Energy-shell bases

Wilkinson, Michael (2001). Energy-shell bases. Journal of Physics A: Mathematical and General, 34(49) pp. 11149–11156.

DOI: https://doi.org/10.1088/0305-4470/34/49/328

Abstract

A new type of basis set for quantum mechanical problems is introduced. These basis states are adapted to describing the dynamics of a Hamiltonian Ĥ which is dependent upon a parameter X. A function f(E) is defined which is an analytic function of E, and which is negligibly small when |E|>>δE, where δE is large compared to the typical level separation. The energy-shell basis set consists of states |ξn(X)} which are derived by applying the operator f(Ĥ(X)-Ēn(X)) to elements of a fixed basis set, where Ēn(X) is an analytic approximation to an eigenvalue En(X). The energy-shell basis states are combinations of states close to energy En, but vary more slowly as a function of X than the eigenfunctions | øn(X) of Ĥ (X). This feature gives the energy-shell basis states some advantages in analysing solutions of the time-dependent Schrödinger equation.

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