The fraction of members of a set of numbers [itex]\{k_i\}_{i=1}^N[/itex] that are equal to a specific number k can be written as(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac 1 N\sum_{i=1}^N\delta_{kk_i}[/tex]

Now consider an ensemble of N identical systems. Because of the above, the operator [itex]f_k^{(N)}[/itex] defined by

[tex]f_k^{(N)}|a_{k_1}\rangle\cdots|a_{k_N}\rangle=\left(\frac 1 N\sum_{i=1}^N\delta_{kk_i}\right)|a_{k_1}\rangle\cdots|a_{k_N}\rangle[/tex]

where the [itex]|a_j\rangle[/itex] are eigenvectors of some observable A, is telling us what fraction of the systems in the ensemble are in state [itex]|a_k\rangle[/itex]. (The eigenvalue is equal to that fraction). It's also telling us what fraction of the results will be [itex]a_k[/itex] if we measure A once on each system in the ensemble (and it's prepared in this particular state). That fraction is equal to the eigenvalue, which is equal to the expectation value of [itex]f_k^{(N)}[/itex] in this state.

Now consider an ensemble such that all the systems are prepared in the state [itex]|s\rangle[/itex], which isn't an eigenvector of A. Does the last interpretation above still hold? I mean, is the expectation value of [itex]f_k^{(N)}[/itex] in an arbitrary state

[itex]|s^{(N)}\rangle=|s\rangle\cdots|s\rangle[/itex]

still equal to the fraction of measurements of A that will yield the result [itex]a_k[/itex]? Is it possible to justify this without using the axiom that the probability of measuring [itex]a_k[/itex] when the state is [itex]|s\rangle[/itex] is [itex]|\langle a_k|s\rangle|^2[/itex].

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Frequency operators

Loading...

Similar Threads for Frequency operators |
---|

I Photons and Frequency |

I Why don't we ever hear about low-frequency photons? |

I Use the Dirac Equation to calculate transition frequencies in Hydrogen |

I Are any electrons ejected below the threshold frequency? |

**Physics Forums | Science Articles, Homework Help, Discussion**