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Non-equilibrium Thermodynamics
Published in Jeffrey Olafsen, Sturge’s Statistical and Thermal Physics, 2019
where the value Ei is the eigenvalue corresponding to the eigenstate Ψ of Schrödinger’s equation. In quantum mechanics, the time-dependent states are then found by the propagator acting on the time-independent states of the system. We wish to bring this to mind for two reasons. First, as we consider relaxing our equilibrium temperature condition, keep in mind that unless the system itself undergoes a physical change, the underlying equilibrium states of the system are not altered. Second, the concept of the total energy of the system, even out of equilibrium, is a valid quantity. Indeed, in the next chapter on biological systems, it is the energy of the system that is modified by the presence of catalysts and enzymes that allow certain changes to occur to take such systems from one state to another.
Concepts of Statistical Physics
Published in P. F. Bortignon, A. Bracco, R. A. Broglia, Giant Resonances, 2019
P. F. Bortignon, A. Bracco, R. A. Broglia
Furthermore, the operator U^=exp(−βH^) appearing in the expression for the partition function, Eq. (7.8), is similar in structure, to a time propagator in quantum mechanics, but for an imaginary time defined by β = it. It is convenient to write the operator U^, as the exponential of a one-body operator. In this case, the expectation value of U^ in a trial wave function of determinant type, gives rise to another determinant (Thouless (1962)). Trace of determinants are simple to calculate making use of the Hubbard-Stratonovich transformation based on the identity
Hidden symmetry reductions and the Ablowitz–Kaup–Newell– Segur hierarchies for nonautonomous solitons
Published in Kuppuswamy Porsezian, Ramanathan Ganapathy, Odyssey of Light in Nonlinear Optical Fibers, 2017
V.N. Serkin, A. Hasegawa, T.L. Belyaeva
Historically, the discovery of squeezed states of a harmonic oscillator dates back to 1927 by E. Kennard [65]. It was in this research that the Greens function or propagator of the Hamiltonian with quantum harmonic oscillator potential was discovered. And it was in this research that the more general class of oscillating wave packets was discovered for arbitrary initial widths and localizations of wave functions not restricted simply by the shifted ground state, but, for example, wider or sharper of it. The Kennard states (known today as squeezed states) are really remarkable for several reasons. The wave packet remains Gaussian at all times, while its width and amplitude oscillate and the wave function can be more highly localized in position space than the coherent state. This feature explains the name “squeezed state” given by Hollenhorst to underscore the increasing sensitivity of gravitational antenna in this state [69]. Over the years, there have been many significant contributions to the development of coherent and squeezed states theory and experiments (see, for example, [70–75] and references therein).
Ab initio study of nitrogen and boron doped dimers
Published in Molecular Physics, 2022
Sandeep Kaur, Hitesh Sharma, V. K. Jindal, Vladimir Bubanja, Isha Mudahar
The transport properties at the steady state are obtained from the propagator through the scattering region, given by, Here and are scattering region overlap and Hamiltonian matrices, respectively; are retarded/advanced self-energies of the leads describing the connection between the leads and the central region. When bias is applied, energy levels of the electrodes are shifted, that is the electrochemical potetials are defined as . The density matrix is expressed in terms of the Keldysh Green's function, where [57], with scattering matrices defined as . The charge distribution of the central region is determined self-consistently within the DFT-NEGF procedure. Once self-consistency is achieved, the current is obtained from where is energy- and voltage-dependent total transmission probability of the contact given by . The numerical implementation including the contours of integration in complex plane are described in [52].
Improved stochastic multireference perturbation theory for correlated systems with large active spaces
Published in Molecular Physics, 2020
James J. Halson, Robert J. Anderson, George H. Booth
The efficiency of FCIQMC is derived from a dual stochastisation of the problem – both the wave function amplitudes are stochastically sampled, as well as the Hamiltonian (or propagator) governing the dynamics. This is particularly efficient, as both quantities can be considered ‘sparse’, and can be especially powerful if this sparsity can be effectively predicted in advance. In the previous work on FCIQMC-NEVPT2, the sparsity in the sampling of higher-body active space quantities followed directly from the sparsity of the wave function amplitudes in terms of walkers. However, in this work, we show that the exact and full accumulation of the higher-body quantity for any pair of stochastically sampled configurations can still lead to a significant and rapidly dominating overhead in computational cost in the stochastic NEVPT2. We improve the FCIQMC-NEVPT2 algorithm to show that the introduction of an additional stochastic sampling in the accumulation of higher-body terms can exploit its sparsity and effectively reduce this cost, with minimal and systematically improvable loss of accuracy in the final results. Furthermore, this approach fits well within the framework and motivation of the FCIQMC dynamic in terms of stochastically sampling sparse, high-dimensional quantities, especially where prior information can be used to enable this to be effective.
Quantum computing methods for electronic states of the water molecule
Published in Molecular Physics, 2019
Teng Bian, Daniel Murphy, Rongxin Xia, Ammar Daskin, Sabre Kais
For each term in a Hamiltonian, H, the propagator, , can be easily constructed in a circuit. However, since most of the time the set of do not commute, the propagator cannot be implemented term by term: i.e. . The first-order Trotter–Suzuki decomposition [27–29] provides an easy way to decompose a propagator for the spin-type Hamiltonian given as a sum of non-commuting terms into a product of each non-commuting term exponentiated for a small time t: Here , and we have an error of order . Here we don't consider time slicing as the original Trotter–Suzuki decompositoin does, as t can be adjusted to be as small as necessary for error control. This method requires only multi-qubit rotations, and therefore U can be implemented easily on a state register.