# Non-invariance under 2-Pi rotations?

• csmcmillion

#### 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!

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.

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.

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.

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

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