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Spin. Griffiths tells me that particle spin is intrinsic angular momentum. This spin isn't actually analogous to the particle spinning in a classical manner, however. Then, later Griffiths says that a spinning charged particle has a magnetic moment. This doesn't make any sense to me, considering that we aren't actually considering the particle to be spinning (in a classical sense).

Is this magnetic moment perhaps similar but different in the same way that spin is 'similar but different' to classical mechanical spin? It seems like there are a great deal of things like this in quantum… it seems kind of like Griffiths is using a lot of classical terms to describe quantum mechanical processees – and like he says, don't take the analogy too far. There are two different kinds of magnetic moments – moments that are intrinsic to particles, like this electron, and moments that are due to the change in current or a flux type situation. (I say griffiths – obviously I mean this as the Quantum voice personified).

Even though the particle isn't actually 'spinning', the fact that there is an intrinsic angular momentum implies (by definition) that some sort of motion is taking place, and Maxwell's equations say that any moving charged particle creates a magnetic moment. So while the electron isn't spinning classically, there is some sort of motion going on that leads to the magnetic moment.

Its my understanding that the quantum and classical magnetic moments are one and the same.

“There are two different kinds of magnetic moments – moments that are intrinsic to particles, like this electron, and moments that are due to the change in current or a flux type situation.”

The intrinsic moment of the electron is due to the intrinsic motion implied by its angular momentum. This motion leads to the change in current/flux situation you mentioned. I have no idea what sort of actual motion the electron must have though and would be very interested to hear some theories.

Griffiths advises to not push the analogy too far: an electron is considered a point particle that doesn't seem to have any internal “structure” and that the spin “cannot be decomposed into orbital angular momenta of constituent parts.” As I see it, when you draw an electron with spin up or down as a dot with a magnetic moment arrow pointing up or down, that's really all you can say about how it's “spinning.”

Multiple sources say that there are two kinds of spin. There may be motion of the electron, but this is not necessarily where the spin comes from. Spin for the electron is an intrinsic property and part of the description of an electron. Unlike the other type of spin which changes and is not an intrinsic property. I agree with Hilbert. I think this is getting pushed too far.

Could we clear up the “eigenspinor” vs. eigenvector thing? I've read elsewhere that eigenspinors are not eigenvectors, but Griffiths seems to be using an eigenspinor for what I would call an eigenvector. But also as a hybrid of what look like eigenvectors. (I refer to pg 175). Could we have a clear and decisive deliniation of all these terms please? spinor, eigenspinor, etc. I'm sure I'm missing something… and I'm kind of tired… but…. y'all know what I mean.

I have no idea what an eigenspinor physically represents.

Neither do I.

This is what I found on Wikipedia:

Eigenspinors are thought of as basis vectors representing the general spin state of a particle. Strictly speaking, they are not vectors at all, but in fact spinors.

Spinors are elements of a complex vector space introduced to expand the notion of spatial vector. They are needed because the full structure of the group of rotations in a given number of dimensions requires some extra number of dimensions to exhibit it. Specifically, spinors are geometrical objects constructed from a vector space endowed with a quadratic form, such as a Euclidean or Minkowski space, by means of an algebraic procedure, through Clifford algebras, or a quantization procedure. A given quadratic form may support several different types of spinors.

For a single spin 1/2 particle, they can be defined as the eigenvectors of the Pauli matrices.

Eigenspinors have a similar place as do rotational quantities in classical mechanics do(for instance omega, which is indicated by the axis it is spinning on). Although rotational quantities in classical mechanics are given a direction, by the axis on which it spins around, this does not satisfy the definition of a vector, so we let it be a pseudo-vector. Sometimes we call rotational quantities vectors, because it fits for that situation and we forget to add that is actually on vector like, which is given by “pseudo”.

Similarly, in QM, eigenvectors found from our spin operators are pseudo-like in nature, to that of its close relative of the eigenstates, which we get from non-spin operators. We call this a Eigenspinors or just Spinors. I don't know what the difference is between the eigenstates and eignespinors and I don't get anything from the classical analogue, since spin isn't the actual “spin” we talk about in classical mechanics. But I think the subtle difference between spinors and eigenstates are don't affect us in the topic we are discussing in Griffths, which is why he calls it a eigenstate, like we calling a rotational quantity in classical mechanics a “vector”.

Hope that helps, there is a lot of math behind this, but I don't think it is really important.

I looked at how Griffths calculated the probability when given a state, but how do you figure out the coefficients of psi+ and psi-? Without this, I am at a lost of calculated probability for certain spin states.

Do you mean ψ or χ?

As far as I can tell, a and b are always given or just some constants, so you can use [4.139] as well as the corresponding eigenvalue for whatever operator you're looking at. It is done in example 4.2 in the book for <math> S_{z} </math> and <math> S_{x} </math>.

χ

I'm looking for a better way to conceptualize what we are doing in class, can anyone help me understand how a particle can have 1/2 of a spin? I know “spin” is a more or less a made up term when it comes to electrons or gravitons and the like, but why don't we just multiply all the spins by 2 and make it easier to understand classically (which is the reason for calling it spin in the first place, if I'm not mistaken)? This is confusing because how can something have half a spin? I guess it's all relative because gravitons have spin of 2, which also makes no sense. Is there a specific reason for there being 1/2 spins instead of just calling everything by twice that?

If you look at equation 4.135, it seems to say that for those operators the observable values must be given in terms of s and <math> \hbar </math>. So you can't really arbitrarily change the eigenvalues/observables for spin because they must obey this relation with the respective operators.

Your question seems to be similar to one thing I was thinking of, which is how are these operators and observables grounded to reality? If you're given some operator, which you apply in a lab setting by taking a measurement, the observable value is the eigenvalue - so I guess you can't just set all spins to be integers instead of half-integers, because integer values might not properly satisfy these eigenvalue equations or you leave out observable quantities but excluding half-integers.

This is a question relative to the discussion and problem 2.38, back when we were working with the infinite square well. If you recall, that was the problem where an infinite square well doubled in size, and the probability of measuring the same energy as before the expansion was 1/2 (I think). That problem was a mathematical curiosity, but our discussion problem last week with the cubical well approximated the FCC system we were describing fairly accurately. My questions regarding this concept are:

1. If you expanded the dimensions, or deformed them like in the extra-credit problem, would you get a similar result in terms of having a non-unity probability of measuring the same energy?

2. Would the energy change between the expectation value of the first and second measurement be the energy required to deform the infinite cubic well (i.e., the FCC structure)?

3. Does the validity of our approximation of the structure as an infinite cubic well break down if it's not cubic or if it's deformed much? I guess the extra credit problem proved that is does deviate to some extent.

I may have to work this out if I have some extra time. I was just wondering about it because we can deform these structures (at least slightly) and the 1-D infinite square well expansion led to an anti-intuitive result.

I like this question.

Does the value of the quantum number s depend on m, or is it completely independent of the other terms and deals with purely spin alone?

Equation 4.137 should solve the question.

I looks like s is a fixed value for a particular particle while m can change depending on the state of that particle. From what I can tell m is actually dependent on s which is independent of the other numbers. It just has to do with the properties of the particle.

Usually I think of s as being like a constant for each particle; all electrons have spins of 1/2, photons have spin of 1, and so on for every particle. Then when you measure it, spin can be up or down, so the <math> m_{s}</math> value can be plus or minus s, or any integer value in between (like <math> m_{l} </math> is for l). So the s value itself tells you something like the magnitude and the m value tells you which direction it's pointing in.

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