Spin-1/2 rotational symmetry

In summary: However you do not get perfect cancellations but only cancellations to a certain degree. The reason is that the 2\pi rotation converts neutrons to antineutrons, and you only get perfect cancellation when you have equal numbers of each. However the state of a spinor is not changed by a 4\pi rotation, so after two full rotations the system is back to the original state. Therefore there are two possible paths between the same initial and final states, one involving a 2\pi rotation and the other involving a 4\pi one.In summary, spin states have a double rotation symmetry due to the mathematical properties of spinor wavefunctions. This means that a 4pi rotation is equivalent to a 2
  • #1
dudy
18
0
Hey,
We saw in class that rotating a spin state with an angle of 2pi returns minus the state, and so it has to be rotated 4pi rad in order to return to the original state.
However, we also saw that the expected value of the spin DOES return to its original value after a rotation of 2pi rad.
My question is- Being that we can only measure expected values, and not the states themselves, what physical meaning is there to the "double rotation" symmetry of a spin state?
I mean, how is this theoretical result manifested in experiments ?
thanks
 
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  • #2
Spin I think has to do with vector states. It has to do with the sort of pattern of how a particle oscillates. If your familiar with polar graphs or parametric, you'll notice that depending on how you make a trigonometric equation, it will take more or less "time" to repeat. It isn't much more than a mathematical pattern, since no one knows exactly what spin is in the real world.
 
  • #3
We saw in class that rotating a spin state with an angle of 2pi returns minus the state, and so it has to be rotated 4pi rad in order to return to the original state.
Not exactly. Spinor wavefunctions are double-valued functions. That is, ψ(x,t) and -ψ(x,t) denote the same state. For a normal wavefunction we would say that ψ and -ψ differ by a phase factor, but for spinors it is even stronger: they are the same. When you rotate a half-integer spin state by 2π, the wavefunction changes sign but the state is the same.
 
  • #4
The reason behind this, as far as I know, is purely mathematical. The mapping between SO(3), the group of real rotations in 3 space (rotating real 3-vectors), and SU(2) the group of rotations in this 2 state spin space (rotating complex 2-spinors) is not one to one. SU(2) double covers SO(3), and so literally there are 2 elements in SU(2) that correspond to anyone element in SO(3). This property carries over to quantum mechanics in the definition of a spinor.

Since a spinor is a 2 component complex vector, then the group of isometries preserving the inner product is SU(2), and we have to use SU(2) matrices to denote rotations of the spinor.

Real observables MUST return to the original configuration after a rotation of 360 degrees. After all, I don't change the universe simply by spinning in a 360 degree circle.
 
  • #5
I suppose if you think entirely relativistically (purely local considerations) rather than considering global connections you would miss a physical manifestation of the 4pi rotation invariance.

In "Rotations, Quaternions, and Double Groups" Simon Alltmann takes up the question of what physical situations lead to a 4pi rather than 2pi invariance. He summarizes the analysis of 2 or 3 papers from other authors and comes to the conclusion that it is a consequence of a connection to an external object that shows the 4pi dependence. That fact is demonstrated in the various "scissors" illustrations invented by Dirac such as this one:

http://www.youtube.com/watch?v=O7wvWJ3-t44&NR=1
 
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  • #6
Spinor wavefunctions are double-valued functions.http://www.bosin.info/g.gif [Broken]
 
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  • #7
dudy said:
My question is- Being that we can only measure expected values, and not the states themselves, what physical meaning is there to the "double rotation" symmetry of a spin state?
I mean, how is this theoretical result manifested in experiments ?
I vaguely recall that this can be demonstrated with a neutron interferometer (similar idea to an ordinary interferometer for light, but using neutrons instead). It's possible (using certain configurations of magnetic fields to rotate a neutron. So performing such a [itex]2\pi[/itex]
on one arm of the interferometer, gives cancellations when the two sides are recombined.
 

1. What is Spin-1/2 rotational symmetry?

Spin-1/2 rotational symmetry is a type of symmetry that describes the behavior of particles with half-integer spin, such as electrons. This symmetry means that the properties of these particles are unchanged when they are rotated by 360 degrees, unlike particles with integer spin which remain unchanged when rotated by 180 degrees.

2. How is Spin-1/2 rotational symmetry related to quantum mechanics?

In quantum mechanics, particles are described by their spin, which is a fundamental property that determines how they behave under rotations and other transformations. Spin-1/2 rotational symmetry is a key concept in quantum mechanics as it helps us understand the behavior of particles with half-integer spin.

3. What is the significance of Spin-1/2 rotational symmetry?

Spin-1/2 rotational symmetry is significant because it explains the behavior of particles with half-integer spin, which are essential building blocks of matter and play a crucial role in the laws of physics. It also helps us understand the fundamental principles of quantum mechanics and the nature of particles at a subatomic level.

4. How is Spin-1/2 rotational symmetry experimentally observed?

Spin-1/2 rotational symmetry is experimentally observed through measurements of particles with half-integer spin, such as electrons, in different orientations and directions. These measurements can be done using specialized equipment, such as a Stern-Gerlach apparatus, which can detect the spin state of a particle and its response to rotations.

5. Can particles with Spin-1/2 rotational symmetry violate the laws of physics?

No, particles with Spin-1/2 rotational symmetry cannot violate the laws of physics. This symmetry is a fundamental aspect of quantum mechanics and is consistent with the laws of nature. While particles with half-integer spin may behave differently compared to particles with integer spin, they still follow the same laws and principles of physics.

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