Can particle quantum spin be described with a wave function? [closed]

I'm a little confused about the idea of spin. It's been non-technically described to me as "like magnetic dipole moment", except only two possible "directions". But I feel like that's a bad analogy, because direction is relative and 3 dimensional.

I understand that under the Ising model, spin has +1 or -1 values, and in the case of a pi meson, 0 (is that because of an alignment of quark-antiquark spins?)

So, then, spin of a particle would be the sum of spin of its quarks, which explains half-spin?

And in light of a recent observation of photons behaving as both a wave and a particle in a single experiment, could one say that spin direction is the direction of the valley and the peak of a wave, and the angular momentum is simply the two dimensional gradient of the wave itself? In other words, is the angular momentum of spin measured as the "up" and "down" "motion" of the compression of energy fields where the spin direction is the alignment of the "up" and "down" peaks/valleys, relative to whatever dimension on which the wave propagates?

I guess what I'm getting at is I'm trying to understand whether particles are just wave interference patterns (actual, not probability) of compressed and decompressed energy across the extra dimensions which we do not call home.

closed as unclear what you're asking by Norbert Schuch, ACuriousMind♦, user36790, CuriousOne, GertApr 27 '16 at 1:44

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• Just to clarify, I'm not proposing anything, I'm asking a genuine question because I'm trying to get a better grasp on "spin", and at the same time I'm exploring particle-wave duality. – Dexter Apr 24 '16 at 2:50
• As far as wave-particle duality is concerned, there is nothing to explore, unless you are interested in the history of science. It's a bad idea on par with the phlogiston and aether and you can forget about it. The treatment of angular momentum in quantum mechanics is a couple of non-trivial chapters in a serious textbook, so I am not sure what you can expect from an answer here... you will have to work trough a textbook if you want to understand how it works. – CuriousOne Apr 25 '16 at 6:05
• Dexter, see my answer here concerning spin and the Einstein-de Haas effect. I don't know of any extra dimensions, but I'd say particles are "wave configurations". When a 511keV wave moves linearly we call it a photon. When it's going round and round in a standing-wave standing-field spin-half bispinor chiral path, we call it an electron or a positron. Also see this and this. – John Duffield Apr 25 '16 at 16:11
• @JohnDuffield: Particles are a type of approximation in classical mechanics that is useless in QM. Quanta, on the other hand, are local field states. – CuriousOne Apr 26 '16 at 6:41
• @JohnDuffield I was simply interested in understanding how spin is represented in formulas concerning the energy of a particle, and I think as far as that is concerned, I've gotten my answer, I'm just studying through all the comments here – Dexter Apr 26 '16 at 13:33

1 Answer

Within the context of first quantization (the Schrodinger equation), spin itself is not characterized by the spatial or momentum space wave function (the function that defines a complex quantity over all space). It is instead treated with a separate Hilbert space that quantifies the spin states 'tacked on' to this wavefunction by an outer product. It is a matter of semantics whether you consider 'wavefunction' to refer to the complex field alone, or the composite field/spin hilbert spaces.

More advanced field equations incorporate spin more naturally. The Dirac equation, for example, no longer describes the evolution of a complex scalar field, but the evolution of a four-vector, that naturally describes spin states. For this system, spin $\textit{is}$ characterized in the wavefunction itself.

• Of course spin is characterized by the wave function. It is just a wave function defined on a spatial and a spin degree of freedom. – Norbert Schuch Apr 24 '16 at 11:04
• @NorbertSchuch hm. This is definitional, but I thought wavefunction implied that which could be represented in, e.g., configuration or momentum space, with distinction made in calling a quantum state the complete quantum description. This doesn't match wikipedia's definition, So I'll consider revising. – anon01 Apr 25 '16 at 3:09
• A wave function assigns to each possible classical state (such as position, or position and spin) of a particle a complex number. – Norbert Schuch Apr 27 '16 at 13:29
• @NorbertSchuch what? spin is not a classical state – anon01 Apr 27 '16 at 13:31
• "spin up" and "spin down" can be understood as completely classical properties. We're not talking about relativistic QM here -- it's just another property of an object, such as its position. – Norbert Schuch Apr 27 '16 at 16:20