Xiao-Yin Pan

Integral coalescence conditions in D≥2, dimension space

We have derived the integral form of the cusp and node coalescence conditions satisfied by the wave function at the coalescence of two charged particles in D⩾2 dimension space. From it we have obtained the differential form of the coalescence conditions. These expressions reduce to the well-known integral and differential coalescence conditions in D=3 space. It follows from the results derived that the approximate Laughlin wave function for the fractional quantum Hall effect satisfies the node coalescence condition. It is further noted that the integral form makes evident that unlike the electron–nucleus coalescence condition, the differential form of the electron–electron coalescence condition cannot be expressed in terms of the electron density at the point of coalescence. From the integral form, the integral and differential coalescence conditions for the pair-correlation function in D⩾2 dimension space are also derived. The known differential form of the pair function cusp condition for the uniform el...

Quantal density functional theory of degenerate states

The treatment of degenerate states within Kohn-Sham density functional theory is a problem of long-standing and current interest. We propose a solution to this mapping from the interacting degenerate system to that of the noninteracting fermion model whereby the equivalent density and energy are obtained via the unifying physical framework of quantal density functional theory. We describe the quantal theory of both ground and excited degenerate states, and for the cases of both pure state and ensemble v-representable densities. The quantal description further provides a rigorous physical interpretation of the corresponding Kohn-Sham energy functionals of the density, ensemble density, bidensity and ensemble bidensity, and of their respective functional derivatives. We conclude with examples of the mappings within the quantal theory.

Hohenberg-Kohn theorems in electrostatic and uniform magnetostatic fields

The Hohenberg-Kohn (HK) theorems of bijectivity between the external scalar potential and the gauge invariant nondegenerate ground state density, and the consequent Euler variational principle for the density, are proved for arbitrary electrostatic field and the constraint of fixed electron number. The HK theorems are generalized for spinless electrons to the added presence of an external uniform magnetostatic field by introducing the new constraint of fixed canonical orbital angular momentum. Thereby, a bijective relationship between the external scalar and vector potentials, and the gauge invariant nondegenerate ground state density and physical current density, is proved. A corresponding Euler variational principle in terms of these densities is also developed. These theorems are further generalized to electrons with spin by imposing the added constraint of fixed canonical orbital and spin angular momenta. The proofs differ from the original HK proof and explicitly account for the many-to-one relationship between the potentials and the nondegenerate ground state wave function. A Percus-Levy-Lieb constrained-search proof expanding the domain of validity to N-representable functions, and to degenerate states, again for fixed electron number and angular momentum, is also provided.

Determination of a wave function functional.

We propose expanding the space of variations in traditional variational calculations for the energy by considering the wave function psi to be a functional of a set of functions chi:psi=psi[chi], rather than a function. A constrained search in a subspace over all functions chi such that the functional psi[chi] satisfies a sum rule or leads to a physical observable is then performed. An upper bound to the energy is subsequently obtained by variational minimization. The rigorous construction of such a constrained-search-variational wave function functional is demonstrated.

Criticality of the electron-nucleus cusp condition to local effective potential-energy theories

Local(multiplicative) effective potential energy-theories of electronic structure comprise the transformation of the Schroedinger equation for interacting Fermi systems to model noninteracting Fermi or Bose systems whereby the equivalent density and energy are obtained. By employing the integrated form of the Kato electron-nucleus cusp condition, we prove that the effective electron-interaction potential energy of these model fermions or bosons is finite at a nucleus. The proof is general and valid for arbitrary system whether it be atomic, molecular, or solid state, and for arbitrary state and symmetry. This then provides justification for all prior work in the literature based on the assumption of finiteness of this potential energy at a nucleus. We further demonstrate the criticality of the electron-nucleus cusp condition to such theories by an example of the hydrogen molecule. We show thereby that both model system effective electron-interaction potential energies, as determined from densities derived from accurate wave functions, will be singular at the nucleus unless the wave function satisfies the electron-nucleus cusp condition.

Electron Correlations in Local Effective Potential Theory

Local effective potential theory, both stationary-state and time-dependent, constitutes the mapping from a system of electrons in an external field to one of the noninteracting fermions possessing the same basic variable such as the density, thereby enabling the determination of the energy and other properties of the electronic system. This paper is a description via Quantal Density Functional Theory (QDFT) of the electron correlations that must be accounted for in such a mapping. It is proved through QDFT that independent of the form of external field, (a) it is possible to map to a model system possessing all the basic variables; and that (b) with the requirement that the model fermions are subject to the same external fields, the only correlations that must be considered are those due to the Pauli exclusion principle, Coulomb repulsion, and Correlation–Kinetic effects. The cases of both a static and time-dependent electromagnetic field, for which the basic variables are the density and physical current density, are considered. The examples of solely an external electrostatic or time-dependent electric field constitute special cases. An efficacious unification in terms of electron correlations, independent of the type of external field, is thereby achieved. The mapping is explicated for the example of a quantum dot in a magnetostatic field, and for a quantum dot in a magnetostatic and time-dependent electric field.

Wave function for time-dependent harmonically confined electrons in a time-dependent electric field

The many-body wave function of a system of interacting particles confined by a time-dependent harmonic potential and perturbed by a time-dependent spatially homogeneous electric field is derived via the Feynman path-integral method. The wave function is comprised of a phase factor times the solution to the unperturbed time-dependent Schrödinger equation with the latter being translated by a time-dependent value that satisfies the classical driven equation of motion. The wave function reduces to that of the Harmonic Potential Theorem wave function for the case of the time-independent harmonic confining potential.

Particle number and probability density functional theory and A-representability

In Hohenberg-Kohn density functional theory, the energy E is expressed as a unique functional of the ground state density rho(r): E = E[rho] with the internal energy component F(HK)[rho] being universal. Knowledge of the functional F(HK)[rho] by itself, however, is insufficient to obtain the energy: the particle number N is primary. By emphasizing this primacy, the energy E is written as a nonuniversal functional of N and probability density p(r): E = E[N,p]. The set of functions p(r) satisfies the constraints of normalization to unity and non-negativity, exists for each N; N = 1, ..., infinity, and defines the probability density or p-space. A particle number N and probability density p(r) functional theory is constructed. Two examples for which the exact energy functionals E[N,p] are known are provided. The concept of A-representability is introduced, by which it is meant the set of functions Psi(p) that leads to probability densities p(r) obtained as the quantum-mechanical expectation of the probability density operator, and which satisfies the above constraints. We show that the set of functions p(r) of p-space is equivalent to the A-representable probability density set. We also show via the Harriman and Gilbert constructions that the A-representable and N-representable probability density p(r) sets are equivalent.

Schrödinger Theory of Electrons in Electromagnetic Fields: New Perspectives

The Schrodinger theory of electrons in an external electromagnetic field is described from the new perspective of the individual electron. The perspective is arrived at via the time-dependent “Quantal Newtonian” law (or differential virial theorem). (The time-independent law, a special case, provides a similar description of stationary-state theory). These laws are in terms of “classical” fields whose sources are quantal expectations of Hermitian operators taken with respect to the wave function. The laws reveal the following physics: (a) in addition to the external field, each electron experiences an internal field whose components are representative of a specific property of the system such as the correlations due to the Pauli exclusion principle and Coulomb repulsion, the electron density, kinetic effects, and an internal magnetic field component. The response of the electron is described by the current density field; (b) the scalar potential energy of an electron is the work done in a conservative field. It is thus path-independent. The conservative field is the sum of the internal and Lorentz fields. Hence, the potential is inherently related to the properties of the system, and its constituent property-related components known. As the sources of the fields are functionals of the wave function, so are the respective fields, and, therefore, the scalar potential is a known functional of the wave function; (c) as such, the system Hamiltonian is a known functional of the wave function. This reveals the intrinsic self-consistent nature of the Schrodinger equation, thereby providing a path for the determination of the exact wave functions and energies of the system; (d) with the Schrodinger equation written in self-consistent form, the Hamiltonian now admits via the Lorentz field a new term that explicitly involves the external magnetic field. The new understandings are explicated for the stationary state case by application to two quantum dots in a magnetostatic field, one in a ground state and the other in an excited state. For the time-dependent case, the evolution of the same states of the quantum dots in both a magnetostatic and a time-dependent electric field is described. In each case, the satisfaction of the corresponding “Quantal Newtonian” law is demonstrated.

Dissipation-induced transition of a simple harmonic oscillator

We investigate the dissipation-induced transition probabilities between any two eigenstates of a simple harmonic oscillator. Using the method developed by Yu and Sun [Phys. Rev. A 49, 592 (1994)], the general analytical expressions for the transition probabilities are obtained. The special cases: transition probabilities from the ground state to the first few excited states are then discussed in detail. Different from the previous studies in the literature where only the effect of damping was considered, it is found that the Brownian motion makes the transitions between states of different parity possible. The limitations of the applicability of our results are also discussed.