Schrodinger stationary equation

In a line of reasoning that many of the younger quantum physicists regarded as reactionary, Schrodinger built his treatment of the electron on the well-understood mathematical techniques of wave equations as partial differential equations involving second derivatives. Schrodinger s equation for stationary electron states, as written in the Annalen der Physik in 1926, took the form... 

The proper description of a microobject (atom, molecule, cluster) is given by quantum mechanics an operator is attributed to each observable the operators act on the state vectors (kets) the state vectors bear all the physical information. The Schrodinger equation for stationary states is of key interest. 

However, there is a stationary variation principle of precisely the type employed in the quantum chemical linear variation method. In the derivation of the Roothaan equations based on finite basis set expansions of Schrodinger wavefimctions, one insists only that the Rayleigh quotient be stationary with respect to the variational parameters, and then assumes that the variational principle guarantees an absolute minimum. In the corresponding linear equations based on the Dirac equation, the stationary condition is imposed, but no further assumption is made about the nature of the stationary point. 

Time-independent Schrodinger Equation for Stationary States... 

The postulates constitute the foundation of quantum mechanics (the base of the TREE trunk). One of their consequences is the Schrodinger equation for stationary stales. Thus, we begin our itinerary with the TREE. The second part of this chapter is devoted to the time-dependent Schrodinger equation, which, from the pragmatic point of view, is outside the scope of this book (which is why it is a side branch on the left side of the TREE). 

It goes through the Schrodinger equation for stationary states, thus far the most important equation in quantum chemical applications. 

Using the Schrodinger stationary equation formalism in the SCE approximation, a number of investigations of the globular state of macromolecules have been carried out... 

Although the time-independent Schrodinger equation is heavily utilized in this chapter, it is not the fundamental form of the Schrodinger equation. Only stationary states—wavefunctions whose probability distributions do not vary over time— provide meaningful eigenvalues using the time-independent Schrodinger equation. 

In principle, a description of the electronic structure of many-electron atoms and of polyatomic molecules requires a solution of a Schrodinger equation for stationary states quite similar to equation 3.36 [2]. Even for a simple molecule like, say, methane, however, such an equation would be enormously more complicated, because the hamiltonian operator would include kinetic energy terms for all electrons, plus coulombic terms for the electrostatic interaction of all electrons with all nuclei and of all electrons with all other electrons. The QM hamiltonian operator for the electrons in a molecule reads ... 

The first-order variation in the energy functional [6] therefore vanishes whenever 0) corresponds to one of the eigenstates 0), thus demonstrating that the eigenstates of the Schrodinger equation represent stationary points of the energy functional. 

For the sake of simplicity, we will here confine ourselves to consider a system of N electrons moving in a given nuclear framework. The stationary states of such a system are described by the solutions to the Schrodinger equation... 

The u and v representations are sometimes distinguished as the Schrodinger and the Heisenberg representation. For stationary operators P, then, the Heisenberg equation of motion is... 

The physical interpretation of the quantum mechanics and its generalization to include aperiodic phenomena have been the subject of papers by Dirac, Jordan, Heisenberg, and other authors. For our purpose, the calculation of the properties of molecules in stationary states and particularly in the normal state, the consideration of the Schrodinger wave equation alone suffices, and it will not be necessary to discuss the extended theory. 

Most semi-empirical models are based on the fundamental equations of Hartree-Fock theory. In the following section, we develop these equations for a molecular system composed of A nuclei and N electrons in the stationary state. Assuming that the atomic nuclei are fixed in space (the Born-Oppenheimer approximation), the electronic wavefunction obeys the time-independent Schrodinger equation ... 

Strictly speaking, a chiral species cannot correspond to a true stationary state of the time-dependent Schrodinger equation H

time scale for such spontaneous racemization is extremely long. The wavefunction of practical interest to the (finite-lived) laboratory chemist is the non-stationary Born-Oppenheimer model Eq. (1.2), rather than the true T of Eq. (1.1). 

Nevertheless, very-long-lived quasi-stationary-state solutions of Schrodinger s equation can be found for each of the chemical structures shown in (5.6a)-(5.6d). These are virtually stationary on the time scale of chemical experiments, and are therefore in better correspondence with laboratory samples than are the true stationary eigenstates of H.21 Each quasi-stationary solution corresponds (to an excellent approximation) to a distinct minimum on the Born-Oppenheimer potential-energy surface. In turn, each quasi-stationary solution can be used to construct an alternative model unperturbed Hamiltonian //(0) and perturbative interaction L("U),... 

---