Next: 3. Fuzzy projections Up: 2. The algorithm Previous: 1. The Hilbert space

2. Spin directions

We choose the Pauli matrices to represent spin directions along axes respectively.
 (3)

 (4)

 (5)

Together with the identity matrix
 (6)

they span the whole complex matrix algebra. In computations it is important to make use of the fact that Pauli matrices (after multiplication by "-i") represent the quaternion algebra, that is:
 (7)

and
 (8)

To each direction in space there is associated spin matrix
 (9)

satisfying automatically and with eigenvalues . Vectors and are eigenvectors of to eigenvalues and respectively and thus correspond to "North" and "South" spin orientations respectively. Let denote the projection operator that projects onto eigenstate of to the eigenvalue Then is given by the formula:
 (10)

Indeed, is Hermitian and has eigenvalues or - thus it is the orthogonal projection, and it projects onto the eigenstate of with spin direction

Next: 3. Fuzzy projections Up: 2. The algorithm Previous: 1. The Hilbert space

2002-04-11