Schrodinger equation eigenstate solutions

The idea of the LvN method for quantum systems first introduced by Lewis and Riesenfeld (H.R. Lewis et.al., 1969) is to solve Eq. (17) and then find the solution to the Schrodinger equation as an eigenstate of the operator in Eq. (17). In quantum field theory the wave functional to the Schrodinger equation is directly given by the wave functional of the operator... 

One simple form of the Schrodinger equation—more precisely, the time-independent, nonrelativistic Schrodinger equation—you may be familiar with is Hx i = ty. This equation is in a nice form for putting on a T-shirt or a coffee mug, but to understand it better we need to define the quantities that appear in it. In this equation, H is the Hamiltonian operator and v i is a set of solutions, or eigenstates, of the Hamiltonian. Each of these solutions,... 

To proceed, we use the relation between the Green function and the eigenfunctions Z a of a system, which are solutions of the Schrodinger equation (6). Let us define Exio) = c in the eigenstate A) in the sense of definition (5), then... 

As a result of its dependence on the density (pa), the one-electron operator H is a pseudo-Hamiltonian, and the corresponding Schrodinger equation is nonlinear, so that its solution (for a fixed pin) must be self consistently adjusted to (e.g., by iteration) [3,53], In the case of full equilibrium, when pm = pa, both optical and inertial potentials (4>RF) depend on pa. As discussed below, the eigenstates of H (i.e., the electronically adiabatic states) are distinct from the diabatic states used to characterize the ET process (see footnote 5). 

The exact Schrodinger equation of motion, equation (1), may be equivalently Stated in a manner which shows the neglected terms arising from the assumption of the product form for the wavefunction, equation (4). The exact eigenstate Yj(x, R) is expanded in terms of the complete orthonormal set of functions y>i(x R) obtained from the solutions of the electronic equation, equation (5), in which case the nuclear wavefunctions (R) appear as the coefficients in the expansion. This procedure yields the following infinite set of coupled equations for the x (R)6... 

Thus Uk is the average energy of the Hamiltonian in the state 4>k ). and if this state is the solution of the equations described in Sect. 4, ak = e on the other hand the coefficient hi is simply a normalization constant for the state i ). We can look at (5-17) firom the point of view of the time-dependent Schrodinger equation evidently if k ) were an eigenstate of H, h = 0. Thus if l " ) is some approximation to the true eigenstate, the magnitude of hi provides an estimate of how good the approximation is because it measures the rate at which l " ) decays. Finally, note that (5-18 c) can be multiplied out and simplified to yield. 

An isolated molecule can be prepared in any stationary or nonstationary pure state. There is thus no reason to restrict oneself to eigenstates of the Hamiltonian, i.e., solutions of the time-independent Schrodinger equation. Hence isolated molecules do not show chemical structure. A derivation of chemical structure from the Schrodinger equation for an isolated molecule can work only by tricks or approximations hidden in the complicated mathematics. 

Here /Tel is the Hamiltonian of the electronic subsystem, —that of the nuclear subsystem (each a sum of kinetic energy and potential energy operators) and kei-N(r, R) is the electrons-nuclei (electrostatic) interaction that depends on the electronic coordinates r and the nuclear coordinates R. The BO approximation relies on the large mass difference between electron and nuclei that in turn implies that electrons move on a much faster timescale than nuclei. Exploring this viewpoint leads one to look for solutions for eigenstates of H of the form iA ,v(r,R) = < (r,R)/ R), or a linear combination of such products. Here (r, R) are solutions of the electronic Schrodinger equation in which the nuclear configuration Ris taken constant... 

The infinite number of Schrodinger s equations (6) in the BO scheme is replaced by a countable infinite set of Schrodinger equations (8) in the R-BO scheme. Whether or not the eigenstates of this latter set provide a complete basis to represent any quantum mechanical state solution to eq.(l) cannot be mathematically proven. However, we assume that subspaces of physically relevant states can be approximated in this manner. 

In the absence of accidental degeneracies among the molecular eigenstates, // , (2.5) or (2.6) are the appropriate solutions of the Schrodinger equation for the isolated molecular system in a real world where spontaneous emission processes are admissible. Assuming that the and < >, do not radiate to any set of common levels, it is possible to evaluate the radiative decay rates of the molecular eigenstates, as... 

The solution of the time-dependent Schrodinger equation leads to the TO states which are eigenstates of the translation by a unit cell length, T ... 

Hence, the states labeled by a are in our case the actual eigenstates of the imaginary time Schrodinger equation. These u e localized tad states centered at position Ra with an associated weight Wa- Thus the 1-step RSB solution approximates the t ul states by a fixed Gaussian form. 

The above results show that the 1-step RSB solution correctly predicts some important features of the eigenvalue distribution. More importantly, we have shown that the 1-step RSB solution can be interpreted in terms of the eigenstates of the Schrodinger equation with a random potential. However, there are differences and these reveal the limitations of the 1-step RSB solution. For example, all the localized states are approximated by the same Gaussian profile when in fact the localization lengths should increase with energy. 

One solution to this problem is that implemented in the dynamical equations (6) and (7), which couple the classical Langevin equation for the atoms in the protein-water system to the stationary Schrodinger equation for the quantum excitations (thus ensuring that only quantum eigenstates are considered). One limitation of this solution is that it is only valid when the quantum excitation responds very fast to any changes in the classical conformation, something which is assumed to be true here. 

The Floquet theorem, when apphed to the quantum mechanics [370], implies the stationarity of Floquet states imder a perfectly periodic Hamiltonian. We define the electronic Floquet operator as 7ft = Hf — ihdt and the Floquet states as its periodic eigenstates which satisfy Tlt x t)) = A] A(f))- The above mentioned stationarity states that the solution of time-dependent Schrodinger equation ] t) can be expanded as... 

In a number of cases, it can be advantageous to compute the absorption spectrum without any reference to the eigenstates of the system but rather using the time-dependent wavepacket computed through the solution of the time-dependent Schrodinger equation. Using the integral form of the Dirac delta function, Eq. (4.55) can be recast as... 

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