4. Jumps are implemented by fuzzy projections

Next: 5. Transition probabilities Up: 2. The algorithm Previous: 3. Fuzzy projections

4. Jumps are implemented by fuzzy projections

Let us now discuss the algebraic operation that is associated with each quantum jump. Suppose before the jump the state of the quantum system is described by a projection operator $P({\bf r})$ , ${\bf r}$ being a unit vector on the sphere. That is, suppose, before the detector flip, the spin "has" direction ${\bf r}$ . Now, suppose the detector $P({\bf n},\alpha )$ flips, and the spin right after the flip has some other direction, ${\bf r}^\prime$ . What is the relation between ${\bf r}$ and ${\bf r}^\prime$ ? It is easy to see that the action of the operator $P({\bf n},\alpha )$ on a quantum state vector is given, in terms of operators, by the formula:

$\begin{displaymath} \lambda(\alpha,{\bf n},{\bf r}) P({\bf r}^\prime) =P({\bf n},\alpha )P({\bf r})P({\bf n},\alpha ), \end{displaymath}$

(12)

where $\lambda(\alpha,{\bf n},{\bf r})$ is a positive number. It is a simple (though somewhat lengthy) matrix computation that leads to the following result:

$\begin{displaymath} \lambda(\alpha,{\bf n},{\bf r})=\frac{1+\alpha^2+2\alpha ({\bf n}\cdot{\bf r})}{4} \end{displaymath}$

(13)

$\begin{displaymath} {\bf r}^\prime=\frac{(1-\alpha^2){\bf r}+2\alpha(1+\alpha({\... ...ot{\bf r})){\bf n}}{1+\alpha^2+2\alpha ({\bf n}\cdot{\bf r} )} \end{displaymath}$

(14)

where $({\bf n}\cdot{\bf r})$ denotes the scalar product

$\begin{displaymath} {\bf n}\cdot{\bf r}=n_1r_1+n_2 r_2+n_3 r_3.\end{displaymath}$

(15)

Next: 5. Transition probabilities Up: 2. The algorithm Previous: 3. Fuzzy projections

2002-04-11