Simultaneous Specification of Several Properties

In this chapter we discuss angular momentum, and in the next chapter we show that for the stationary states of the hydrogen atom the magnitude of the electron s angular momentum is constant. As a preliminary, we consider what criterion we can use to decide which properties of a system can be simultaneously assigned definite values. 

In Section 3.3 we postulated that if the state function is an eigenfunction of the operator A with eigenvalue s, then a measurement of the physical property A is certain to yield the result s. If is simultaneously an eigenfunction of the two operators A and B, that is, if and then we can simultaneously assign definite val- 

Recall that the commutator of A and B is defined d [A,B] = AB - BA [Eq. (3.7)]. The following commutator identities are helpful in evaluating commutators these identities are easily proved by writing out the commutators in detail (Problem 5.1)  

For a state function that is not an eigenfunction of A, we get various possible outcomes when we measure A in identical systems. We want some measure of the spread or dispersion in the set of observed values A,-. If (a) is the average of these values, then the deviation of each measurement from the average is A, — (a). If we averaged all the deviations, we would get zero, since positive and negative deviations would cancel. Hence to make all deviations positive, we square them. The average of the squares of the deviations is called the variance of A, symbolized in statistics by o- and in quantum mechanics by (AA)  

The positive square root of the variance is called the standard deviation, or AA. The standard deviation is the most commonly used measure of spread, and we shall take it as the measure of the uncertainty in the property A. 

EXERCISE Show that for a one-particle, three-dimensional system, 

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Valuable data on the properties of active centres are obtained from kinetic measurements. They reveal the simultaneous existence of several centre types. The stable centres are active during the whole course of polymerization in addition, some fraction of decaying centres is also present. Isotactic centres exhibit stereoregulating ability and are, moreover, extremely active. Centres may oscillate between active and inactive (dormant) forms and some centres selectively polymerize enantiomers from a racemate. External effects, caused by specific properties of centres, will be discussed in subsequent chapters. In addition, the centres on which dienes are polymerized will be treated in Chap. 5, Sect. 4. The structure of these centres is a function of the coordinated diene, and it is therefore better presented together with propagation. 

The classical kinetic analytical methods [1-3] are mainly appUed in two versions (1) kinetic catalytic method based on catalytic reactions and (2) kinetic differential method based on the use of systems with simultaneous reactions of a reagent with several mixture components with similar properties. These versions are recommended for enzyme reactions with a view to determining enzymes, inhibitors and substrates. These reactions are highly sensitive and specific their use is without any doubt of particular interest for some systems for the selective and highly sensitive determination of some components of systems [3] to which GC can be applied. 

Based on the laboratory experimental and field application results, electrokinetic remediation technology has been shown to be a promising method for simultaneously recovering multiple metal contaminants. However, the process is accompanied by limitations on the removal of several specific heavy metals. In this last section, the previous investigations, which focused on the removal of heavy metals, including Cr, As, and Hg, will be examined in more detail. Heavy metals have some reasonable properties, which should be considered more specifically ... 

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