Operator Methods in Classical Mechanics, II

Paul Richard Halmos, John von Neumann

Olaf Stenzel

Abdullah Verçin

Lloyd A. Young

It is shown that in the limit $h\ensuremath{\rightarrow}0$ the quantity local momentum becomes equal to the classical momentum and the equations involving local momentum become equations of classical mechanics. Operators are found for this quantity corresponding to the operator $(\frac{h}{2\ensuremath{\pi}i})(\frac{d}{\mathrm{dx}})$.

Sina Ataollah Khorasani

The success of quantum physics in description of various physical interaction phenomena relies primarily on the accuracy of analytical methods used. In quantum mechanics, many of such interactions such as those found in quantum optomechanics and quantum computing have a highly nonlinear nature, which makes their analysis extraordinarily difficult using classical schemes. Typically, modern quantum systems of interest nowadays come with four basic properties: (i) quantumness, (ii) openness, (iii) randomness, and (iv) nonlinearity. The newly introduced method of higher-order operators targets analytical solutions to such systems, and while providing at least mathematically approximate expressions with improved accuracy over the fully linearized schemes, some cases admit exact solutions. Many different applications of this method in quantum and classically nonlinear systems are demonstrated throughout. This review is purposed to provide the reader with ease of access to this recent and well-established operator algebra, while going over a moderate amount of literature review. The reader with basic knowledge of quantum mechanics and quantum noise theory should be able to start using this scheme to his or her own problem of interest.

Roberto Beneduci, James A. Brooke, Ray Curran +1 more