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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
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