Hamiltonian coordinate representation

A time-varying wave function is also obtained with a time-independent Hamiltonian by placing the system initially into a superposition of energy eigenstates ( n)), or forming a wavepacket. Frequently, a coordinate representation is used for the wave function, which then may be written as... 

Separating the Hamiltonian into parts describing a free rotator and the orienting effect of a liquid cage, we have in a coordinate representation... 

The density operator in the coordinate representation is given by the functions T q, Q, q Q, <), and can be expressed for Hamiltonian systems in terms of its amplitudes, which become the wavefunctions Q, <) I convenient to introduce the formally exact eikonal representation. 

The solute-solvent system, from the physical point of view, is nothing but a system that can be decomposed in a determined collection of electrons and nuclei. In the many-body representation, in principle, solving the global time-dependent Schrodinger equation with appropriate boundary conditions would yield a complete description for all measurable properties [47], This equation requires a definition of the total Hamiltonian in coordinate representation H(r,X), where r is the position vector operator for all electrons in the sample, and X is the position vector operator of the nuclei. In molecular quantum mechanics, as it is used in this section, H(r,X) is the Coulomb Hamiltonian[46]. The global wave function A(r,X,t) is obtained as a solution of the equation ... 

The theoretical view advocated here focus attention on the quantum states relevant to the description of particular phenomena. The concept of stationary Hamiltonians follows from the coordinate representation and leads to a numerical determination of relevant quantum states. He will denote this class of Hamiltonians. Thus, given a system that can be decomposed into n-electrons and m-nuclei in different dispositions, the set of He =Ea(l/tna)[pa]2+Vcoui are written in terms of fluctuation coordinates around particular... 

A complementary approach to the parabolic barrier problem is obtained by considering the Hamiltonian equivalent representation of the GLE. If the potential is parabolic, then the Hamiltonian may be diagonalized" using a normal mode transformation. One rewrites the Hamiltonian using mass weighted coordinates q Vmd. An orthogonal transformation matrix... 

The definition of a reduced dimensionality reaction path starts with the full Cartesian coordinate representation of the classical A-particle molecular Hamiltonian,... 

In the following we present the axioms or basic postulates of quantum mechanics and accompany them by their classical counterparts in the Hamiltonian formalism. We begin the presentation with a brief summary of some of the mathematical background essential for the developments in the following. It is by no means a comprehensive presentation, and the reader is supposed to have some basic knowledge about quantum mechanics that may be obtained from any of the many introductory textbooks in quantum mechanics. The focus here is on results of particular relevance to the subjects of this book. We consider, for example, a derivation of a formal expression for the flux density operator in quantum mechanics and its coordinate representation. A systematic way of generating any representation of any combination of operators is set up, and is of immediate usage for the time autocorrelation function of the flux operator used to determine the rate constants of a chemical process. 

It is convenient to convert to the coordinate representation of the density matrix of the vibration system to take into account the possible change of the set of normal coordinates. The Hamiltonian of the harmonic vibration system in the initial and final states may be expressed in the form ... 

Having expressed the Hamiltonian in terms of the canonical momenta, we can readily quantize the particles dynamics. To do so we replace each particle s canonical momentum by the momentum operator in the coordinate representation,... 

For a non-linear triatomic molecule (ABC) there are three degrees of freedom. The representation of the three degrees of freedom depends on how many reaction channels are open/relevant. A review of the choice of coordinates and the corresponding Hamiltonian is given in Ref. [75]. It is common to use either the Jacobi or the hyper-spherical coordinate representation. 

The elements of the Hamiltonian matrix are written as integrals by returning to the coordinate representation. 

It is convenient to eliminate the factor Tq from the coordinate representation of the projectile state and use the radial Hamiltonian (4.10). 

The exciton-photon interaction is written in such a form that 2y yields the Rabi splitting energy in the perfect system. We chose to use the same number N of photon modes, and the wavevectors k are discrete with 2ir/Na increments. Our approach is to straightforwardly find the normalized polariton eigenstates Tj) (i is the state index) of the Hamiltonian (10.50) and then use them in the site-coordinate representation ... 

To estimate the value of <[>, using the results of the previous section, it is necessary to compare the form of the present Hamiltonian, written in a coordinate representation, to that in the previous section, written in an algebraic representation. Facilitating the comparison, the creation and annihilations operators for the Morse oscillator are known to satisfy (34)... 

To calculate the necessary DCFs one must first choose a coordinate representation for the anharmonic force field. In order to simplify the ultimate numerical work, it is often convenient to employ the set of coordinates that diagonalizes the harmonic component of the vibrational Hamiltonian. For this purpose, the intramolecular potential is written as the sum of purely harmonic (V, ) and anharmonic (Knh) components. 

Conversely, one can derive the path integral from the Schrddinger formulation of quantum mechanics. The amplitude given in equation (I) is the quantum mechanical propagator, i.e., the coordinate representation of the Green function for the time-dependent Schrddinger equation. For a time-independent Hamiltonian H = T -irV, where T and V are the kinetic and potential energy operators, respectively. 

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