Rotating an Electron: Why Different Degrees of Freedom?

[Khashishi]

How come we can rotate a molecule at many multiples of the angular momentum quantum (rotational degrees of freedom), but we can only rotate an electron at 2 different steps (spin degree of freedom)?

 

[mfb]

Electrons cannot be rotated at all, they are (probably) point-like.
Spin is not a rotation - while it shares some common features, the available quantum numbers are not the same.

 

[ApplePion]

Electrons cannot be rotated at all, they are (probably) point-like.
Spin is not a rotation - while it shares some common features, the available quantum numbers are not the same.

The electron's spin has a direction. If you change that direction through continuous transformations you are rotating the electron.

Even if it did not have spin, if it were in a state with non-zero angular momentum you could still rotate it.

Indeed, ther3 are things known as "rortation operators" that can operate on an electron.

 

[mfb]

Those rotations are different from the rotations of molecules - they are rotations of the spin direction (or angular momentum direction, if the electron could have that).

 

[Meselwulf]

Electrons cannot be rotated at all, they are (probably) point-like.
Spin is not a rotation - while it shares some common features, the available quantum numbers are not the same.

Oh they can be rotated even if they are pointlike, difference is, is that you would need to make a 720 degree turn just to bring it back to its original orientation... If my memory serves, it would need to spin faster than light for a pointlike particle, which is forbidden.

 

[Khashishi]

The "spinning faster than light" comes from an incorrect mixture of ideas. In special relativity, there's a maximum speed, but there's no maximum momentum or angular momentum.

 

[ApplePion]

Those rotations are different from the rotations of molecules - they are rotations of the spin direction (or angular momentum direction, if the electron could have that).

Huh?

When you rotate a molecule all you are doing is rotating the electron wave3function and the nuclear wavefunction. There is no special magic.

And even if electrons had no spin, they still could be rotated. If an electron were in a state with a momentum of 8 in the x direction and no other momentum, and I shifted it so it had a momentum of 8 in the y direction and no other momentum, it would have undergone a rotation of 90 degrees about the z axis.

 

[pervect]

How come we can rotate a molecule at many multiples of the angular momentum quantum (rotational degrees of freedom), but we can only rotate an electron at 2 different steps (spin degree of freedom)?

There is an important sense in which you CAN rotate electrons (or other fermions) by small amounts. This becomes important in, for instance, nuclear magnetic resonance, which can be thought of as rotating the classical spin axis of protons by small amounts via an RF field. You can also imagine rotating the spin by a simple, passive coordinate change corresponding to a physical rotation, and imagine that this simple passive coordinate change must be equivalent to some active rotation, for instance physically rotating the object that contains the electron.

What happens is that the quantum mechanical measurement of spin of a single electron (or other fermion) is either |up> or |down>. So when you measure the spin, it's always in one of these states. The problem is not so much being able to rotate the spin, there are a couple of useful notions that correspond to that as I mentioned previously, it's being able to measure the spin that's the problem.

The formal mathematical description of electron spin is \alpha \, |up> + \beta \, |down>, where \alpha and \beta are complex numbers. This is mathematically described by the group called SU(2). The squared magnitude alpha and beta, obtained by multiplying by the complex conjugate, i.e \alpha \, \alpha^* and \beta \, \beta^* describe the probability that you'll get a result of |up> or |down>, and the two probabilities sum to unity.

The formal mathematical description of classical rotation is by a three component real vector, \omega, for example \left[ \omega_x \, \omega_y \, \omega_z \right]. Normalizing the vector means that the sums of the squares of the real components are unity. This is mathematically described by the group called SU(3).

The groups are not identical! It takes 2 "copies" of the classical group SU(3) to "cover" SU(2). Rotating an electron through 360 degrees inverts the sign of the wavefunction, you don't restore the initial state until you rotate the electron through 720 degrees. This is why electrons and other fermions are calls "spin 1/2" particles. You'll see similar results with quaternions, which are a representation of SU(2). Mathematically , SU(2) is a double cover of SO(3). Quaternions are used in many graphics programs and games (notably, second life, for instance) and share the property that rotating a quaternion through 360 degrees changes its sign, so that you need two complete rotations to restore the quaternion to its initial state.

 

[mfb]

When you rotate a molecule all you are doing is rotating the electron wave3function and the nuclear wavefunction. There is no special magic.

When you rotate the electron spin, it rotates while you apply a magnetic field. Stop the field, and it stops changing the spin direction.
The absolute spin of the electron is constant all the time, just its direction changes.

This is completely different from the rotation of molecules Khashishi mentioned: Transfer energy to a molecule in an appropriate way, and it rotates - stop any interaction, and it keeps rotating. The total angular momentum of the molecule is variable. The analog procedure here would be to change the direction of rotation, while keeping the total angular momentum conserved.

And even if electrons had no spin, they still could be rotated. If an electron were in a state with a momentum of 8 in the x direction and no other momentum, and I shifted it so it had a momentum of 8 in the y direction and no other momentum, it would have undergone a rotation of 90 degrees about the z axis.

No, this is just a change of the velocity (expressed in some specific reference frame).

 

[ApplePion]

When you rotate the electron spin, it rotates while you apply a magnetic field. Stop the field, and it stops changing the spin direction.
The absolute spin of the electron is constant all the time, just its direction changes.
This is completely different from the rotation of molecules Khashishi mentioned: Transfer energy to a molecule in an appropriate way, and it rotates - stop any interaction, and it keeps rotating. The total angular momentum of the molecule is variable.

You can do the same thing to a single electron. You can apply a magnetic field and give it orbital angular momentum.

You can change the intrinsic spin directions in a molecule--something you can do with a single electron, and you can give the molecule orbital angular momentum--something you can also do with a single electron.

 

[mfb]

You can apply a magnetic field and give it orbital angular momentum.

Orbit around another particle? That was not the original question.

You can change the intrinsic spin directions in a molecule

Again, that was not the original question.