Understanding the Contradiction: Spin Rotation in Quantum Systems

[haibane90]

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.

 

[Bill_K]

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|m_z=+1/2> + β|m_z=-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.

 

[K^2]

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.

 

[andrien]

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:

 

[K^2]

That's what Bill_K said.

 

[haibane90]

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).