Spectral filters in quantum mechanics: A measurement theory perspective

We present the time-domain theory of spectral filters, starting with the ba sic propositions of the theory of measurement in quantum mechanics, and dev elop its parameter-free implementation in the traditional correlation funct ion as well as the filter diagonalization (FD) form; The present study unif ies all the time-domain spectral filter algorithms in the literature, under a single theme which is based on the notion of selective measurements; For specific numerical purposes, we have selected Chebyshev polynomials for de veloping the time propagator and this permits us to carry out the relevent time integrals fully analytically and obtain FD equations in a numerically convenient form. We also argue that the FD method is a particular realizati on of the general spectral filter goal and it is constrained, in general, b y the time-energy uncertainty regime at least as much as the correlation-fu nction-based method. To contrast the performance of the correlation functio n and the FD methods, we have carried out the detailed numerical experiment s on a model system, which suggest that the FD method needs almost as much time propagation as the correlation function method, in order to identify t he correct spectrum. The difference lies in the procedure for the exact loc ation of eigenvalue positions, for which the FD method employs a diagonaliz ation step while the correlation function method involves the location of z eros.