Schrodinger equation for stationary states

It may be instructive to see how Erwin Schrodinger invented his famous equation (1.13) for stationary states ijf of energy E (fi denotes the Hamiltonian of the system) 

Sehrodinger surprised the contemporary quantum elite (associated mainly with 

Sctnodingef s cinkulum vitae found in Breslau (now Wroclaw)  

Presumably the spirit of Ludwig Boltzmann (deceased in 1906), operating especially intensively in Vienna, directed me first towards the probability theory in physics. Then, (...) a closer contact with the experimental works of Exner and Rohrmuth oriented me to the physiological theory of colours, in which I tried to confirm and develop the achievements of Helmholtz. In 1911-1920 I was a laboratory assistant under Franz Exner in Vienna, of course, with 4 years long pause caused by war. I have obtained my habilitation in 1914 at the University of Vienna, while in 19201 accepted an offer from Max Wien and became his assistant professor at the new theoretical physics department in Jens This lasted, unfortunately, only one semester, because I could not refuse a professorshp at the Technical University in Stuttgart. I was there also only one semester, because April 

1921 I came to the University of Hessen in succession la Klemens Schrafer. I am almost ashamed la confess, that at the moment I sign the present curriculum vitae I am no longer a professor at the University of Breslau, because on Oct. 15.1 received my nomination to the University of Zurich. My instability may be recognized exclusively as a sign of my ingratitude  

---

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. 

Time-independent Schrodinger Equation for Stationary States... 

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

The postulates constitute the foundation of quantum mechanics (the base of the TREE trunk). One of their consequences is the Schrodinger equation for stationary states. Thus we begin our itinerary on 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 main theme of this book (this is why it is a side branch on the left side of the TREE). 

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 ... 

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... 

Introducing the potential of the harmonic oscillator (eq. 3.2) in the monodimensional equivalent of equation 3.9 (i.e., the Schrodinger equation for onedimensional stationary states see eq. 1.9), we obtain... 

Bardeen considers two separate subsystems first. The electronic states of the separated subsystems are obtained by solving the stationary Schrodinger equations. For many practical systems, those solutions are known. The rate of transferring an electron from one electrode to another is calculated using time-dependent perturbation theory. As a result, Bardeen showed that the amplitude of electron transfer, or the tunneling matrix element M, is determined by the overlap of the surface wavefunctions of the two subsystems at a separation surface (the choice of the separation surface does not affect the results appreciably). In other words, Bardeen showed that the tunneling matrix element M is determined by a surface integral on a separation surface between the two electrodes, z = zo. 

A central problem in physics and chemistry has always been the solution of the Schrodinger equation (SE) for stationary states. Such stationary states may relate to electronic structure problems, in which case one is primarily interested in bound states, or to scattering problems, in which case the stationary solutions are continuum states. In both cases, one of the most powerful tools in the theoretical arsenal for solving such problems is the partitioning technique (PT), which has been developed in a series of papers prominently by Per-Olov Lowdin [1-6] and Herman Feshbach [7-9]. 

Although the above explanation relied on a crude semiclassical estimate (with exponential accuracy), it can easily be refined either by exactly solving the Schrodinger equation for the one-dimensional potential (7.1) (see, for example, Press [1981]) or, for sufficiently high barriers (V0/h(o0> 2), by employing the WKB approximation. The eigenfunctions of stationary states A and E... 

V-clcctron state T, correlation energy can be defined for any stationary state by Ec = E — / o, where Eo = ( //1) and E = ( // 4 ). Conventional normalization ) = ( ) = 1 is assumed. A formally exact functional Fc[4>] exists for stationary states, for which a mapping — F is established by the Schrodinger equation [292], Because both and p are defined by the occupied orbital functions occupation numbers nt, /i 4>, E[p and E[ (p, ] are equivalent functionals. Since E0 is an explicit orbital functional, any approximation to Ec as an orbital functional defines a TOFT theory. Because a formally exact functional Ec exists for stationary states, linear response of such a state can also be described by a formally exact TOFT theory. In nonperturbative time-dependent theory, total energy is defined only as a mean value E(t), which lies outside the range of definition of the exact orbital functional Ec [ ] for stationary states. Although this may preclude a formally exact TOFT theory, the formalism remains valid for any model based on an approximate functional Ec. 

It may be worthwhile to write the Schrodinger equation for one electron moving in the potential U(x,y,z) having the (negative) eigen-value E for stationary states with the real wave-function ip... 

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. 

From the Schrodinger equation for the stationary unperturbed state... 

A rigged BO approach is developed and used to describe a chemical system calculated with present day advanced electronic methods. Chemical species are determined by electronic wave functions that are independent from the nuclear configuration space. This is the fundamental hypothesis [11]. Boundary conditions in the global electronic wave function are introduced via the solution of electronic Schrodinger equations for systems of external Coulomb sources (Cf. Eq.(8)). The associated stationary arrangement of external Coulomb sources allows for the introduction of molecular frames. This approach naturally leads to a state-to-state description particularly useful in gas phase reactions. A chemical reaction is described as if it were an electronic spectroscopy event or series of events. 

The stationary-state wave functions and energy levels of a one-particle, one-dimensional system are found by solving the time-independent Schrodinger equation (1.19). In this chapter, we sdlve the time-independent Schrodinger equation for a very simple system, a particle in a one-dimensional box (Section 2.2). Because the Schrodinger equation is a differential equation, we first review the mathematics of differential equations (Section 2.1). 

For stationary states, the time-independent Schrodinger equation holds ... 

---