a representation of the group of rotation operators. When rotation operators act on quantum states, it forms a representation of the Lie group SU(2) (for R and Rinternal), or SO(3) (for Rspatial). From the relation between J and rotation operators, When angular momentum operators act on quantum states, it forms a representation of the Lie algebra or . (The Lie algebras of SU(2) and SO(3) are identical.) The ladder operator derivation above is a method for classifying the representations of the Lie algebra SU(2). Connection to commutation relations Classical rotations do not commute with each other: For example, rotating 1° about the x-axis then 1° about the y-axis gives a slightly different overall rotation than rotating 1° about the y-axis then 1° about the x-axis. By carefully analyzing this noncommutativity, the commutation relations of the angular momentum operators can be derived. (This same calculational procedure is one way to answer the mathematical question "What is the Lie algebra of the Lie groups SO(3) or SU(2)?") Conservation of angular momentum The Hamiltonian H represents the energy and dynamics of the system. In a spherically symmetric situation, the Hamiltonian is invariant under rotations: where R is a rotation operator. As a consequence, , and then due to the relationship between J and R. By the Ehrenfest theorem, it follows that J is conserved. To summarize, if H is rotationally-invariant (spherically symmetric), then total angular momentum J is conserved. This is an example of Noether's theorem. If H is just the Hamiltonian for one particle, the total angular momentum of that one particle is conserved when the particle is in a central potential (i.e., when the potential energy function depends only on ). Alternatively, H may be the Hamiltonian of all particles and fields in the universe, and then H is always rotationally-invariant, as the fundamental laws of physics of the universe are the same regardless of orientation. This is the basis for saying conservation of angular momentum is a general principle of physics. For a particle without spin, J = L, so orbital angular momentum is conserved in the same circumstances. When the spin is nonzero, the spin–orbit interaction allows angular momentum to transfer from L to S or back. Therefore, L is not, on its own, conserved. Angular momentum coupling Often, two or more sorts of angular momentum interact with each other, so that angular momentum can transfer from one to