Schrodinger equation time-dependent form

The. itarting point for any discussion of quantum mechanics is, of course, the Schrodinger t-qualion. The full, time-dependent form of this equation is... 

In what I broadly regard as structure (essentially quantum theory), the equation that epitomizes the transition from classical mechanics to quantum mechanics, is the de Broglie relation, k = hip, for it summarizes the central concept of duality. Stemming from duality is the aspect of reality that distinguishes quantum mechanics from classical mechanics, namely superposition y = y/A + y/R with its implication of the roles of constructive and destructive interference. Then of course, there is the means of calculating wavefixnctions, the Schrodinger equation. For simplicity I will write down its time-independent form, Hip = Eip, but it is just as important for a physical chemist to be familiar with its time-dependent form and its ramifications for spectroscopy and reaction. 

The //-electron wave function T is an antisymmetric function of N sets of spatial and spin coordinates r,, v, for individual electrons, all evaluated at a common timer. In postulating a time-dependent Schrodinger equation of the form... 

Fortunately there is a simple mathematical formalism that gives us the best of the quantum and classical approaches. By recasting the time-dependent Schrodinger equation into a form using a so-called density operator, physicists have long been able to follow the development of a quantum system with time. This formalism... 

We start by writing the general solution of the time-dependent Schrodinger equation in the form... 

Here it is the level 0) that is taken as the driving state, and the flux is carried through another level jl) coupled to two continua, L = / and R = r. Looking for a solution to the time-dependent Schrodinger equation of the form... 

One approach to solving the Schrodinger equation in a strong field is to make a transformation to the so-called Kramers-Henneberger or wiggling frame. Starting from the time-dependent Schrodinger equation in the form ... 

We must, therefore, use the Schrodinger equation in its time-dependent form to describe the motion of the molecule, with the wave packet being initially localized on the PES, in space and time. If discrete travelling-wave solutions of the Schrodinger wave equation are combined, then they can be used to construct the required wave packet, which localizes it to a transient pulse. Assuming that a single-frequency wave solution of the time-dependent Schrodinger equation can be written as y(r, t) =A sin( r — rot), then the superposition wave-packet solution is Y (r, t) =... 

The earliest appearance of the nonrelativistic continuity equation is due to Schrodinger himself [2,319], obtained from his time-dependent wave equation. A relativistic continuity equation (appropriate to a scalar field and formulated in terms of the field amplitudes) was found by Gordon [320]. The continuity equation for an electron in the relativistic Dirac theory [134,321] has the well-known form [322] ... 

When the wave function is completely general and pennitted to vary in the entire Hilbert space the TDVP yields the time-dependent Schrodinger equation. However, when the possible wave function variations are in some way constrained, such as is the case for a wave function restricted to a particular functional form and represented in a finite basis, then the corresponding action generates a set of equations that approximate the time-dependent Schrodinger equation. 

It was stated above that the Schrodinger equation cannot be solved exactly for any molecular systems. However, it is possible to solve the equation exactly for the simplest molecular species, Hj (and isotopically equivalent species such as ITD" ), when the motion of the electrons is decoupled from the motion of the nuclei in accordance with the Bom-Oppenheimer approximation. The masses of the nuclei are much greater than the masses of the electrons (the resting mass of the lightest nucleus, the proton, is 1836 times heavier than the resting mass of the electron). This means that the electrons can adjust almost instantaneously to any changes in the positions of the nuclei. The electronic wavefunction thus depends only on the positions of the nuclei and not on their momenta. Under the Bom-Oppenheimer approximation the total wavefunction for the molecule can be written in the following form ... 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

We next address the question as to whether equation (3.59) is actually the most general solution of the time-dependent Schrodinger equation. Are there other solutions that are not expressible in the form of equation (3.59) To... 

From the time-dependent Schrodinger equation in the matrix form of Eq. (58), it can be shown [54] that if S(R) is chosen so that... 

This mixed classical/quantum expression is valid for classical nuclear behavior and, strictly speaking, for the case of direct two-site interaction rather than superexchange, as the Landau-Zener expression was derived from the time-dependent Schrodinger equation assuming a two-state (reactant/product) electronic system with direct coupling. Nevertheless, it becomes clear on physical grounds that the form of Eqs. 4-5 can serve to define an effective A in the superexchange case in terms of the Rabi precession frequency characteristic of the two trap sites embedded in the complex system wherein 2A/h would be computed from this net effective Rabi precession frequency. 

We now make the ansatz that the form of this partial differential equation does not change if V = V(x, t) rather than a constant. Generalizing eqn (2.28) to three dimensions, we find the time-dependent Schrodinger equation,... 

Equations 2.85 and 2.86 may be considered the Schrodinger representation of the absorption of radiation by quantum systems in terms of spectroscopic transitions between states i) and /). In the Schrodinger picture, the time evolution of a system is described as a change of the state of the system, as implemented here in the form of the time-dependent perturbation theory. The results hardly resemble the classical relationships outlined above, compare Eqs. 2.68 and 2.86, even if we rewrite Eq. 2.86 in terms of an emission profile. Alternatively, one may choose to describe the time evolution in terms of time-dependent observables, the Heisenberg picture . In that case, expressions result that have great similarity with the classical expressions quoted above as we will see next. 

Messina et al. consider a system with two electronic states g) and e). The system is partitioned into a subset of degrees of freedom that are to be controlled, labeled Z, and a background subset of degrees of freedom, labeled x the dynamics of the Z subset, which is to be controlled, is treated exactly, whereas the dynamics of the x subset is described with the time-dependent Hartree approximation. The formulation of the calculation is similar to the weak-response optimal control theory analysis of Wilson et al. described in Section IV [28-32], The solution of the time-dependent Schrodinger equation for this system can be represented in the form... 

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