New Proof Of HPT And Analytical Solutions Of Two-electron Problem By Path Integral Approach

By using the path integral approach, we investigate the problem of Hooke's atom (two electrons interacting with Coulomb potential in an external harmonic-oscillator potential) in an arbitrary time-dependent electric field. For a certain infinite set of discrete oscillator frequencies, we obtain the analytical solutions. The ground state polarization of the atom is then calculated. The same result is also obtained through linear response theory. The Harmonic-Potential-Theorem(HPT) provides the solution of the time-dependent Schr?dinger equation for a system of interacting electrons confined in an external harmonic potential and subject to a spatially homogenous time-dependent force field.We provide a proof of the theorem via the Feynman path integral method.What distinguishes this proof from prior proofs is that no apriori ansatz as to the structure of the wave function need be made.The solution and thereby the proof reveal themselves as a consequence of the derivation.This paper contains four parts:In the first part:we give particular review of Hooke's atom about its discovery history,its Hamiltonian format ,its analytical solutions by path integral approach and its important application in density function theory(DFT),time-dependent density function theory(TDDFT),current density function theory(CDFT),quantal density function theory(QDFT) and quantum entanglement(QE). We also give a detailed recommend of HPT about its foundation background,its concrete content,its proof in Schr?dinger representation and its widely application in density function theory(DFT),time-dependent density function theory(TDDFT),current density function theory(CDFT),quantal density function theory(QDFT).In the second part: by using the path integral approach, we investigate the problem of Hooke's atom (two electrons interacting with Coulomb potential in an external harmonic-oscillator potential) in an arbitrary time-dependent electric field. In the third part:we provide a proof of the theorem via the Feynman path integral method.In the last part: the ground state polarization of the atom is then calculated with the above exact ground wavefunction. The same result is also obtained through linear response theory.