Elementary considerations

9 Quantum dynamics of the harmonic oscillator 2.9.1 Elementary considerations 

The classical Hamiltonian for a one-dimensional harmonic oscillator of mass m centered about x = 0, 

It is convenient for future reference to define the dimensionless position f and momentum (p 

In terms of these variables the Hamiltonian (2.125) takes the form 

The solutions of the time-independent Schrodinger equation Hij/ = Exl are the (orthonormal) eigenfunctions and the corresponding eigenvalues 

Quantum dynamics using the time-dependent Schrodinger equation 

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Flow Maldistribution. One of the principal reasons for heat exchangers failing to achieve the expected thermal performance is that the fluid flow does not foUow the idealized anticipated paths from elementary considerations. This is referred as a flow maldistribution problem. As much as 50% of the fluid can behave differently from what is expected based on a simplistic model (18), resulting in a significant reduction in heat-transfer performance, especially at high or a significant increase in pressure drop. Flow maldistribution is the main culprit for reduced performance of many heat exchangers. 

Notice that one event has units of per-demand and the others have a per-unit-time dimension. From elementary considerations, the top event can only have dimensions of per-demand (pure probability) or per-unit-time dimensions. Which dimensions they have depends on the application. If the fault tree provides a nodal probability in an event tree, it must have per-demand dimensions, if the fault tree stands alone, to give a system reliability, it must have per-unit-time dimensions. Per-unit-time dimensions can be converted to probability using the exponential model (Section 2.5.2.6). This is done by multiplying the failure rate and the "mission time" to give the argument of the exponential which if small may be... 

Atmospheres containing HjO vapour, or SO2 are technically very important, but much more fundamental work is needed, and there is space here for only elementary considerations. 

Elementary considerations indicate that with appropriate substitutions some of the reactions mentioned above can be eliminated. Indeed, when 5-methyl-2-vinyl-furan was used, no alkylation was observed, the positions C-3 and C-4 being rather unreactive16, and the polymer was a mixture of linear chains with polyunsaturations and linear saturated chains, i.e. only structures like 21, 23 and 26 were present, with a 5-methyl ring instead of the 5-unsubstituted one. When 2-isopropenylfuran was used, no hydride transfer took place since this requires a hydrogen atom in the a-position to the ring, which this monomer does not have the polymers were white and gave electronic spectra transparent down to 280 nm. Alkylation at C-5, how-... 

The activities of the various components 1,2,3. .. of an ideal solution are, according to the definition of an ideal solution, equal to their mole fractions Ni, N2,. . . . The activity, for present purposes, may be taken as the ratio of the partial pressure Pi of the constituent in the solution to the vapor pressure P of the pure constituent i in the liquid state at the same temperature. Although few solutions conform even approximately to ideal behavior at all concentrations, it may be shown that the activity of the solvent must converge to its mole fraction Ni as the concentration of the solute(s) is made sufficiently small. According to the most elementary considerations, at sufficiently high dilutions the activity 2 of the solute must become proportional to its mole fraction, provided merely that it does not dissociate in solution. In other words, the escaping tendency of the solute must be proportional to the number of solute particles present in the solution, if the solution is sufficiently dilute. This assertion is equally plausible for monomeric and polymeric solutes, although the... 

Percent extraction as the term implies provides directly a measure of effectiveness a solvent or a strippant used in solvent extraction. The mathematical expression of percent extraction is quite simple to derive simply from elementary considerations. This fact, therefore, spares the need to deduce. The expression for percent extraction, P, is shown as ... 

Elementary considerations show that of all spectroscopic methods only the UV need be considered for olefin polymerisations. The reasons for the success of this method in the... 

Nearly all kinetic isotope effects (KIE) have their origin in the difference of isotopic mass due to the explicit occurrence of nuclear mass in the Schrodinger equation. In the nonrelativistic Bom-Oppenheimer approximation, isotopic substitution affects only the nuclear part of the Hamiltonian and causes shifts in the rotational, vibrational, and translational eigenvalues and eigenfunctions. In general, reasonable predictions of the effects of these shifts on various kinetic processes can be made from fairly elementary considerations using simple dynamical models. 

The basic nature of the turbulent exchange process is not yet well enough known to allow accurate prediction of behavior without recourse to experiment. Correlation of the growing body of experimental knowledge in this field, however, offers the possibility of evaluating time-averaged point values of thermal and material transport for many conditions of industrial interest. It is the purpose of this discussion to present some of the more elementary considerations of the nature of turbulent flow with particular emphasis upon thermal and material transport. 

This can be seen from elementary considerations. The distortion of the surrounding medium that leads to the intermediate state with energy WH can be described by simple harmonic motion, with a parameter x for the displacement, a potential energy px2 and a wave function of the form r=const xexp(—ax2). The probability P 2 of a configuration with potential energy WH is thus... 

What makes this problem different Is it not possible to find the exponents from elementary considerations in the case of impulse ... 

In fact the redshift can be calculated from the following elementary considerations. If Wo = Kvo be the initial energy of an electromagnetic wave (say, of a single photon) and W = Kv be residual energy after a path r, we have... 

In any case, obvious questions remain on the theoretical side, such as the nature of the B-H bonds in this deceptively simple molecule. Are they really the straightforward sigma bonds elementary considerations would suggest On the basis of the OBS-GMCSC results, the present author will argue that the molecule s electronic structure is not quite that simple. 

According to this equation the rate constant k of a substitution reaction is determined by the polarizability (P) of the nucleophile and the pKa of its conjugate acid. We have developed (5) a similar interpretation of nucleophilic reactivity based on an elementary consideration of the energy factors determining the activation energy of a displacement reaction, and this procedure may be used for equilibria in solution. The reaction... 

The dipole moments of the positive and negative ions in each layer are opposed and alternate in sign from layer to layer for ions of the same type. This behavior is to be expected from very elementary considerations of the electric field at the lattice sites of the surface region. The net dipole moment of layer A (per ion pair), defined as... 

The kinetics of physical adsorption were reviewed by Brunauer in 1943 (5). A 1 careful literature survey (6) reveals that the present status of the subject is substantially unchanged. Elementary considerations indicate that the process of physical adsorption should proceed very rapidly and be practically complete within the order of magnitude of one minute, and indeed, experiment usually confirms this. Deviations from this behavior have been accounted for by the time required to dissipate the heat of adsorption or by a very slow surface diffusion into a microporous structure. 

We now turn to devising a procedure for estimating the maximum allowable temperature rise. Equation (8) is not useful in a quantitative sense because the parameters G and R cannot be measured or estimated. However, it is possible to deduce from elementary considerations how they might vary with reaction conditions. 

Especially pleasing is the circumstance that our model gives a unified framework which includes bilayers with no additional assumptions. Various physical properties of bilayer vesicles are readily accounted for. Even the Young s modulus of bilayers appears to be given correctly from elementary considerations. 

In all these cases, the correct design must grow from the equations of mass, energy, and momentum balance to which we now turn in the next few sections. From these we proceed to the design problem (Sec. 9.5) and hence to elementary considerations of optimal design (Sec. 9.6). The stability and sensitivity of a tubular reactor is a vast and fascinating subject. Since the steady state equations are ordinary differential equations, the equations describing the transient behavior are partial differential equations. This... 

The partly ionic, complex iodide salts discussed in the next section did show far higher conductivity, as expected from the elementary considerations in Section 2.2. 

This result is known as the Stokes-Einstein relation. It can also be derived from elementary considerations Let a system of noninteracting charged particles be in thermal equilibrium with a uniform electric field Ex = — 9charged particles satisfies p(x) exp(—/f diffusion flux, —Ddp/dx = —flDqpEx and the drift flux, uqlfp should add to zero. This yields (11.66). 

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