Raising operators, definition

That the wave function for an orbital theoretically has no outer limit as one moves outward from the nucleus raises interesting philosophical questions regarding the sizes of atoms. Chemists have agreed on an operational definition of atomic size, as we will see in later chapters. 

If we take the isospin raising current J+ = J -I- iJ, the associated charge operator is now the isospin raising operator T+ (see Section 1.3) whose effect on a state of definite isospin is given in (1.3.15). FVom the above one deduces... 

Remark. A logician might raise the following objection. In section 1 stochastic variables were defined as objects consisting of a range and a probability distribution. Algebraic operations with such objects are therefore also matters of definition rather than to be derived. He is welcome to regard the addition in this section and the transformations in the next one as definitions, provided that he then shows that the properties of these operations that were obvious to us are actually consequences of these definitions. 

The effect of the spherical operators on individual spin states is actually opposite to this for example 1+ j8 > -> la >. It is the magnetic quantum number that is raised by the operator —1/2 (j8 state) to +1/2 (a state). In this book, we will reverse the definition for convenience so that the operators make intuitive sense I" " raises the spin state from a to jS. 

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

This optimization is achieved by exploiting to the utmost the correlations established during definition of the initial catalytic formulas It should not only take into account the problems raised by the study of use but also the need for a simple and economical preparation that can be expanded to industrial scale Therefore, the problems of extrapolating to industrial scale the various unit operations perfected in the laboratory have to be resolved in the pilot plant This study consists of... 

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... 

Once the form is recognized as received by the system, a data entry operator can start entry into the database. In the data definition process, screen layouts will have been defined to facilitate the accurate and speedy entry. Some validation or discrepancy checks can be designed to trigger at entry. For example, if a data entry operator attempts to violate the criteria defined to the system for a particular data field (e.g., entering character information into a numerically defined field), a discrepancy can be raised to alert the operator for acceptance or correction to the data. If the entry correctly reflects what is written on the CRF, the value can be accepted and a discrepancy noted for later follow up. 

The above naturally raises a question as to what exactly is meant by the term char in the context of bulk pyrolysis experiments. The definition can clearly only be operational, and is simply the mass that is left when an apparently steady state condition is achieved, regardless of how incomplete the pyrolysis is or however much carbon has resulted from processes involving secondary carbonization reactions of tars and their precursors. Thus all reports of properties of cellulosic chars must be viewed in this light. 

Since we have defined the product of two operators, we have a definition for the powers of an operator. An operator raised to the nth power is the operator for n successive applications of the original operator ... 

Not only can the amount of substance formed in a chemical reaction during any given interval of time be expressed in this manner, but all sorts of varying magnitudes can be subject to a similar operation. The distance passed over by a train travelling with a known velocity, can be represented in terms of a definite integral. The quantity of heat necessary to raise the temperature,... 

Attention was also called to the fact that a definite changeover from traditional cars to electric cars with fuel cells would be associated with the need for a fundamental restructuring not only of the car industry and of the oil production and refining operations, but also of many other economic sectors related to them. It must be acknowledged that even today no solution is in sight for many questions raised in the mentioned article. 

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