- #1

- 63

- 2

Thanks!

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter csmcmillion
- Start date

- #1

- 63

- 2

Thanks!

- #2

- 13,194

- 763

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

- 4,157

- 202

- #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

- 643

- 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:

Share: