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Chapter 3. Spin Systems
be dissected into a sequence of two-dimensional rotations, whereby in each rotation one axis remains
invariant. Here, all rotations are counter clockwise (right-handed, mathematically positive sense).
In order to simplify the problem, let us start with a two-dimensional rotation:
Suppose the coordinates, (x,y), of a point in the two-dimensional XY system are known, but we are
actually interested in knowing the coordinates of this point in another coordinate system, X’Y’, which
is related to the XY system by a counter-clockwise rotation by an angle ϕ.
As the figure indicates, the coordinates of the given point in the new coordinate system will be:
x0
y
0
= xcosϕ + ysinϕ
= − xsinϕ + ycosϕ
(3.1)
or, in matrix notation:
Now, transferred to a three-dimensional problem, the goal will be to describe the coordinates in a final
rotated system (x,y,z) which is related to some initial coordinate system (X,Y,Z) by the Euler angles.
The final system is developed in three steps, each step involving a rotation described by one Euler
angle. At the start, both coordinate systems, (X,Y,Z) and (x(1), y(1), z(1)), shall be coincident.
76 [January 6, 2009]