Hamiltonian operators dynamical equation

The appearance of the Hamiltonian operator in equation (3.55) as stipulated by postulate 5 gives that operator a special status in quantum mechanics. Knowledge of the eigenfunctions and eigenvalues of the Hamiltonian operator for a given system is sufficient to determine the stationary states of the system and the expectation values of any other dynamical variables. 

The second method is based on a representation of the 4 x 4 Hamiltonian operator matrix, equation (11), which accomplishes by means of a unitary transformation (which, as it turns out, must itself depend necessarily on the dynamic degrees of freedom, i.e energy and momentum of the particles) to remove terms which couple upper and lower components. If this transformation is parametrized in the form... 

As shown above in Section UFA, the use of wavepacket dynamics to study non-adiabatic systems is a trivial extension of the methods described for adiabatic systems in Section H E. The equations of motion have the same form, but now there is a wavepacket for each electronic state. The motions of these packets are then coupled by the non-adiabatic terms in the Hamiltonian operator matrix elements. In contrast, the methods in Section II that use trajectories in phase space to represent the time evolution of the nuclear wave function cannot be... 

We can make further approximations to simplify the NRF of the Hamiltonian presented in equation (75) for non-dynamical properties. For such properties, we can freeze the nuclear movements and study only the electronic problem. This is commonly known as the clamped nuclei approximation, and it usually is quite good because of the fact that the nuclei of a molecule are about 1836 times more massive than the electrons, so we can usually think of the nuclei moving slowly in the average field of the electrons, which are able to adapt almost instantaneously to the nuclear motion. Invocation of the clamped nuclei approximation to equation (75) causes all the nuclear contributions which involve the nuclear momentum operator to vanish and the others to become constants (nuclear repulsion, etc.). These constant terms will only shift the total energy of the system. The remaining terms in the Hamiltonian are electronic terms and nuclear-electronic interaction contributions which do not involve the nuclear momentum operator. 

Once this discussion of the space-inversion operator in the context of optically active isomers is accepted, it follows that a molecular interpretation of the optical activity equation will not be a trivial matter. This is because a molecule is conventionally defined as a dynamical system composed of a particular, finite number of electrons and nuclei it can therefore be associated with a Hamiltonian operator containing a finite number (3 M) of degrees of freedom (variables) (Sect. 2), and for such operators one has a theorem that says the Hamiltonian acts on a single, coherent Hilbert space > = 3 (9t3X)51). In more physical terms this means that all the possible excitations of the molecule can be described in . In principle therefore any superposition of states in the molecular Hilbert space is physically realizable in particular it would be legitimate to write the eigenfunctions of the usual molecular Hamiltonian, Eq. (2.14)1 3 in the form of Eq. (4.14) with suitable coefficients (C , = 0. Moreover any unitary transformation of the eigen-... 

This chapter is intended to present an integrated description of this general approach to quantum dynamics. Applications of the equations and strategies both to scattering and bound state problems will be discussed. In the next section, we begin with a detailed summary of the salient features of the DAFs as they are used to represent the Hamiltonian operator. Then in Sec. Ill, we discuss the TIWSE and some of the choices that can be made in solving for bound states and scattering information. Included in this is a discussion of the polynomial representations of various operators involved in the TIW form of quantum mechanics. Finally, in Sec. IV we briefly summarize some of the applications made to date of this overall approach. 

This contribution deals with the description of molecular systems electronically excited by light or by collisions, in terms of the statistical density operator. The advantage of using the density operator instead of the more usual wavefunction is that with the former it is possible to develop a consistent treatment of a many-atom system in contact with a medium (or bath), and of its dissipative dynamics. A fully classical calculation is usually suitable for a many-atom system in its ground electronic state, but is not acceptable when the system gets electronically excited, so that a quantum treatment must then be introduced initially. The quantum mechanical density operator (DOp) satisfies the Liouville-von Neumann (L-vN) equation [1-3], which involves the Hamiltonian operator of the whole system. When the system of interest, or object, is only part of the whole, the treatment can be based on the reduced density operator (RDOp) of the object, which satisfies a modified L-vN equation including dissipative rates [4-7]. 

Here, it is usual to make the Bom-Oppenheimer approximation that allows a classical treatment of the nuclei to be separated from a quantum mechanical description of the electrons. In this case, the wave function becomes just that of the electrons, and the nuclear-nuclear interaction is added to the energy as a sum over point particles. Consequently, the Hamiltonian operator H includes the kinetic energy of the electrons, the electron-electron interactions, and the electron-nuclei interactions. The wave function determined by solving this eigenproblem consists of a Slater determinant of the molecular orbitals for a molecule or, alternatively, the band structure of a solid. Unfortunately, direct solution of this equation is complicated by the electron-electron interactions. Often, it is necessary to introduce a mean-field approximation that neglects the individual dynamical electron-electron correlations but instead treats the electrons as moving in the average field created by the other electrons. Various corrections have been developed to improve upon this approximation [160, 167, 168]. 

Thus, for a = —we obtain the system of equations of the dynamics of a rigid body with a fixed point. Moreover, among the operators pat, of normal series there exists the classical operator rpX = IX 4- XI, where / is a real diagonal matrix. Indeed, we set b = -ta, then PabEij = y = (a + ay)Eclassical Hamiltonian system of equations of motion of a rigid body with a fixed point (without potential) into the... 

Provided the Hamiltonian operator can be written in a suitable form and an efficient integration scheme is used to solve the equations of motion (see Sects. 4.2.3 and 4.2.4), the MCTDH method allows one to perform accurate quantum dynamics calculations for realistic molecular systems with more than twenty degrees of freedom [33-45], and on simple model systems with up to roughly eighty degrees of freedom [46-48]. 

To prepare quantum mechanical equations, dynamic variables must be associated with operators according to the rules given in Table 3.1. The classical hamiltonian, or the sum of kinetic and potential energies of a body, transforms into the hamiltonian operator. 

A simple, non-selective pulse starts the experiment. This rotates the equilibrium z magnetization onto the v axis. Note that neither the equilibrium state nor the effect of the pulse depend on the dynamics or the details of the spin Hamiltonian (chemical shifts and coupling constants). The equilibrium density matrix is proportional to F. After the pulse the density matrix is therefore given by and it will evolve as in equation (B2.4.27). If (B2.4.28) is substituted into (B2.4.30), the NMR signal as a fimction of time t, is given by (B2.4.32). In this equation there is a distinction between the sum of the operators weighted by the equilibrium populations, F, from the unweighted sum, 7. The detector sees each spin (but not each coherence ) equally well. 

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

Equation (31) is known as Heisenberg s equation of motion and is the quantum-mechanical analogue of the classical equation (17). The commutator of two quantum-mechanical operators multiplied by 2mfh) is the analogue of the classical Poisson bracket. In quantum mechanics a dynamical quantity whose operator commutes with the Hamiltonian, [A, H] = 0, is a constant of the motion. 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

Standard Green s function techniques are used in the following [46] to describe the dynamics of the protons and the ionic displacements. The equations of motions for the retarded Green s functions [[A (q) S (q))) are obtained from the Hamiltonian Eq. 1 where the operator A denotes 0p, 0k> u or... 

The general relations among the coefficients - and Dy are presented elsewhere [179]. The quantities yj and y2 are the damping constants for the fundamental and second- harmonic modes, respectively. In Eq.(82) we shall restrict ourselves to the case of zero-frequency mismatch between the cavity and the external forces (ff>i — ff> = 0). In this way we exclude the rapidly oscillating terms. Moreover, the time x and the external amplitude have been redefined as follows x = Kf and 8F =. The s ordering in Eq.(80) which is responsible for the operator structure of the Hamiltonian allows us to contrast the classical and quantum dynamics of our system. If the Hamiltonian (77)-(79) is classical (i.e., if it is a c number), then the equation for the probability density has the form of Eq.(80) without the s terms ... 

Here, Q is the projector on the bound subspace and P projects onto the open, continuum channels. The intramolecular coupling is written as V+ U so that, as before, U is any additional coupling brough about by external perturbations. The equation H = Hq + V+U, where Ho is the zero-order Hamiltonian of the Rydberg electron and so includes only the central part of the potential due to the core plus the motion (vibration, rotation) of the core, uncoupled to the electron. The perturbations V + U can act within the bound subspace, as the operator Q(V+l/)Q is not necessarily diagonal and is the cause of any intramolecular dynamics even in the absence of coupling to the continuum. The intramolecular terms can also couple the bound and dissociative states. 

---