Consider an orbital angular momentum I (I is an integer and positive or zero) and a...

   

Transcribed Image Text:

Consider an orbital angular momentum I (I is an integer and positive or zero) and a spin s with s = 1/2. The tensor product II, ml; s. ms > = l, ml> |s, ms> is then a basis of the state space, which is formed from common eigenstates of 12, 1, s and s. We now define a total angular momentum j = 1 + s and construct a new basis of the state space, which this time consists of simultaneous eigenstates of j and j. Now there must be a basis of common eigenfunctions of the four operators j, j, s, 1. In the following we will call this the "total angular momentum basis". We now want to construct this new basis from the tensor product basis. a First show: jz |I, ml; s, ms > = hbar(ml + ms) |l, ml; s, ms > ! b Now justify that II, ml = 1; s, ms = +1/2> from the tensor product basis is the basis vector lj = 1 + 1/2, mj = 1 + 1/2> of the total angular momentum basis! Consider an orbital angular momentum I (I is an integer and positive or zero) and a spin s with s = 1/2. The tensor product II, ml; s. ms > = l, ml> |s, ms> is then a basis of the state space, which is formed from common eigenstates of 12, 1, s and s. We now define a total angular momentum j = 1 + s and construct a new basis of the state space, which this time consists of simultaneous eigenstates of j and j. Now there must be a basis of common eigenfunctions of the four operators j, j, s, 1. In the following we will call this the "total angular momentum basis". We now want to construct this new basis from the tensor product basis. a First show: jz |I, ml; s, ms > = hbar(ml + ms) |l, ml; s, ms > ! b Now justify that II, ml = 1; s, ms = +1/2> from the tensor product basis is the basis vector lj = 1 + 1/2, mj = 1 + 1/2> of the total angular momentum basis!

Answer rating: 100% (QA)

First lets clarify the notation mi represents the eigenvalue of the zcomponent of the orbital angula... View the full answer