We present an investigation of Hermite polynomials as a basic paradigm for quantum dynamics, and make a thorough comparison with the well-known Chebyshev method. The motivation of the present study is to develop a compact and numerically efficient formulation of the spectral filter problem. In particular, we expand the time evolution operator in a Hermite series and obtain thereby an exponentially convergent propagation scheme. The basic features of the present formulation vı̀s a vı̀s Chebyshev scheme are as follows: (i) Contrary to the Chebyshev scheme Hamiltonian renormalization is not needed. However, an arbitrary time scaling may be necessary in order to avoid numerical hazards, and this time scaling also provides a leverage to accelerate the convergence of the Hermite series. We emphasize the final result is independent of the arbitrary scaling. (ii) As with the Chebyshev scheme the method is of high accuracy but not unitary by definition, and thus any deviation from unitarity may be used as a guideline for accuracy. The calculation of expansion coefficients in the present scheme is extremely simple. To contrast the convergence property of present method with that of the Chebyshev one for finite time propagation, we have introduced a time–energy scaling concept, and this has given rise to a unified picture of the overall convergence behavior. To test the efficacy of the present method, we have computed the transmission probability for a one-dimensional symmetric Eckart barrier, as a function of energy, and shown that the present method, by suitable time–energy scaling, can be very efficient for numerical simulation. Time–energy scaling analysis also suggests that it may be possible to achieve a faster convergence with the Hermite based method for finite time propagation, by a proper choice of scaling parameter. We have further extended the present formulation directed toward the spectral filter problem. In particular, we have utilized the Gaussian damping function for the purpose. The Hermite propagation scheme has allowed all the time integrals to be done fully analytically, a feature not completely shared by the Chebyshev based scheme. As a result, we have obtained a very compact and numerically efficient scheme for the spectral filters to compute the interior eigenspectra of a large rank eigensystem. The present formulation also allows us to obtain a closed form expression to estimate the error of the energies and spectral intensities. As a test, we have utilized the present spectral filter method to compute the highly excited vibrational states for the two-dimensional LiCN (J=0)$\text{(J=0)}$ system and compared with the exact diagonalization result.