Lowering operators, definition

We follow here the treatment by D. D. Fitts (1995) J. Chem. Educ. 72, 1066. However, the definitions of the lowering operator and the constants axi and b i have been changed. 

Recall that the coherent states are by definition the eigenkets of the lowering operator ... 

By definition, a coherent state a) is the eigenvector of the non-Hermitean lowering operator a of the quantum harmonic oscillator. Thus, the basic equation and its conjugate are, respectively,... 

By adding and subtracting them, we get the definition of Ix and Iv in terms of the raising and lowering operators... 

It is not immediately obvious why IXSX should represent zero and double quantum coherence, but expression of this product in terms of raising and lowering operators makes the realtionship clear. From the definitions in Eq. 2.7, we obtain... 

Electrolytic treatment technologies have definite advantages over these more common treatment processes. The primary benefit is that chemical change in an electrochemical process is brought about by the ability to add or remove electrons from species to be treated. This eliminates the use of redox agents to treat wastes and also removes the need to treat spent redox streams. Other, equally important, benefits of electrochemical processes include close control of reactions through control of the applied potential or current lower operating temperatures and hence lower costs increased possibility of on-site treatment, especially in small-scale use possible simultaneous use of the anode and cathode for waste minimization and the ability to... 

Initial economic comparison of the dissociation-extraction type process with the chemical-consuming process of caustic wetsh, acidification, euid solvent extraction indicates that the dissociation extraction process, which has lower operating costs but may have higher capital cost for contactors and ctssociated equipment, may be less costly overall. A process optimisation cuid more detciiled economic assessment would have to be made, however, before a definite decision could be made between the two processes. 

Four of the most frequently used reversed-phase (RP) solvents will be discussed in detail water, acetonitrile, methanol, and THF. From a UV absorbance point of view, the perfect UV solvent would not absorb from 195 nm (a typical lower operating limits for detectors) all the way through the entire UV range. From this definition alone, water is the only commonly used solvent that fits into this category. However, some solvents exhibit very low absorbances below 220 nm. Acetonitrile is one such... 

In film deposition, the situation is often complicated by the fact that the vapor and the substrate are not phases of the same material, and by the fact that the temperature of the substrate is usually lower than that of the vapor. The definition of equilibrium vapor pressure is not so clear in this case. However, in most cases there will be some level of vapor pressure below which deposition of film material onto the growth surface will not occur this serves as an operational definition of Pe for film growth. 

Several steps are required to find the integral that is equal to c. From the definition of the raising and lowering operators, each is seen to be the complex conjugate of the other. And because these are Hermitian operators, they can be rearranged in the integral we are seeking to evaluate. 

From the definition of specific speed (eqs. 9 and 10), it follows that reciprocating pumps operate at high pressures and low flow rates. Conversely, centrifugal pumps are appHed at lower pressures and higher flow rates. Many rotary pumps are selected for viscous Hquids having pressures equal to or less than, and capacities lower than, centrifugal pumps. However, these limits are relative and a gray area exists as some pump types cross boundaries into the domain of other types. 

As microprocessor-based controls displaced hardwired electronic and pneumatic controls, the impac t on plant safety has definitely been positive. When automated procedures replace manual procedures for routine operations, the probability of human errors leading to hazardous situations is lowered. The enhanced capability for presenting information to the process operators in a timely manner and in the most meaningful form increases the operator s awareness of the current conditions in the process. Process operators are expected to exercise due diligence in the supervision of the process, and timely recognition of an abnormal situation reduces the likelihood that the situation will progress to the hazardous state. Figure 8-88 depicts the layers of safety protection in a typical chemical jdant. 

Definition of Absolute Temperature.— The temperatures of two bodies are proportional to the quantities of heat respectively taken in and given out in localities at one temperature and at the other, respectively, by a material system subjected to a complete cycle of perfectly reversible thermodynamic operations, and not allowed to part with or take in heat at any other temperature or, the absolute values of two temperatures are to one another in the proportion of the heat taken in to the heat rejected in a perfect thermodynamic engine working with a source and refrigerator at the higher and lower of the temperatures respectively. ... 

While solving the operator equations (2) we establish the basic properties of the operator A such as self-adjointness, positive definiteness, the lower bound of the operator and its norm and more. The operator A constructed in Example 1 will be frequently encountered in the sequel. Before stating the main results, will be sensible to list its basic properties. 

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