On the meaning and mathematics of rotating spin 1/2 particles

[snoopies622]

TL;DR Summary

How to interpret 1/2 spin

The other day I found a fascinating video on geometric algebra:



At 34:50, after showing how to rotate a vector in three dimensions, he says, "wait a minute, this looks like a spinor from quantum mechanics. The way that spinors rotate is always said to be a part of so-called 'quantum weirdness', but in fact it's just based on the fact that the best way to represent rotations involves applying the rotation twice."

I've never understood what is meant by this idea that one has to rotate an electron 720 degrees in order to have it arrive back at its original orientation. Is this the meaning? This makes it seem like only a mathematical construct rather than some kind of physical reality.

 

[PeroK]

I've never understood what is meant by this idea that one has to rotate an electron 720 degrees in order to have it arrive back at its original orientation. Is this the meaning? This makes it seem like only a mathematical construct rather than some kind of physical reality.

The basic idea of QM is that things are described by a state vector. The spin state of a spin-1/2 particle is described by a two-dimensional complex vector. Now, a rotation of the system in ordinary 3D space corresponds to an operator (matrix) that acts on spin states. You rotate the system and the spin state is operated on by the relevant matrix. But, like the example of the rotor in your video, this operator uses half the angle. That means that a physical 360 degree rotation transforms the spin state into the negative of what it was. In one sense, that doesn't matter, as the factor of $-1$ is just a phase factor and doesn't change the value of any measured observable. Anything you choose to measure will be the same as before.

But, if you create a superposition of states, rotate one by 360 degrees, then recombine the superposition, they should cancel out (an extraordinary example of quantum interference). This experiment has been conducted successfully. In a demonstrable way, therefore, the state of an electron is changed by a 360 degree rotation and it only returns to the original state after a 720 degree rotation.

 

[snoopies622]

Thank you PeroK for that thorough explanation! I have only one point of confusion: When you say that a rotation of the system corresponds to an operator matrix, I thought that in QM, operators correspond to physical observables, as in, "this matrix operator corresponds to the particle's momentum in the x direction". and with the matrix we have its eigenvectors and eigenvalues which correspond to the possible resulting states and values of performing the (in this case) momentum in the x direction measurement.

I understand how in pure mathematics, multiplying a vector by a matrix creates a new vector, and one can choose a matrix that will rotate the vector in any desirable fashion. In this electron spin area, are the matrices that correspond to the physical observables (spin in the x,y or z direction) the same as the matrices that will rotate a vector by . . some angle?

 

[PeroK]

I have only one point of confusion: When you say that a rotation of the system corresponds to an operator matrix, I thought that in QM, operators correspond to physical observables, as in, "this matrix operator corresponds to the particle's momentum in the x direction". and with the matrix we have its eigenvectors and eigenvalues which correspond to the possible resulting states and values of performing the (in this case) momentum in the x direction measurement.

Observables are represented by Hermitian operators, as you say. But, operators play a wider role in QM. E.g. rotations and space and time translations are represented by unitary operators.

 

[snoopies622]

Thanks again PeroK, I think I understand. Mentioning interference was very helpful. It seems to me now that the fellow in the video is incorrect to dismiss the "quantum weirdness" of rotating spin 1/2 particles as a mere manifestation of a particular mathematical technique.

 

[vanhees71]

But, if you create a superposition of states, rotate one by 360 degrees, then recombine the superposition, they should cancel out (an extraordinary example of quantum interference). This experiment has been conducted successfully. In a demonstrable way, therefore, the state of an electron is changed by a 360 degree rotation and it only returns to the original state after a 720 degree rotation.

Do you have a reference to that experiment? Maybe it's a good idea to implement it in my QM lecture notes :-).

 

[PeroK]

Do you have a reference to that experiment? Maybe it's a good idea to implement it in my QM lecture notes :-).

I got that from Sakurai (page 162: Neutron Interferometry Experiment).

 

[DrClaude]

Do you have a reference to that experiment? Maybe it's a good idea to implement it in my QM lecture notes :-).

Theoretical proposal:
Y. Aharonov and L. Susskind, Phys. Rev. 158, 1237 (1967)

Experimental realization:
S. A. Werner et al., Phys. Rev. Lett. 35, 1053 (1975)
H. Rauch et al., Phys. Lett. A 54, 425 (1975)