Postulate 4 and Expectation Values

The first postulate of quantum mechanics asserts that the wave function of a system contains all available information about the values of mechanical variables for the state corresponding to the wave function. The third postulate provides a mathematical operator for each mechanical variable. The fourth postulate provides the mathematical procedure for obtaining the available information  

The expectation value is the predicted mean of a set of many measurements of the variable, given that the system is in the state corresponding to the wave function I at the time of each measurement. As is the case with all quantum mechanical integrations, the integrals in Eq. (16.4-1) extend over all values of the coordinates, which are abbreviated by. We will see that Eq. (16.4-1) leads to the famous fact that quantum mechanics often provides only statistical information. 

Consider first the case of a wave function b that is a product of a coordinate wave function and a time-dependent factor. The complex conjugate of the time-dependent factor can be obtained by changing the sign in front of the i symbol in the exponent (see Appendix B)  

Since the operator A contains no time dependence, the time-dependent factors cancel  

The expectation value can be obtained from the coordinate wave function and is time-independent if the wave function is the product of a coordinate factor and a time factor. A state corresponding to such a wave function is called a stationary state, and corresponds to a standing wave. 

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For Examples 5, 8, 12, 21, and 28, all first order, the logL values for Steps 1, 4, and 6 are too large. With Examples 8 and 12 one can obtain reasonable logL values by postulating (as was done in connection with similar examples listed in Table V) that the gas molecule does not lose all of its entropy upon adsorption. For Example 28 Kuriacose and Jewur (56) postulated a bimolecular surface mechanism, that is. Step 4. The L value for that step is very large the authors claimed that an intermediate is ferric acetate. If this is correct, one would indeed expect the L calculation to indicate that the reaction is more complex than any of our models. For... 

If, now, the n-fructose 6-phosphate, labeled as predicted from the operation of Reaction (4), is introduced as a substrate into Reactions (1), (2), and (3), carbon atoms 2,3, and 4 of the tetrose produced will have more isotope from G-3,2,1 than previously calculated (see Table I) the G-3/G-2 ratio in Cl of the tetrose will be unaffected. It may be seen from Table I that the concentration of G-3,2,1 in the bottom three carbon atoms of the tetrose phosphate, expectedfrom Reactions (1), (2), and (3) alone, is 1/(1 + 5), or 17 per cent. The concentration of G-3,2,1 in carbon atoms 4,5, and 6 of shikimate is 1/(1 -h 2.5), or approximately 29 per cent. If it be assumed that this increase in concentration of G-3,2,1 (to nearly double the expected value) is due to the exchange postulated in Reaction (4), it follows that almost half of the n-fructose 6-phosphate had been affected by this reaction. The relatively high incorporation of G-3,2,1 into carbon atoms... 

The discrepancy from the experimental values is due to the fact that H atoms bound to different types of C atoms are replaced by chlorine at different rates. The substitution of Cfcrt— H takes place via a tertiary radical. The substitution of Csec—H takes place via the somewhat less stable secondary radical, and the substitution of Cprjm—H takes place via even less stable primary radicals (for the stability of radicals, see Table 1.2). According to Hammond s postulate, the rate of formation of these radicals should decrease as the radical s stability decreases. Hydrogen atoms bound to Ctert should thus be substituted more rapidly than H atoms bound to Csec, and these should in turn be substituted by Cl more rapidly than H atoms bound to Cprjm. As the analysis of the regioselectivity of the monochlorination of isopentane carried out by means of Table 1.4 shows, the relative chlorination rates of C —H, C —H, and C. —H are 4.4 33 1, in agreement with this expectation. 

The Hammett postulate in essence is that the quantity yB yB H+/yB H+yB" is unity, independent of the medium and of the particular indicator in the series. For solvents of high dielectric constant and for the closely similar nitroaniline indicators, both theoretical expectation and experimental justification uphold this postulate. By the use of (4-120), values of pK for the various nitroaniline indicators were measured up to a pK value of —10.10 for 2,4,6-trinitroaniline. 

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