**5**

**THE DENSITY OPERATOR**

In many cases of practical interest, rather than considering a single quantum system, we need to study a large number or collection of systems called an *ensemble*. Furthermore, rather than being in a single state, members of the ensemble can be found in one of two or more different quantum states. There is a given probability that a member of the ensemble is found in each of these states. We make this more concrete with a simple example.

Consider a two-dimensional Hilbert space with basis vectors {*|x*〉, *|y*〉}. We prepare a large number *N* of systems, where each member of the system can be in one of two state vectors

These states are normalized, so |*α*|^{2} + |*β*|^{2} = |*γ*|^{2}+|*δ*|^{2} = 1, and the usual rules of quantum mechanics apply. For a system in state *|a*〉, if a measurement is made, then there is a probability |*α| ^{2}* of finding

*|x*〉 while there is a probability |

*β|*of finding

^{2}*|y*〉, and similarly for state

*|b*〉.

Now suppose that we prepare *n _{a}* of these systems in state

*|a*〉 and

*n*of the systems in state

_{b}*|b*〉. Since we have

*N*total systems, then

If we divide by *N* ,

This relation tells us that if we randomly select a member of the ensemble, the probability that it is found in state ...

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