Ladder operators and SU(2) representation

In summary, the uniqueness of the eigenvector at the top and bottom of the ladder in the SU(2) representation is necessary for the finite dimensional representation to allow for all spin-z components. This is due to the fact that all invariant subspaces are one-dimensional and the matrix shifts these subspaces one place higher or lower. For a more thorough mathematical explanation, see [6] in the sources and the underlying theorem (7.1.) at the provided link.
  • #1
kelly0303
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Hello! I read in many places the derivation of the representation for SU(2) using ladder operators and in all of the places they say that, due to the fact that we are looking for a finite dimensional representation, the ladder must end at a point, hence why we have an eigenvector of ##L_3## (usually) such that, when acted on with the raising operator gives zero, and the same for the lowering operator. I didn't find an explanation as to why this must be unique. From a physics point of view it makes sense, as you have an irreducible representation of a spin J particle, so in order for the representation to allow for all the spin-z component, all the eigenvectors must have a different eigenvalue, hence why you have just one eigenvector at the top and at the bottom, and in general for any eigenvalue of ##L_3##, but I am not sure I understand mathematically, why can't you have more than one vector with the same eigenvalue in an irrep of SU(2). Can someone help me? Thank you!
 
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  • #2
kelly0303 said:
Hello! I read in many places the derivation of the representation for SU(2) using ladder operators and in all of the places they say that, due to the fact that we are looking for a finite dimensional representation, the ladder must end at a point, hence why we have an eigenvector of ##L_3## (usually) such that, when acted on with the raising operator gives zero, and the same for the lowering operator. I didn't find an explanation as to why this must be unique. From a physics point of view it makes sense, as you have an irreducible representation of a spin J particle, so in order for the representation to allow for all the spin-z component, all the eigenvectors must have a different eigenvalue, hence why you have just one eigenvector at the top and at the bottom, and in general for any eigenvalue of ##L_3##, but I am not sure I understand mathematically, why can't you have more than one vector with the same eigenvalue in an irrep of SU(2). Can someone help me? Thank you!
You can find the underlying theorem (7.1.) here: https://www.physicsforums.com/insights/journey-manifold-su2-part-ii/
For the mathematical derivation I recommend [6] in the sources. The main reason is, that all invariant subspaces are one-dimensional, and the upper triangular matrix shifts those subspaces one place higher, the lower triangular one place less.
 

Related to Ladder operators and SU(2) representation

What are ladder operators and how do they relate to SU(2) representation?

Ladder operators are mathematical operators used in quantum mechanics to describe the behavior of quantum systems. They are closely related to the group representation theory of SU(2), which is a mathematical framework used to study the symmetries of quantum systems.

How do ladder operators create eigenstates in a quantum system?

Ladder operators act on a quantum state to raise or lower its energy level by a fixed amount. Repeatedly applying these operators can create a series of eigenstates, which are states that have a fixed energy value and are invariant under the action of the operator.

Can ladder operators be used to find the energy spectrum of a quantum system?

Yes, by using the commutation relations between ladder operators, it is possible to determine the energy spectrum of a quantum system. This is because the energy eigenvalues are related to the eigenvalues of the operators.

What are the applications of ladder operators and SU(2) representation in physics?

Ladder operators and SU(2) representation are used in various fields of physics, such as quantum mechanics, quantum field theory, and solid state physics. They are particularly useful for describing the behavior of particles with spin, such as electrons and protons.

Are there other mathematical representations that can be used instead of SU(2) representation?

Yes, SU(2) representation is just one of many group representations that can be used to describe the symmetries of quantum systems. Other examples include SU(3) representation and SO(3) representation.

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