- #1

- 63

- 2

Thanks!

- Thread starter csmcmillion
- Start date

- #1

- 63

- 2

Thanks!

- #2

- 13,024

- 579

Anyways, Ballentine's book on QM (or any group theory book for physicists) explains how that (-1)^s comes from.

- #3

Bill_K

Science Advisor

- 4,155

- 199

- #4

- 189

- 0

[tex]\vec{n} = ( \sin\theta \cos\varphi, \, \sin\theta \sin\varphi, \, \cos\theta )[/tex]

Of course by 2pi rotation, this vector doesn't change,

[tex]\varphi \to \varphi + 2\pi, \qquad \vec{n} \to \vec{n}[/tex]

The n component of the spinor operator is

[tex] \hat{S}_n = \vec{n}\cdot \vec{S} = \frac{\hbar}{2} \left[ (\sin\theta \cos\varphi) \sigma_1 + (\sin\theta \sin\varphi) \sigma_2 + (\cos\theta) \sigma_3 \right][/tex]

where sigma is Pauli matrices. So,

[tex] \hat{S}_n = \frac{\hbar}{2} \left( \begin{array}{cc} \cos\theta & e^{-i\varphi} \sin\theta \\ e^{i\varphi} \sin\theta & -\cos\theta \end{array} \right) [/tex]

The

[tex]\alpha_n = \left( \begin{array}{c} \cos \frac{\theta}{2} e^{-i\varphi /2} \\ \sin\frac{\theta}{2} e^{i\varphi /2} \end{array} \right) \quad \hat{S}_n \alpha_n = \frac{\hbar}{2} \alpha_n [/tex]

If we rotate the direction of "n" by 2pi, the unit vector "n" doesn't change, as shown above.

[tex]\vec{n} \to \vec{n} \quad (\varphi \to \varphi + 2\pi)[/tex]

But its eigenstate change from +1 to -1 by 2pi rotation.

[tex]\alpha_n \to - \alpha_n \quad (\varphi \to \varphi + 2\pi) [/tex]

- #5

- 642

- 15

Another way to think of it is that if an object shares an elastic force or field interaction with other objects that have spatial extension, spinning the object [itex]2\pi[/itex] radians gives a different topological situation compared with [itex]4\pi[/itex] radians.

Dirac is said to have invented this type of example to explain the concept:

Spinor rotated twice

http://www.youtube.com/watch?v=O7wvWJ3-t44&NR=1

Dirac is said to have invented this type of example to explain the concept:

Spinor rotated twice

http://www.youtube.com/watch?v=O7wvWJ3-t44&NR=1

Last edited:

- Replies
- 3

- Views
- 3K

- Replies
- 6

- Views
- 4K

- Replies
- 13

- Views
- 1K

- Last Post

- Replies
- 4

- Views
- 2K

- Last Post

- Replies
- 5

- Views
- 4K

- Replies
- 2

- Views
- 2K

- Replies
- 1

- Views
- 1K

- Replies
- 6

- Views
- 1K

- Replies
- 14

- Views
- 297

- Replies
- 2

- Views
- 851