2. Spin directions

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

2. Spin directions

We choose the Pauli matrices $\sigma_x,\sigma_y,\sigma_z$ to represent spin directions along

axes respectively.

$\begin{displaymath} \sigma_1=\sigma_x=\pmatrix{0,&1\cr 1,&0}\end{displaymath}$

(3)

$\begin{displaymath} \sigma_2=\sigma_y=\pmatrix{0,&{\mathrm{i}}\cr -{\mathrm{i}},&0}\end{displaymath}$

(4)

$\begin{displaymath} \sigma_3=\sigma_z=\pmatrix{1,&0\cr 0,&-1} \end{displaymath}$

(5)

Together with the identity matrix

$\begin{displaymath} \sigma_0=I=\pmatrix{1,&0\cr 0,&1} \end{displaymath}$

(6)

they span the whole $2\times 2$ 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:

$\begin{displaymath} \sigma_1^2=\sigma_2^2=\sigma_3^2=I \end{displaymath}$

(7)

and

$\begin{displaymath} \sigma_1\sigma_2=-\sigma_2\sigma_1={\mathrm{i}}\sigma_3,\ \s... ...a_1,\ \sigma_3\sigma_1=-\sigma_1\sigma_3={\mathrm{i}}\sigma_2 \end{displaymath}$

(8)

To each direction ${\bf n}$ in space there is associated spin matrix

$\begin{displaymath} \sigma({\bf n})=n_1\sigma_1+n_2\sigma_2+n_3\sigma_3=\pmatrix{n_3,&n_1+{\mathrm{i}}n_2\cr n_1-{\mathrm{i}}n_2,&-n_3} \end{displaymath}$

(9)

satisfying automatically $\sigma({\bf n})^2=I,$ and with eigenvalues

. Vectors $\pmatrix{1\cr 0}$ and $\pmatrix{0\cr 1}$ are eigenvectors of $\sigma_3$ to eigenvalues

and

respectively and thus correspond to "North" and "South" spin orientations respectively. Let $P({\bf n})$ denote the projection operator that projects onto eigenstate of $\sigma({\bf n})$ to the eigenvalue

Then $P({\bf n})$ is given by the formula:

$\begin{displaymath} P({\bf n})=\frac{1}{2}(I+\sigma({\bf n})). \end{displaymath}$

(10)

Indeed, $P({\bf n})$ is Hermitian and has eigenvalues $\frac{1}{2}(1\pm 1)=$

- thus it is the orthogonal projection, and it projects onto the eigenstate of $\sigma({\bf n})$ with spin direction ${\bf n}.$

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

2002-04-11