Chapter 9 The Two-Body Problem

1.6. Rotations in Coordinate Space and in Spin Space
If the state of a particle in a given coordinate system is described by , and in a rotated system the state is , then these two are related to each other by a unitary transformation . Let us consider the simple case of a rotation about the axis by an angle , where the rotated coordinates are related to the original coordinates by a set of linear transformation;

From these relations we find the following derivatives

Now let us expand the transformed state as a power series in

The coefficient of in (9.151) can be calculated in terms of the derivatives;

where is the component of the angular momentum operator. In a similar way we can determine the coefficients etc. The resulting infinite series can be summed up as an exponential operator

The result for the rotation about the -axis can be generalized to a rotation about a given axis , and in this case the general form of the unitary operator for this rotation is

where is the angle of rotation.
Rotations and Spin Space - The average value of the spin must transform as a vector, therefore we expect that the averages of the components of spin transform according to

Let us assume that the average value of in a state with spin up amplitude and spin down amplitude , i.e.

is . Then

To this relation we add the normalization condition

By solving Eqs. (9.157) and (9.158) for and we find two solutions for which we denote by and ;

where and are real phases to be determined. Again for the state we find and and equate them with and , of Eq. (9.155).

and

By substituting for and from (9.159) in (9.160) and (9.161) we find that . Therefore and will be given in terms of and by the matrices

For becomes spin up state and becomes spin down state
Suppose that a spinor is in the state given by , Eq. (9.159), then a rotation by an angle about the axis changes to . In analogy with rotation in coordinate space, Eq. (9.152), we expect that the generator of such a transformation to be

i.e. the transformed state be given by

Noting that

we find the expansion of in powers of ;

Thus

which, apart from the physically unobservable phase factor, is the same as except for the change in azimuthal angle as is expected.