Spin portrays the matrices of complex numbers that represent rotations in quantum mechanics.

**Click on the applet** to pause the animation, or redraw with two new random spins.

A quantum mechanical particle with a given amount of total spin, *j*, is described by
a state vector in a complex vector space of dimension 2*j*+1. The two matrices in the
**bottom row** of the diagram represent rotations for particles with two randomly
chosen total spins, *j*_{1} and *j*_{2}.

These matrices show the effect of the current rotation (portrayed in the inset) on
a basis of vectors {|*j**m*>}, *m*=–*j*, –*j*+1, ..., *j*–1, *j*, where *m* is
the component of the particle’s spin measured in the direction of the *z*-axis.
Successive columns of the matrix represent the rotated version of each of these vectors,
with the brightness of the squares proportional to the magnitude of the vector’s
complex coordinates, and the colour indicating their phase.

The **top left-hand** matrix shows the effect of the current rotation on the
combined, two-particle system, which is described by state vectors in a complex vector
space of dimension (2*j*_{1}+1)(2*j*_{2}+1). This matrix is constructed by taking each element
of the *j*_{1} matrix, multiplying the entire *j*_{2} matrix by that value, and substituting
the result in place of the original *j*_{1} element. The **top right-hand** matrix
portrays the same transformation in a different basis, which is chosen to show how
the two-particle state space can be split into a number of subspaces, each of which
contains vectors that transform like single particles of definite spins, ranging from
|*j*_{1}–*j*_{2}| to *j*_{1}+*j*_{2}.