Why can the spin and the angular momentum transform to each other?

[ndung200790]

In relativistic limit the spin and the angular momentum are not of conservation because of spin-orbit interaction.Then the symmetry SU(2) is broken because vector spin does not commute with the interaction Hamintonian.The SO(3) symmetry is also broken for the same reason.So I do not understand why the sum of spin and angular momentum is of conservation in relativistic limit(then the spin and the angular momentum transform to each other).

 

[jarod765]

I had a similar question in:

https://www.physicsforums.com/showthread.php?t=642283

The answer is, I think, that in high energies spin is a good quantum number because the states we are considering are asymptotically non-interacting particles. In lower energies, where spin-orbit is more important, the bound states are elements of the full symmetry group which has "j" as a good quantum number.

 

[ndung200790]

What is the full symmetry?Is it the direct product SU(2)xSO(3) symmetry?Is quantum number j the result of this product symmetry?

 

[jarod765]

The full symmetry is some gauged lie group, for example SU(3)xSU(2)xU(1), *and* the Poncaire group, which is a continuous global symmetry.

The intrinsic spin of particles arises because particles live in non-scalar representations of the Poncaire group. Thus, the symmetry group which is associated with spin is the rotation subgroup of the Poncaire group.

One must keep in mind that the rotation subgroup contains a finite dimensional matrix representation as well as an infinite dimensional rep. The finite dimensional rep is associated with the intrinsic spin S while the infinite dimensional rep corresponds to orbital angular momentum L. The generator of the this rotation group is L+S and it's casimir operator is J^2=(L+S)^2. Thus it is the eigenvalues of J which are good quantum numbers.

 

[ndung200790]

As you say intrinsic spin is associated with finite dim rep of rotation group.Then I am very confused because as we know the spin is also associated with representations of SU(2) group.Then is the rep of SU(2) to be the rep of rotation group?

 

[DrDu]

In relativistic limit the spin and the angular momentum are not of conservation because of spin-orbit interaction.Then the symmetry SU(2) is broken because vector spin does not commute with the interaction Hamintonian.The SO(3) symmetry is also broken for the same reason.So I do not understand why the sum of spin and angular momentum is of conservation in relativistic limit(then the spin and the angular momentum transform to each other).

I don't understand what you are saying. Rotation symmetry SU(2) is not broken in relativistic QM. The commutation of vector spin (whatever this may be) is not important. Decisive is that total angular momentum commutes with the hamiltonian. I also don't understand what you mean by sum of spin and angular momentum. Spin is part of total angular momentum. Total angular momentum is conserved hence it can't transform into spin. The absolute value of spin is a scalar and commutes with both total angular momentum and the hamiltonian. Hence it is conserved, too.

 

[DrDu]

As you say intrinsic spin is associated with finite dim rep of rotation group.Then I am very confused because as we know the spin is also associated with representations of SU(2) group.Then is the rep of SU(2) to be the rep of rotation group?

Basically yes. Rotation group is SO(3). However in QM, we are not only interested in true representations of groups but also in projective ones. Projective representations of the rotation group can be shown to be true representations of the group SU(2) which is the global covering group of SO(3).

 

[ndung200790]

Thanks DrDu very much!Now I can understand the problem.

 

[jarod765]

The absolute value of spin is a scalar and commutes with both total angular momentum and the hamiltonian.

This is only true in some systems. For example, in the relativistic treatment of the hydrogen atom, this is not the case. In fact both the angular and spin angular momentum are not good quantum numbers in this system.

 

[DrDu]

This is only true in some systems. For example, in the relativistic treatment of the hydrogen atom, this is not the case. In fact both the angular and spin angular momentum are not good quantum numbers in this system.

Spin is angular momentum in the rest frame. You don't want to tell me that for a hydrogen atom the angular momentum in it's rest frame is not a good quantum number. However, it does not equal the sum of the spins of the individual electon and nucleus which is also sometimes called the spin of the hydrogen atom although this is not quite correct.

 

[ndung200790]

S_{z}and L_{z} do not commute with spin-orbit interacting Hamontonian,but the sum S_{z}+L_{z} commutes with the Hamintonian.Then S_{z}and L_{z} separately considering are not of conservation,but the sum is of conservation and the spin and angular momentum can transform to each other.

 

[ndung200790]

So spin is ''intrinsic'' angular momentum only in rest frame.

 

[DrDu]

S_{z}and L_{z} do not commute with spin-orbit interacting Hamontonian,but the sum S_{z}+L_{z} commutes with the Hamintonian.Then S_{z}and L_{z} separately considering are not of conservation,but the sum is of conservation and the spin and angular momentum can transform to each other.

In the last sentence, you should write "orbital angular momentum" ant not angular momentum.
The second point I wanted to make is that in relativistic qm S_z is not really spin. E.g. in a hydrogen atom, the spin of the total system includes also orbital angular momentum in the rest frame.
So neither S_z nor L_z are conserved in a hydrogen atom once relativistic effects like spin orbit interaction are taken into account. Nevertheless total angular momentum J and total spin are conserved quantities and commute with the hamiltonian.

 

[DrDu]

So spin is ''intrinsic'' angular momentum only in rest frame.

http://en.wikipedia.org/wiki/Pauli-Lubanski_pseudovector