Finite Rotations: Prove D^(1/2)[R]

  • Thread starter Norman
  • Start date
  • Tags
    Finite
In summary, we have proven that 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}} for the given values of J_i^{\frac{1}{2}} and \sigma_i.
  • #1
Norman
897
4
Finite Rotations

Problem:
PROVE:
[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]

where:
[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]
therefore:
[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.
Thanks,
Norm
 
Last edited:
Physics news on Phys.org
  • #2
aSolution: Starting with your last expression, we can simplify it as follows: 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}}) = cos(\frac{\theta}{2}) (I + \frac{2}{\hbar} \hat{\theta} \cdot \mathbf{J}^{\frac{1}{2}}) -i sin(\frac{\theta}{2}) (I + \frac{2}{\hbar} \hat{\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}} + \frac{2}{\hbar} sin(\frac{\theta}{2}) \hat{\theta} \cdot \mathbf{J}^{\frac{1}{2}}-i sin(\frac{\theta}{2}) I -\frac{2i}{\hbar}sin(\frac{\theta}{2}) \hat{\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}} Thus, the proof is complete.
 
  • #3
an


Dear Norman,

Thank you for sharing your work so far. It seems like you are on the right track, but there are a few errors in your calculations.

First, when you expanded the exponential term, you only need to consider the first two terms because for j=1/2, the sum only goes up to n=1. However, your expansion goes up to n=3. This is why you have extra terms in your final expression.

Secondly, when you let \sum_{i=1}^3 \sigma_i = \frac{2}{\hbar} \hat{\theta} \cdot \mathbf{J}^{\frac{1}{2}}, you are missing a factor of i in the right-hand side. It should be \frac{2i}{\hbar} \hat{\theta} \cdot \mathbf{J}^{\frac{1}{2}}. This is why you end up with the incorrect expression for D^{1/2}[R] at the end.

To correct these errors, let's start from the beginning. We have:

D^{\frac{1}{2}}[R] = e^{\frac{-i}{\hbar} \mathbf{\theta} \cdot \mathbf{J}^{\frac{1}{2}}}

= cos(\frac{\theta}{2}) I -i sin(\frac{\theta}{2}) \mathbf{\theta} \cdot \mathbf{J}^{\frac{1}{2}}

= cos(\frac{\theta}{2}) I -i sin(\frac{\theta}{2}) \frac{2i}{\hbar} \hat{\theta} \cdot \mathbf{J}^{\frac{1}{2}}

= cos(\frac{\theta}{2}) I - \frac{2}{\hbar} sin(\frac{\theta}{2}) \hat{\theta} \cdot \mathbf{J}^{\frac{1}{2}}

= cos(\frac{\theta}{2}) I - \frac{2}{\hbar} sin(\frac{\theta}{2}) (\frac{\hbar}{2} \sigma_x \hat{\theta}_x + \frac{\hbar}{2} \sigma_y \hat{\theta}_y + \frac{\hbar}{2} \sigma_z \hat{\theta
 

1. How do you define finite rotations?

Finite rotations are defined as rigid body movements where an object moves along a circular path, with a fixed center of rotation, and returns to its original position after a specific angle of rotation.

2. What is the purpose of proving D^(1/2)[R] in finite rotations?

The purpose of proving D^(1/2)[R] in finite rotations is to understand the mathematical principles behind these movements and to find a way to represent them accurately.

3. What is D^(1/2)[R] in the context of finite rotations?

D^(1/2)[R] is a mathematical representation of the finite rotation, where D represents a differential operator and R represents the rotation matrix.

4. How does the proof of D^(1/2)[R] help in understanding finite rotations?

The proof of D^(1/2)[R] helps in understanding finite rotations by providing a mathematical framework to analyze and represent these movements accurately. It also helps in predicting the behavior of objects undergoing finite rotations.

5. Can D^(1/2)[R] be applied to any type of finite rotation?

Yes, D^(1/2)[R] can be applied to any type of finite rotation, as long as the rotation is rigid and follows the principles of circular motion with a fixed center of rotation.

Similar threads

  • Introductory Physics Homework Help
Replies
1
Views
144
  • Introductory Physics Homework Help
Replies
2
Views
549
  • Introductory Physics Homework Help
Replies
6
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
808
  • Introductory Physics Homework Help
Replies
5
Views
224
  • Introductory Physics Homework Help
Replies
17
Views
309
  • Introductory Physics Homework Help
Replies
13
Views
2K
  • Introductory Physics Homework Help
Replies
6
Views
1K
  • Introductory Physics Homework Help
Replies
21
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
1K
Back
Top