Understanding the Contradiction: Spin Rotation in Quantum Systems

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Discussion Overview

The discussion revolves around the concept of spin rotation in quantum systems, specifically focusing on the behavior of spin-1/2 particles in the presence of an applied magnetic field and radiofrequency (RF) pulses. Participants explore the theoretical underpinnings of spin states, their transformations, and the implications of quantum mechanics versus classical interpretations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about how a spin-1/2 nucleus can rotate when it only takes on discrete values of +1/2 or -1/2 in relation to the magnetic field.
  • Another participant explains that the general state of a spin-1/2 system can be represented as a linear combination of 'spin up' and 'spin down' states, and that transformations between axes are achieved using unitary matrices from SU(2).
  • A third participant notes that every spin state corresponds to a direction in 3-space, but measurements can only yield projections onto chosen axes, resulting in the ±1/2 outcomes.
  • One participant suggests that the spin can also be represented as a linear combination along a chosen z-axis, indicating a potential for different bases in representation.
  • Another participant references a previous explanation by Bill_K, indicating a shared understanding of the topic.
  • A later reply raises the idea that this discussion highlights a classical versus quantum disagreement, particularly in how certain orientations might yield different results when added classically.

Areas of Agreement / Disagreement

Participants express varying interpretations of spin behavior, with some agreeing on the mathematical representation of spin states while others highlight the conceptual differences between classical and quantum perspectives. The discussion remains unresolved regarding the implications of these differences.

Contextual Notes

Participants mention specific mathematical frameworks and concepts, such as unitary transformations and the interaction Hamiltonian, but do not fully resolve the implications of these ideas or their relationship to classical mechanics.

haibane90
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This has been a contradiction in my brain for some time.
If I want to rotate one nuclei (spin 1/2), with say an applied magnetic field B and RF pulse (at the appropriate larmor frequency), how does the spin actually rotate? I thought it can only take on discrete values of 1/2 or -1/2 corresponding to the parallel and anti-parallel directions (with respect to B). I am missing something here, and its killing me.
 
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The general state of a spin-1/2 system is a linear combination of two states, 'spin up' and 'spin down'. It can be quantized along any axis, and the transformation from one axis to another is done by means of a 2-dimensional unitary matrix, an element of SU(2).

Conversely, if you take a state, say |ψ> = α|mz=+1/2> + β|mz=-1/2>, where |α|2 + |β|2 = 1, you can find an axis along which |ψ> is "spin up".

When a B field is applied, the interaction Hamiltonian μ·B adds an additional phase e+iμ·Bt/ħ to the spin up state and e-iμ·Bt/ħ to the spin down state. Thus α changes in time by e+iμ·Bt/ħ and β changes by e-iμ·Bt/ħ. And therefore the axis along which |ψ> is "spin up" changes in time.
 
Every possible state of the spinor corresponds to a particular direction in 3-space. You can come up with a state that corresponds to any direction you like. (Two such states, to be precise.) The problem is that you can't actually measure this direction. You can only measure a projection of this direction vector onto an axis of your choice. And that will correspond to the ±1/2 result you get. The rest follows Bill_K's description.
 
don't you think it can be represented also as a linear combination with suitably chosen two base states along a certain chosen z-axis.
edit::-p
 
Last edited:
That's what Bill_K said.
 
I think I get it. So am I correct in saying that this is one of the classical and qunatum disagreements? Since if we pick certain orientations and add them classically, we would get 0 (I read this is Peres' book).
 

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