sec8 - VIII. Addition of Angular Momenta a. Coupled and...

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VIII. Addition of Angular Momenta a. Coupled and Uncoupled Bases When dealing with two different sources of angular momentum, J ˆ 1 and J ˆ 2 , there are two obvious bases that one might choose to work in. The first is called the uncoupled basis. Here the basis kets are eigenstates of both operators: 2 j , m ; j 2 , m 2 = j ( j + 1 29 j , m 1 ; j 2 , m 2 J ˆ 1 1 1 1 1 1 j , m 1 ; j 2 , m 2 = m 1 j , m 1 ; j 2 , m 2 J ˆ 1 z 1 1 2 j , m ; j 2 , m 2 = j ( j + 1 29 j , m ; j 2 , m 2 J ˆ 2 1 1 2 2 1 1 j , m ; j 2 , m 2 = m 2 j , m ; j 2 , m 2 J ˆ 2 z 1 1 1 1 In the case of spin orbit coupling, this would mean that our basis states would be simultaneous eigenfunctions of orbital and spin angular momentum, and each state would have a particular value for , { m l } and m . Note that this is only possible if (as we assume): s ˆ ˆ [ J , J 2 ] = 0 1 otherwise, the two operators would not have simultaneous eigenfunctions. This is clearly true for L ˆ and S ˆ since the two operators act on different spaces. This really defines what we mean by different angular momenta, since operators that do not commute will, in some sense, define overlapping – and thus not completely distinct – forms of angular momentum. This basis is appropriate if J ˆ 1 and J ˆ 2 do not interact. However, when the Hamiltonian contains an interaction between these two angular momenta (such as J ˆ 1 J ˆ 2 ), the eigenstates will be mixtures of the uncoupled basis functions and this basis becomes somewhat awkward. In these cases, it is easiest to work in the coupled basis, which we now develop. First, note that the total angular momentum is given by:
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J ˆ = J ˆ 1 + J ˆ 2 It is easy to show that this is, in fact, an angular momentum (i.e. ˆ ˆ ˆ [ J , J y ] = J i ). We can therefore associate two quantum numbers, j x z and m , with the eigenstates of total angular momentum indicating its magnitude and projection onto the z axis. The coupled basis states are eigenfunctions of the total angular momentum operator. This specifies two quantum numbers for our basis states ( j and m ). However, as we saw above, the uncoupled basis states were specified by four quantum numbers ( j , j 2 , m 1 and m 2 ) and we 1 therefore need to specify two more quantum numbers to fully specify the coupled states. To specify these last two quantum numbers, we note that 2 ˆ ˆ ˆ 2 ˆ ˆ 2 ˆ ˆ ˆ ˆ [ J , J 1 z ] = [ ( J + 2 J ˆ 1 J + J 2 29 , J ] = [ 2 J ˆ 1 J 2 , J ˆ ] = 2 [ J ˆ 1 , J 1 z ] J 0 1 2 1 z 1 z 2 and similarly for J ˆ 2 z . Thus J ˆ 1 z and J ˆ 2 z do not share common eigenfunctions with J ˆ 2 . To put it another way, to obtain a definite state of the total angular momentum, one must generally mix states with different m 1 and m 2 . All thus leads to the conclusion that neither m 1 nor m 2 can be one of the other quantum numbers that specify the coupled basis. What about
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sec8 - VIII. Addition of Angular Momenta a. Coupled and...

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