Non-invariance under 2-Pi rotations?

[csmcmillion]

I have heard that quantum systems (and therefor all physical systems) are not truly invariant under 2∏ rotations. Something to do w/ the wave function changing sign. Is this true? Can someone point me to an on-line primer on this?

Thanks!

 

[dextercioby]

Well, for non-relativistic systems in which the spin is half integral (an electron from Pauli equation, for example), a rotation of angle 2\pi indeed changes the sign of the electronic wavefunction. However, this is not really important, because in computing the probabilities, we always take a square, thus such a phase factor is eliminated.

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

 

[Bill_K]

There's a difference. Spinors are double-valued functions. When you write a spinor wavefunction there is always understood to be a ± sign in front. A spinor ψ is a single-valued function on the group space of SU(2), but when you go to SO(3) the mapping is twofold, and ψ becomes double-valued. Consequently for a spin one-half object, ψ and -ψ do not just "differ by a phase", they are literally the same.

 

[ytuab]

Here we define unit vector n in the polar coordinate.

\vec{n} = ( \sin\theta \cos\varphi, \, \sin\theta \sin\varphi, \, \cos\theta )
Of course by 2pi rotation, this vector doesn't change,

\varphi \to \varphi + 2\pi, \qquad \vec{n} \to \vec{n}
The n component of the spinor operator is

\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]
where sigma is Pauli matrices. So,

\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)
The eigenstate of this operator (which direction is "n" ) is

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

\vec{n} \to \vec{n} \quad (\varphi \to \varphi + 2\pi)
But its eigenstate change from +1 to -1 by 2pi rotation.

\alpha_n \to - \alpha_n \quad (\varphi \to \varphi + 2\pi)

 

[PhilDSP]

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 2\pi radians gives a different topological situation compared with 4\pi 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