User talk:Stevenj

Hello and Welcome! I hope you like the place. --mav

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Hi, I have a hard time understanding two of your additions to Quaternion and Pauli matrix:

In quantum mechanics, the 2x2 matrices that multiply b/c/d, times i, are called the Pauli matrices (plus the identity matrix for a). Moreover, this representation of a quaternion corresponding to a spatial rotation is equivalent to the rotation operator for spin-1/2 particles.

What do you mean when you say "this representation of a quaternion as a spatial rotation"? We didn't represent it as a spatial rotation, we represented it as a 2-by-2 complex matrix. I don't think you can represent quaternions as spatial rotations; the unit quaternions of course give rise to spactial rotations, but even this representation isn't faithful.

Together with the identity matrix I (which is sometimes written as σ0), the Pauli matrices form a basis for the set of 2 × 2 complex Hermitian matrices. This basis is equivalent to quaternion numbers, and when used as the basis for the spin-1/2 rotation operator it is the same as the corresponding quaternion rotation representation.

In which sense is this basis "equivalent" to quaternion numbers? What "quaternion rotation representation" are you referring to here?

Thanks, AxelBoldt 01:24 Apr 28, 2003 (UTC)

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Hi Axel, part of the misunderstanding here is that you are misquoting me. I didn't say "this representation of a quaternion as a spatial rotation," I said "this representation of a quaternion corresponding to a spatial rotation." What I meant was basically the opposite of what you're thinking: I'm referring to the representation of a rotation by a quaternion, not the other way around. Less succinctly:

Suppose you take a rotation and represent it by a quaternion, which in turn is represented in the 2x2 matrix form. Alternatively, take the same rotation and represent it by the spin-1/2 rotation operator (a 2x2 matrix). The statement is that these two matrices are, in fact, the same. See also e.g. http://www.nyu.edu/classes/tuckerman/quant.mech/lectures/lecture_5/node4

• slight correction: they are slightly different, corresponding to a 90-degree coordinate rotation x -> y, y -> -x. They are the same if you modify the 2x2 matrix formula in the quaternion page to use the Pauli matrix mapping described below.

I think that if you parse my original sentence carefully, the meaning is correct, but perhaps we should rephrase it to be more clear. =)

Regarding the equivalence of the quaternions and the Pauli matrices, I meant that if you take the quaternion a + bi + cj + dk, and map it to the matrix a * sigma_0 - b * i * sigma_1 - c * i * sigma_2 - d * i * sigma_3, you get an isomorphism. That is, you make the identification (1,i,j,k) <-> (sigma_0, -i sigma_1, -i sigma_2, -i sigma_3).

• this representation is slighly different from the 2x2 matrix representation on the quaternions page; the two are related by the isomorphism b -> c, c -> -b.

The basic point is that there is a deep connection between the algebras of quaternions and Pauli matrices, and between the use of quaternions to represent rotations and the rotation operator for spin-1/2 particles.

- Steven G. Johnson, Wed May 28 20:34:39 EDT 2003

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Ah I see you spotted my "spinning particle" reference on magnetic field. Call it my little bit of rebellion -- I'm inclined to think the difference between QM spin and mechanical spin is rather exaggerated. BTW Griffiths calls B the magnetic field and H the "auxillary field", hence the article title. -- Tim Starling 02:10 10 Jun 2003 (UTC)

Hi Tim. There is an important difference between QM spin and a (classical) spinning particle: the axis of a spinning particle is a 3-vector, whereas QM spin is not (there is an observable difference in how they transform under rotations). Regarding the term magnetic induction for B, it is a historical thing. Jackson uses magnetic field initially for B, but switches to magnetic induction when he starts talking about H. It's one of those rules that's observed more in the breach, but it's important to mention. -- Steven G. Johnson, Mon Jun 9 22:31:50 EDT 2003

Just saw your home page... do you know anything about quantum many body techniques, or ab-initio quantum chemistry? -- Tim Starling 02:38 10 Jun 2003 (UTC)

Some, although it's not my speciality (my work more involves solid-state physics applied to classical electromagnetism, not quantum mechanics). -- Steven G. Johnson

Damn. I thought you might be able to help me with my PhD project, because my supervisor seems to be incapable. Your comment about spin hit my misconception dead-on -- let's just say I was rather humbled. You must be a very good lecturer, not to mention an exceptional physicist. Don't waste too much time hanging around Wikipedia, okay? -- Tim Starling 03:11 10 Jun 2003 (UTC)

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Hi there. What's the the handedness of the universe? (seen on Pseudovector) -- Tarquin 20:29 12 Jun 2003 (UTC)

You can't really define an absolute "handedness". If you want a left-hand rule for cross-products, you just slightly change your definitions of things like the magnetic field. The real question is whether the laws of physics are invariant under inversion; for classical physics they are, but for weak interactions they are not. To take a more simplistic example, suppose that some observable quantity depended upon the sum E+B (assuming compatible units). After inversion, this becomes B-E, and this is not just a change of definitions because Maxwell's equations (which are invariant) also relate E and B...so an experiment measuring that quantity could determine whether the inverted or non-inverted version was correct, but you can still always define things to use a right-hand rule. -- Steven G. Johnson

Regarding your cross-product edit on handedness: does that mean that in a LH system, a ^ b is still defined as a RH triple? I know of at least one programming language where that is not the case. -- Tarquin 09:01 13 Jun 2003 (UTC)