On the variational principle for the non-linear Schrödinger equation

While variation of the energy functional yields the Schrödinger equation in the usual, linear case, no such statement can be formulated in the general nonlinear situation when the Hamiltonian depends on its eigenvector. In this latter case, as we illustrate by sample numerical calculations, the points of the energy expectation value hypersurface where the eigenvalue equation is satisfied separate from those where the energy is stationary. We show that the variation of the energy at the eigensolution is determined by a generalized Hellmann–Feynman theorem. Functionals, other than the energy, can, however be constructed, that result the nonlinear Schrödinger equation upon setting their variation zero. The second centralized moment of the Hamiltonian is one example.

Potential Energy Expectation Value for Lithium Excited State (1s2s3s)

The purpose of the present work is to calculate the expectation value of potential energy for different spin states (??? ? ???,??? ? ???) and compared it with spin states (??? , ??? ) for lithium excited state (1s2s3s) and Li- like ions (Be+,B+2) using Hartree-Fock wave function by partitioning techanique .The result of inter particle expectation value shows linear behaviour with atomic number and for each atom and ion the shows the trend ??? < ??? < ??? < ???

AB INITIO CALCULATION OF $^5_\Lambda {\rm He}$ WITH EXPLICIT Σ ADMIXTURE

Variational calculations for s-shell hypernuclei are performed by explicitly including Σ degrees of freedom. Two sets of YN interactions (D2 and SC97e(S)) are used. The bound-state solution of [Formula: see text] is obtained by using each of YN potentials, and a large energy expectation value of the tensor ΛN - ΣN transition part is found by using the SC97e(S). The internal energy of 4 He subsystem changes a lot by the presence of a Λ particle with the strong tensor ΛN - ΣN transition potential.