Time-independent equation

The expression for the force on the nuclei, Eq. (89), has the same form as the BO force Eq. (16), but the wave function here is the time-dependent one. As can be shown by perturbation theory, in the limit that the nuclei move very slowly compared to the electrons, and if only one electronic state is involved, the two expressions for the wave function become equivalent. This can be shown by comparing the time-independent equation for the eigenfunction of H i at time t... 

In cases where the elassieal energy, and henee the quantum Hamiltonian, do not eontain terms that are explieitly time dependent (e.g., interaetions with time varying external eleetrie or magnetie fields would add to the above elassieal energy expression time dependent terms diseussed later in this text), the separations of variables teehniques ean be used to reduee the Sehrodinger equation to a time-independent equation. 

When the Sehrodinger equation ean be separated to generate a time-independent equation deseribing the spatial eoordinate dependenee of the wavefunetion, the eigenvalue E must be returned to the equation determining F(t) to find the time dependent part of the wavefunetion. By solving... 

One of the great difficulties in molecular quantum mechanics is that of actually finding solutions to the Schrodinger time-independent equation. So whilst we might want to solve... 

Suppose that is the lowest energy solution to the Schrodinger time-independent equation for the problem in hand. That is to say,... 

Thus, the approximate total wave function ik(r,X) = vFi(rs,rm X) ik(X) is taken as the solution of the time-independent equation (2) with energy levels Eik The time-dependent equation can be cast in terms of this energy so that jk (t) = exp(-iEikt/ll). An arbitrary quantum state can be expanded on the basis of the Ojkfi Xjexpf-iEjkt/li)... 

The stationary theory deals with time-independent equations of heat conduction with distributed sources of heat. Its solution gives the stationary temperature distribution in the reacting mixture. The initial conditions under which such a stationary distribution becomes impossible are the critical conditions for ignition. 

For an isolated system, H(x) is time independent, eq. (3) is separated as usual leading to the time independent equation H(x) < x d)> = E < x d>>. The structure of H is not known in detail. So far, it is just a symbol, but if this is a hermitian and self-adjoint operator, there exists a complete denumerable set of eigenfunctions. 

We shall not present the detailed analysis of this complication. In essence, it involves the time-dependent Schrodinger equation rather than the time-independent equation that resulted in Equation (22). Casimir and Polder have investigated this situation. They found that for values of r > X, the potential energy of attraction according to the modified London treatment is given by... 

Spectroscopy involves transitions between states. To discuss transitions, we must use the time-dependent Schrodinger equation, which is harder to deal with than the time-independent equation. Rather than attempt an exact solution, we will use an approximate treatment called time-dependent perturbation theory. 

Quantum mechanics describes molecules in terms of interactions between nuclei and electrons and molecular geometry in terms of minimum energy arrangements of nuclei. All quantum-mechanical methods ultimately trace back to Schrodinger s (time-independent) equation, which may be solved exactly for the hydrogen atom. For a multinuclear and multielectron system, the Schrodinger equation may be defined as ... 

After a sufficiently long time the positrons reach equilibrium, so that y(v,t) = f(v) exp(—(Af)t), where (Af) is the equilibrium annihilation rate and f(v) is the associated speed distribution, which is the solution of the time-independent equation... 

Note that wavefunction

time independent. The wave functions obtained from a time-independent equation are called standing (or stationary) waves. To obtain such an equation, we assume

[Pg.10]

The time-independent equations above are elliptic in nature. In a rectangular grid system, the finite difference forms of Eqs. (10.29) and (10.26) can be written as follows ... 

We assume that the molecule is in a stationary state initially, the wave function of which is describable by HF. In the density matrix formalism [9, 10] (which is equivalent to the usual operator form), the Fock F(0) and density matrices D(Cl> satisfy the time-independent equation... 

Obviously, is simply 4 (Z = 0) and is by definition time independent. Equation (2.62) is a unitary transformation on the wavefunctions and the operators at time t. The original representation in which the wavefunctions are time dependent while the operators are not, is transformed to another representation in which the operators depend on time while the wavefunctions do not. The original fonnulation is referred to as the Schrodinger representation, while the one obtained using (2.62) is called the Heisenberg representation. We sometimes use the subscript S to emphasize the Schrodinger representation nature of a wavefunction or an operator, that is. 

The basic problem in the quantum treatment of chemical bonds is obtaining solution to Schrodinger s nonrelativistic time-independent equation ... 

We have now derived the four basic (time-independent) equations of stellar structure. These are mass continuity (Eq. (14)), hydrostatic equilibrium (Eq. (17)), conservation of energy (Eq. (28)), and energy transport (Eq. (33)). These form a set of coupled first order ordinary differential equations relating one independent variable, e.g. r, to four dependent variables i.e., m, /, / //, which uniquely describe the structure of the star, note that any variable could be used as the independent variable. In an Eulerian frame, the spatial coordinate r is the independent variable. For most problems in stellar structure and evolution it is usually more convenient to work in a Lagrangian frame, with mass as the independent variable. Transforming, we obtain ... 

Schrodinger s time-independent equation in one dimension is given by ... 

Let us presume that the electronic equation, Eq. (25), has been solved. Noting that the functions form a complete ortho-normal set, we may develop the solution of the total time-independent equation, Eq. (17), in full generality and exactness as... 

For the moment let us restrict our investigation of the Bom approximation to an elastic collision between two particles without internal stracture as in Section IV-B (2). Then, and in the time-independent equation, Eq. (112c), are nonexistent, and that equation may be rewritten in the present notation as ... 

The equations for time-independent self-consistent perturbation theory are of very similar form to those time-dependent ones we have just studied, but the interpretation of the case of infinitesimal perturbations is, if an)d hing, more problematical. What is the meaning of the fact that there eire non-trivial solutions of the time-independent equations for arbitrarily small perturbations ... 

However, due to the lack of Hermiticity, the spectrum of H includes, in addition to the normal spectrum of the ordinary Hamiltonian H, the possibility of complex eigenvalues. The occurence of such then causes the general operator exp(-iHt/h) to be undefined unless one considers it in a projected subspace (42). Within such a subspace comprised of all physical solutions to the time dependent Schrodinger equation, equation (82) is perfectly well behaved. Let us consider again the time independent equation... 

The electro-thermal model is not solved by merely finding a stationary state solution to Eqs. (6.2) to (6.5). If one solves the time independent equations with numerical methods there exists the danger that one follows one branch of solutions without being aware of the existence of other branches. Furthermore, the solution in question may not be stable. It appears that Spenke (1936a, 1936b) was the first to point out the existence... 

As in the preceding section solutions of the time-dependent and time-independent equations are attempted by assuming the spatial dependence... 

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