Expectation Values, Mean Values, and Measured Values

Jochen Pade

Peter Joshua Atkins, Julio de Paula, Ronald Friedman

Contents Wavefunctions as linear combinations 53 Brief illustration 7.1: The measurement of an observable 54 Mean values as expectation values 54 Example 7.1: Calculating an expectation value 55 The orthogonality of eigenfunctions 55 The expectation value of a linear combination of...

F. Robicheaux, Deepak A. Suresh

Results from higher order mean-field calculations of light interacting with atom arrays are presented for calculations of one- and two-time expectation values. The atoms are approximated as two levels and are fixed in space. Calculations were performed for mean-field approximations that include the expectation value of one operator (mean field), two operators (mean field 2), and three operators (mean field 3). For the one-time expectation values, we examined three different situations to understand the convergence with increasing order of mean field and some limitations of higher order mean-field approximations. As a representation of a two-time expectation value, we calculated the ${g}^{(2)}(\ensuremath{\tau})$ for a line of atoms illuminated by a perpendicular plane wave at several emission angles and two different intensities. For many cases, the mean field 2 will be sufficiently accurate to quantitatively predict the response of the atoms as measured by one-time expectation values. However, the mean-field-3 approximation will often be needed for two-time expectation values.

M. Horibe, A. Hayashi, T. Hashimoto

We investigate the optimal estimation of quantum expectation value of a physical observable, which minimizes a mean error with respect to general measure of deviation, when a finite number of copies of a pure state are prepared. If pure sates are uniformly distributed, the minimum value of mean error for any measure of deviation is achieved by projective measurement on each copy.

Brian P. Anderson