Brief History of a One-Dimensional Ground State Wave Function

Rhett J. Allain

Bill Sutherland

I. Brief History.When I first learned of this conference, and saw the wide range of interests represented by the participants, I was certain that finally a common definition of that most intriguing concept-the soliton-would emerge.And I was anxious that this definition be broad enough to encompass solitons in both their classical and quantum versions.For recent work had convinced me that the general technique known as Bethe's ansatz for solving diverse one-dimensional quantum problems in fact was nothing more nor less than the quantum soliton.However, from conversations with conference participants, I discovered that many were not aware of this accumulated work on exactly soluble quantum systems.I even gained the impression that some feel quantum mechanics to be much harder than classical mechanics-an unnecessary complication.My own feeling, on the other hand, is that in many cases quantum mechanics clarifies matters.It forces one to immediately face problems that would eventually have to be faced in the corresponding classical case, problems such as the counting of states, to determine how important solitons really are.Anyway, the history of the quantum soliton is all to the greater glory of the soliton concept.And quite a respectable history it is.I would date the beginning at 1931 with a paper of H. Bethe [1] on magnetism; that is a total span of 45 years.Quantum mechanics itself has not been around all that much longer!It now seems my hopes for a consensus on a definition of the soliton were premature, and things are still fermenting.Nevertheless, here is my contribution to the conference in the form of a brief history for general interest, and a more original work on the quantization of the Toda lattice.Perhaps next time there will be some consolidation, and we can then decide just how particular and how general a concept is the soliton.1.We begin this historical account with the classic paper of H. Bethe [1], which introduced first the many-body wave function crucial

Bo-Bo Wei

In this brief report, two one-dimensional interacting particles interacting with a delta interaction in a harmonic trap is discussed. For strong interactions, we derive the asymptotic expressions for all the energy levels. In addition, the variational method is used to obtain asymptotic ground state energy and wave function. The variational results are compared with the exact solution and they turn out to agree very well.

Rhett J. Allain

Hefeng Wang

This thesis contains four projects: entanglement and quantum phase transition in one-dimensional quantum dots system; entanglement and electron correlation in molecular systems; quantum teleportation in one-dimensional quantum dots system and simulation of molecular systems on a quantum computer. Chapter 1 presents a general introduction to quantum computing. A brief history and motivation for quantum computing is introduced. In Chapter 2, we studied the entanglement of an array of quantum dots system modeled by one-dimensional Hubbard Hamiltonian and the relation between quantum entanglement and quantum phase transition. In Chapter 3, entanglement in molecular systems is studied. It is proved that the full configuration interaction (FCI) wave function violates Bell’s inequality, thus it is entangled; while the Hartree-Fock (HF) wave function does not violate Bell’s inequality, so it is unentangled. Based on this, we studied the relation between quantum entanglement and electron correlation in molecular systems. We suggested that quantum entanglement might be used as an alternative way to measure electron correlation in molecular systems. The application of entanglement in an array of quantum dots system in quantum teleportation is studied in Chapter 4. We showed that the space-spin entanglement in the quantum dots system can be filtered to obtain pure spin (space) entanglement, and it can be used in quantum teleportation. We also showed that by selecting appropriate Hamiltonian for the C-NOT gate, the spin-based information can be transformed into charge-based information. The pulse sequence for realizing the C-NOT operation is given. In Chapter 5, we studied the simulation of molecular systems on a quantum computer. We proposed a quantum algorithm to solve the Schrödinger equation using the basis set method based on the multi-configuration self-consistent (MCSCF) wave function. By using this algorithm, the potential energy surfaces of molecules, even some complicated regions, can be studied more efficiently than if the simpler HF wave function was employed. This algorithm can give high success probability, almost deterministic results for ground state and excited states. A small increase in the number of configuration state functions in the MCSCF wave function can dramatically increase the success probability of the quantum algorithm. A multi-reference configuration interaction approach is suggested for the treatment of larger systems. We generalized this method to the finite element method and simulated a quantum system of one particle in Yukawa potential. We showed that numerical methods for solving differential equations can be implemented efficiently on a quantum computer.

Mikhail M. Mestechkin