N-Mode Hamiltonians with Coordinate-Momentum Coupling Terms

The Virial theorem, which is related to the Feynman-Hellmann (FH) theorem, usually applies to Hamiltonians without coordinate-momentum coupling terms. In this paper we discuss when there are coordinate-momentum coupling terms in N-mode Hamiltonians, how the Virial theorem should be modified, and the energy contribution arising from the coordinate-momentum coupling is also be discussed. Work supported by the specialized research fund for the doctoral progress of higher education in China: 20070358009.     

A generalization of the virial theorem for strongly singular potentials

Using scale transformations we prove a generalization of the virial theorem for the eigenfunctions of Schrödinger Hamiltonians which are defined as the Friedrichs extension of strongly singular differential operators. Our theorem also applies to situations where the ground state has divergent kinetic and potential energy and thus the usual version of the virial theorem becomes meaningless.     

Deriving Internal Energy by Virtue of Generalized Feynman-Hellmann Theorem for Mixed States

We show how to directly use the generalized Feynman-Hellmann theorem, which is suitable for mixed state ensemble average, to derive the internal energy of Hamiltonian systems. A concrete example, which is a two coupled harminic oscillators, is used for elucidating our approach.     

Deriving Internal Energy by Virtue of Generalized Feynman-Hellmann Theorem for Mixed States

We show how to directly use the generalized Feynman-Hellmann theorem, which is suitable for mixed state ensemble average, to derive the internal energy of Hamiltonian systems. A concrete example, which is a two coupled harminic oscillators, is used for elucidating our approach.     

Virial Theorem for Angular Displacement and Torque

The usual Virial theorem is expressed through the coordinate and the force, 2áT? = áX\fracdVdX? = -áXF?2\langle T\rangle =\langle X\frac{dV}{dX}\rangle =-\langle XF\rangle , F=-\fracdVdXF=-\frac{dV}{dX}, XF is the work done by the force F, T is the kinetic energy. In this paper we extend the usual discussion on the Virial theorem about coordinate-force variables to the case of angular displacement-torque variables. By virtue of introducing the entangled state representation and the bosonic operator realization of the Hamiltonian of quantum pendulum system we derive the Virial theorem for angular variable and torque.     

Some Propagation Properties of the Iwatsuka Model

In this paper we study a two dimensional magnetic field Schr?dinger Hamiltonian introduced in [7]. This model has some interesting propagation properties, as conjectured in [2] and at the same time is a special case of the class of analytically decomposable Hamiltonians [5]. Our aim is to start from a conjugate operator, intimately related to the band structure of the Hamiltonian and to prove existence of an asymptotic velocity in one spatial direction and a theorem giving minimal and maximal velocity bounds for the propagation associated to the Hamiltonian. A simple example of this model, with a very simple conjugate operator, has been given in [9]. At the same time, by using the Virial Theorem, we obtain a generalisation of the hypothesis in [7]. Received: 12 February 1997 / Accepted: 26 February 1997     

Revised Virial Theorem for Hamiltonians with Coordinates-Momentum Coupling Terms

Usually the Virial theorem, which can be derived from the Feynman Hellmann theorem, applies to Hamiltonians without coordinates-momentum coupling. In this paper we discuss when there are such kind of couplings in Hamiltonians then how the Virial theorem should be modified. We also discuss the energy contribution arising from the coordinates-momentum coupling for a definite energy level.     

Energy convexity as a consequence of decoherence and pair-extensive interactions in many-electron systems

Using the concept of self-entanglement, through which a pure state constructed in an augmented Hilbert space can describe a mixed state and through which the effects of physical decoherence can be mapped onto systems separated by an infinite distance, with the role of environmental states assumed by system states in disjoint Hilbert spaces, we show that expectation values of Hamiltonians subscribing to decoherence and satisfying the condition of extensivity, defined in the text, obey the energy convexity relation. The analysis based on self-entanglement also leads to a surprising interpretation of the failure of the convexity relation for model Hamiltonians such as the Hubbard model: The failure is due to the existence of self-entangled states with lower energies than the ground state so that in such models decoherence, i.e., disentangling from the self-entangled states, would cost energy and disallow the observation of the state through measurement. The Hubbard model is discussed extensively in an appendix where we also discuss and resolve some of the counterarguments to the convexity relation that have been advanced in the literature.     

Exact wave functions for generalized harmonic oscillators

We transform the time-dependent Schrödinger equation for the most general variable quadratic Hamiltonians into a standard autonomous form. As a result, the time evolution of exact wave functions of generalized harmonic oscillators is determined in terms of the solutions of certain Ermakov and Riccatitype systems. In addition, we show that the classical Arnold transform is naturally connected with Ehrenfest’s theorem for generalized harmonic oscillators.     

The Quantum N-Body Problem and the Auxiliary Field Method: New Developments

Approximate analytical energy formulas for N-body semirelativistic Hamiltonians with one- and two-body interactions are obtained within the framework of the auxiliary field method. We first review the method in the case of nonrelativistic two-body problems. A general procedure is then given for N-body systems and a connection is presented between the method and the generalized virial theorem. The procedure is applied to the case of baryons in the large-N _c limit.     

Hellmann–Feynman connection for the relative Fisher information

The (i) reciprocity relations for the relative Fisher information (RFI, hereafter) and (ii) a generalized RFI–Euler theorem are self-consistently derived from the Hellmann–Feynman theorem. These new reciprocity relations generalize the RFI–Euler theorem and constitute the basis for building up a mathematical Legendre transform structure (LTS, hereafter), akin to that of thermodynamics, that underlies the RFI scenario. This demonstrates the possibility of translating the entire mathematical structure of thermodynamics into a RFI-based theoretical framework. Virial theorems play a prominent role in this endeavor, as a Schrödinger-like equation can be associated to the RFI. Lagrange multipliers are determined invoking the RFI–LTS link and the quantum mechanical virial theorem. An appropriate ansatz allows for the inference of probability density functions (pdf’s, hereafter) and energy-eigenvalues of the above mentioned Schrödinger-like equation. The energy-eigenvalues obtained here via inference are benchmarked against established theoretical and numerical results. A principled theoretical basis to reconstruct the RFI-framework from the FIM framework is established. Numerical examples for exemplary cases are provided.     

Efficient solvability of Hamiltonians and limits on the power of some quantum computational models

One way to specify a model of quantum computing is to give a set of control Hamiltonians acting on a quantum state space whose initial state and final measurement are specified in terms of the Hamiltonians. We formalize such models and show that they can be simulated classically in a time polynomial in the dimension of the Lie algebra generated by the Hamiltonians and logarithmic in the dimension of the state space. This leads to a definition of Lie-algebraic "generalized mean-field Hamiltonians." We show that they are efficiently (exactly) solvable. Our results generalize the known weakness of fermionic linear optics computation and give conditions on control needed to exploit the full power of quantum computing.     

A time dependent Ginzburg-Landau equation and its application to the problem of resistivity in the mixed state

A time dependent modification of the Ginzburg-Landau equation is given which is based on the assumption that the functional derivative of the Ginzburg-Landau free energy expression with respect to the wave function is a generalized force in the sense of irreversible thermodynamics acting on the wave function. This equation implies an energy theorem, according to which the energy can be dissipated by i) production of Joule heat; ii) irreversible variation of the wave function. The theory is a limiting case of the BCS theory, and hence, it contains no adjustable parameters. The application of the modified equation to the problem of resistivity in the mixed state reveals satisfactory agreement between experiment and theory for reduced temperatures higher than 0.6.     

Convergence of Hamiltonian systems to billiards

We examine in detail a physically natural and general scheme for gradually deforming a Hamiltonian to its corresponding billiard, as a certain parameter k varies from one to infinity. We apply this limiting process to a class of Hamiltonians with homogeneous potential-energy functions and further investigate the extent to which the limiting billiards inherit properties from the corresponding sequences of Hamiltonians. The results are mixed. Using theorems of Yoshida for the case of two degrees of freedom, we prove a general theorem establishing the "inheritability" of stability properties of certain orbits. This result follows naturally from the convergence of the traces of appropriate monodromy matrices to the billiard analog. However, in spite of the close analogy between the concepts of integrability for Hamiltonian systems and billiards, integrability properties of Hamiltonians in a sequence are not necessarily inherited by the limiting billiard, as we show by example. In addition to rigorous results, we include numerical examples of certain interesting cases, along with computer simulations. (c) 1998 American Institute of Physics.     

Formal aspects of supersymmetric embedding of Hamiltonians in quantum scalar field theories

It is formally shown that Hamiltonians in a quantum multicomponent scalar field theory are embedded into supersymmetric Hamiltonians if they have a strictly positive zero energy state.Supported by the Grant-in-Aid, No. 62740072 and No. 62460001 for science research from the Ministry of Education, Japan.     

Thermodynamics under Dynamic Forcing

Beginning with a system that is governed by an arbitrary time-dependent Hamiltonian, we exhibit an existence proof for a unitary generator that has an arbitrary initial value and yet contact transforms the representation to one governed by any given kinematically equivalent Hamiltonian. By choosing the initial value of the unitary operator to be unity, we are able to compare the behaviour of the same system under two different Hamiltonians and the same initial state vector. We thus are able to establish that the eventual physical states evolving from two distinct initial quantum state vectors will become practically indistinguishable under one of the two Hamiltonians if and only if they do so under the other. For the restricted class of systems for which one of the two Hamiltonians is a time-independent energy operator, and also generates equilibrium thermodynamics, then the condition for merging under the time-dependent Hamiltonian is the same as under the time-independent one. The two states must have the same initial energy. As a special case of the above, we choose the time-independent Hamiltonian to be the relativistic energy measuring operator for the time-dependent Hamiltonian, as associated with the chosen initial time. If the system under the time-dependent Hamiltonian is such that its relativistic energy measuring operator for any fixed time generates equilibrium thermodynamics, then we are led rigorously to the conclusion that the instantaneous relativistic energy for the system under the time-dependent Hamiltonian is simply a well-defined function of time and depends only on the initial energy and not on any other initial conditions. For a composite system that is of the above type, and in addition consists of one very small system in contact with a very large one, which is called a generalized reservoir, we consider a specific initial physical state for the large system, and various states for the small one. The eventual dynamic state of the composite system is essentially independent of the initial state of the small system which has almost no influence on the total composite energy. Hence the eventual dynamic state of the small system is shown rigorously to be independent of its initial state. For a forced system with a time-dependent Hamiltonian, we discuss the assignment of equilibrium thermodynamic potentials to a representation with a time-independent Hamiltonian. We discuss the concept of a process under a time-dependent Hamiltonian. Such a process is a natural generalization of the static and quasi-static processes. Also, we verify all of the theory with both general and specific examples of electromagnetic interactions.     

Generalized Hellmann-Feynman Theorem and Its Applications

The Hellmann-Feynman(H-F) theorem is generalized from stationary state to dynamical state.The generalized H-F theorem promotes molecular dynamics to go beyond adiabatic approximation and clears confusion in the Ehrenfest dynamics.     

Energy landscape of 3D spin Hamiltonians with topological order

We explore the feasibility of a quantum self-correcting memory based on 3D spin Hamiltonians with topological quantum order in which thermal diffusion of topological defects is suppressed by macroscopic energy barriers. To this end we characterize the energy landscape of stabilizer code Hamiltonians with local bounded-strength interactions which have a topologically ordered ground state but do not have stringlike logical operators. We prove that any sequence of local errors mapping a ground state of such a Hamiltonian to an orthogonal ground state must cross an energy barrier growing at least as a logarithm of the lattice size. Our bound on the energy barrier is tight up to a constant factor for one particular 3D spin Hamiltonian.     

On bound states in the continuum of N-body systems and the virial theorem

We prove generalized versions of the quantum mechanical virial theorem and apply them to the investigation of the spectrum of N body Hamiltonians. We show, in particular, that for N particles interacting through 2-body potentials which may have singularities but “don't wiggle too much,” no positive energy bound state can exist. We also prove results on the absence of bound states with energy bigger than some value E⁰ ? − ∞ and extend them to the case of N particles interacting through ν-body forces (ν = 1, 2,…, N) and with an external electromagnetic field. Also some remarks for the case of a Dirac electron in an external potential are given as well as for some problems with boundary conditions. A by-product of this investigation is the unitarity of the S matrix and the strong asymptotic completeness for systems of N particles interacting by 2-body forces which are not restricted to be purely repulsive.     

Maximum Speed of Quantum Evolution

In this paper, we discuss the question of the minimum time needed for any state of a given quantum system to evolve into a distinct (orthogonal) state. This problem is relevant to deriving physical limits in quantum computation and quantum information processing. Here, we consider both cases of nonadiabatic and adiabatic evolution and we derive the Hamiltonians corresponding to the minimum time evolution predicted by the Margolus–Levitin theorem.