Time-independent perturbation for bound states

Jan Rais, Hendrik van Hees, Carsten Greiner

We study the temporal formation of quantum-mechanical bound states within a one-dimensional attractive square-well potential by first solving the time-independent Schr\"odinger equation and then studying a time-dependent system with an external time-dependent potential. For this we introduce Gaussian potentials with different spatial and temporal extensions and generalize this description also for subsequent pulses and for random noisy potentials. Our main goal is to study the timescales in which the bound state is populated and depopulated. Particularly, we clarify a likely connection between the uncertainty relation for energy and time and the transition time between different energy eigenstates. We demonstrate that the formation of states is not delayed due to the uncertainty relation but follows the pulse shape of the perturbation. In addition we investigate the (non-)applicability of first-order perturbation theory on the considered quantum system.

Geoffrey C Milward, Colin Wilkin C

The time-independent bound-state perturbation theory derived for the quantum probability equation (Milward G C and Wilkin C 2000 J. Phys. A: Math. Gen. 33 7353 ) is extended to treat the scattering domain. Formulae for the changes in the partial wave phase shifts are derived to third order in the perturbing potential. The results are compared with those obtained for analytically soluble problems.

S. Rajasekar, R. Velusamy

In chapters 13 and 14 we focused on perturbation theories for time-independent and time-dependent perturbations. For time-independent perturbations we obtained first-order and second-order corrections to energy eigenvalues and eigenfunctions. For time-dependent perturbations we obtained the probability for a system to make a transition from an ith initial state to fth final state. We have also discussed how to study the effect of adiabatic and sudden perturbations. In the present chapter we consider the variational method. The variational method can be used to construct an approximate ground state eigenfunction. In some cases the method can be used for finding approximately the energy of first few discrete states of a bound system. The perturbation theory uses a set of unperturbed states and the perturbation Hamiltonian H(1) to construct the eigenfunctions and eigenvalues through highly formalized approximations. In contrast, the variational method is based on a guess for the state-a trial state which is used to determine the corresponding energy. Any well behaved trial state introduced will yield an appropriate energy, although some trial states will be better than other, but whatever the guess, we can always make systematic improvements. We can construct a reasonably accurate eigenfunction often by a physical insight and ingenuity.