: The general problem : lecture_8 : lecture_8

# A simple example of angular momentum addition

Given two spin-1/2 angular momenta, and , we define

The problem is to find the eigenstates of the total total spin operators and and identify the allowed total spin states.

In order to solve this problem, we recognize that the eigenvectors we seek will be expressible in terms of tensor products of the spin eigenstates of the individual spins. There will be four such vectors:

Although this is a valid set of basis vectors for the full spin Hilbert space, these are not eigenvectors of or . Thus, we seek a unitary transformation among these vectors that generates a new set of four vectors that are eigenvectors of and .

The four new vectors will correspond to adding the spins in either a parallel or anti-parallel fashion:

for a total of four states, as expected.

To find these states, begin by noting that the state

is, in fact, an eigenstate of and . To see this note that

Also, since

The action of on is

However, the second term on the right vanishes because each involves a raising operator acting on a state for either the first or second spin, which annihilates the state. Thus,

Thus, the eigenvalue of is , corresponding to a value of with a corresponding value of . These facts make it clear that is a total spin-1 state of the total spin:

In order to find the the states corresponding to and , we can use the total lowering operator:

Recall the general relation for the action of raising and lowering operators on general spin states:

The procedure is, then, to act on both sides of

with :

On the left, we obtain so that

Finally, the state with is obtained by acting again with the lowering operator:

The last state is and is obtained by recognizing that it must be composed of those states that have opposite values of and . So we include these states with arbitrary coefficients:

The coefficients and are then determined by the conditions that must be normalized and that it must be orthogonal to the other three states. The first condition yields:

In order to fulfill the second, we recognize that is manifestly orthogonal to and . However, orthogonality to needs to be enforced:

From the normalization condition, it is clear, then, that so that . The choice of sign is arbitrary, so we choose and so that

Thus, the four new basis vectors are:

We can write the transformation from the old basis to the new basis as a matrix equation:

The matrix

can easily be shown to be unitary. Note that the elements of the matrix can be computed from the following overlaps:

These are examples of what are known as Clebsch-Gordan coefficients. In the next section, we will study the general forms of these coefficients.

Finally, note that the new basis vectors are, in fact, eigenvectors of and . Thus, they satisfy

The states , , form what are called the triplet states, corresponding to . Note that these are symmetric with respect to exchange of the two spins. The state is known as the singlet state as it is antisymmetric (changes sign) upon exchange of the spins.

: The general problem : lecture_8 : lecture_8
Mark Tuckerman 平成17年2月18日