... I get so close but the solution is incorrect:

    [tex] = e^{\frac{-i \theta}{2}} e^{\sum_{i=1}^3 \sigma_i} [/tex]
    [tex] = (cos(\frac{\theta}{2})-i sin(\frac{\theta}{2})) e^{\sum_{i=1}^3 \sigma_i} [/tex]
    [tex] = (cos(\frac{\theta}{2})-i sin(\frac{\theta}{2})) \sum_{n=0}^\infty \frac{(\sum_{i=1}^3 \sigma_i)^n}{n!} [/tex]

    for j=1/2 the sum over n only needs to go from 0 to 2j (1) so the last line only pics up the first 2 terms.

    [tex] = (cos(\frac{\theta}{2})-i sin(\frac{\theta}{2}))(I + \sum_{i=1}^3 \sigma_i)[/tex]
    Now let:
    [tex] \sum_{i=1}^3 \sigma_i = \frac{2}{\hbar} \hat{\theta} \cdot \mathbf{J}^{\frac{1}{2}} [/tex]
    [tex] D^{\frac{1}{2}}[R]= (cos(\frac{\theta}{2})-i sin(\frac{\theta}{2}))(I + \frac{2}{\hbar} \hat{\theta} \cdot \mathbf{J}^{\frac{1}{2}}) [/tex]

    Now I know this is isn't correct... but it is sooo close that I am having a hard time finding where I went wrong and how else to get the cosine and sine terms to show up. Please help, I am horribly frustrated.
    Last edited: Feb 16, 2004
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted