I can't visualize spin I know that its not just a particle rotating

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Discussion Overview

The discussion centers around the concept of quantum spin, particularly the challenges of visualizing it and understanding its implications in quantum mechanics. Participants explore the nature of spin as a property of particles, its mathematical representation, and the absence of classical analogues.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants express difficulty in visualizing spin, acknowledging that it is not simply a particle rotating about an axis.
  • One participant asserts that spin is a purely quantum phenomenon without a classical analogue for visualization.
  • Another participant suggests thinking of spin as a property of particles, which is supported by others.
  • A participant elaborates that spin relates to the number of rotations of a particle's wavefunction in the complex plane, clarifying that this does not imply literal rotation of the particle.
  • One participant introduces a mathematical perspective, explaining that for a spin 1/2 particle, the Hilbert space is a 2-dimensional complex vector space, and relates this to the Riemann sphere in complex analysis.
  • There is mention of a potential connection between the geometry of the Riemann sphere and the probabilities of measuring spin states, although this is not fully understood by all participants.

Areas of Agreement / Disagreement

Participants generally agree that spin is a property of particles and that it cannot be visualized in classical terms. However, there are differing views on how it can be mathematically represented and understood, particularly regarding the implications of the Riemann sphere and higher spin systems.

Contextual Notes

Some limitations in understanding are noted, including the dependence on mathematical representations and the complexity of higher spin systems, which remain unresolved in the discussion.

Ezio3.1415
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I can't visualize spin... I know that its not just a particle rotating about its axis... Then how do I think of it?
 
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Unfortunately, you can't. It's a purely quantum phenomenon that doesn't have any classical analogue to use for visualization.

And as you say, it certainly isn't a particle rotating about an axis.
 


Then we just think of it as a property of the particles...
 


Ezio3.1415 said:
Then we just think of it as a property of the particles...

Correct.
 
Ezio3.1415 said:
Then we just think of it as a property of the particles...

Yes. Essentially, it refers to the number of times you must rotate a particle's wavefunction through the complex plane to bring it back to it's original value.

A misconception is that it refers to the number of times you must literally rotate a particle to bring it back to it's original state, but this is false. 'Rotation' in the above context refers to an operator on the wavefunction in Hilbert Space.
 


Thanks... :)
 


It's a purely quantum phenomenon that doesn't have any classical analogue to use for visualization.

It might not have a classical analogue, but that doesn't mean that it can't be visualized mathematically, and this is particularly true for a spin 1/2 particle.

If you have a spin 1/2 particle, like the electron that is can be spin up or spin down, the Hilbert space is a 2-dimensional complex vector space. When you consider than multiplying by a constant give you the same physical state, you get a 1-dimensional complex projective space. This is the nothing but the Riemann sphere of complex analysis. Thus, the Riemann sphere can be associated to any system with a 2-d state space, which each point in the Riemann sphere corresponding to a physical state. And what's really interesting is that the geometry of the Riemann sphere is related to the probability of measuring spin up or spin down. For details, see The Road to Reality, by Roger Penrose, Chapter 22, I believe.

There seems to be some kind of similar picture for higher spin that he talks about, but I don't understand it yet.
 

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