Understanding the Contradiction: Spin Rotation in Quantum Systems

In summary, the two states that a spin-1/2 system can be in are spin up and spin down. When a magnetic field is applied, the state changes by adding an extra phase. The direction that the state is in can only be determined by measuring a projection of the state onto an axis.
  • #1
haibane90
4
0
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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  • #2
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.
 
  • #3
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.
 
  • #4
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::tongue2:
 
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  • #5
That's what Bill_K said.
 
  • #6
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).
 

1. What is the meaning of "spin 1/2" in reference to a system?

Spin 1/2 refers to the intrinsic quantum mechanical property of a particle or system, which can have two possible values: +1/2 or -1/2. This is often used to describe the spin of subatomic particles, such as electrons, protons, and neutrons.

2. How does the rotation of spin 1/2 systems differ from classical rotation?

In classical mechanics, rotation refers to the physical spinning of an object around an axis. However, in quantum mechanics, the concept of rotation is used to describe the change in orientation of a particle's spin. This is a purely mathematical concept and does not involve physical movement.

3. What is the significance of the rotation of spin 1/2 systems in quantum computing?

The rotation of spin 1/2 systems is crucial in quantum computing as it is used to manipulate and control the state of qubits, which are the basic units of quantum information. By applying rotations to qubits, quantum computers can perform complex calculations and solve problems that are not feasible with classical computers.

4. Can the rotation of spin 1/2 systems be observed or measured?

No, the rotation of spin 1/2 systems cannot be directly observed or measured. However, its effects can be observed indirectly through measurements of other properties, such as energy levels or magnetic moments, which are affected by the rotation of the system.

5. How is the rotation of spin 1/2 systems related to the concept of superposition?

The rotation of spin 1/2 systems is closely related to the concept of superposition, which refers to the ability of a quantum system to exist in multiple states simultaneously. By applying rotations to a qubit, it is possible to create and manipulate superposition states, which are essential for quantum computing and other quantum applications.

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