Spectral filters in quantum mechanics: A measurement theory perspective

We present the time-domain theory of spectral filters, starting with the basic propositions of the theory of measurement in quantum mechanics, and develop its parameter-free implementation in the traditional correlation function as well as the filter diagonalization (FD) form. The present study unifies 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 developing 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 realization of the general spectral filter goal and it is constrained, in general, by the time-energy uncertainty regime at least as much as the correlation-function-based method. To contrast the performance of the correlation function and the FD methods, we have carried out the detailed numerical experiments 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 the correct spectrum. The difference lies in the procedure for the exact location of eigenvalue positions, for which the FD method employs a diagonalization step while the correlation function method involves the location of zeros.