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Ideas of Quantum Chemistry P8

Ideas of Quantum Chemistry P8 shows how quantum mechanics is applied to chemistry to give it a theoretical foundation. The structure of the book (a TREE-form) emphasizes the logical relationships between various topics, facts and methods. It shows the reader which parts of the text are needed for understanding specific aspects of the subject matter. Interspersed throughout the text are short biographies of key scientists and their contributions to the development of the field. | 36 1. The Magic of Quantum Mechanics where X means the mean value of many measurements of the quantity X. The standard deviation AA represents the width of the distribution of A i.e. measures the error made. Eq. 1.20 is equivalent to AA 2 A- Ji 2 1.21 because Ji - A 2 1 - 2Ji l A 2 A2 - 2 A A 2 A2 - A 2. Consider the product of the standard deviations for the operators A and B taking into account that u denotes Postulate IV the integral T u T according to 1.19 . One obtains denoting A A - A and B B - B of course A B A B AA 2 AB 2 ll .-l 1 1 1 A T A B B where the Hermitian character of the operators A and B is used. Now let us use the Schwarz inequality Appendix B fi 1 2f2 f1 f2 2 AA 2 AB 2 AT .IT 2- T 2- T .IT 2- T 2. Next A B AB A B BAYV i C B Jt i c B A i ct Jt Z5 . Hence i O 2i Im A T B This means that Im A T B T IZICH which viws f A T B T 2 1 ICd This means that im Ai Bi yj 2 which gives Ai b i 2 . Hence AA 2 AB 2 .IT BT 2 T CT 2 1.22 or taking into account that CT TAA B T we have 1 AA AB A B . 1.23 There are two important special cases a C 0 i.e. the operators A and B commute. We have AA AB 0 i.e. the errors can be arbitrarily small. Both quantities therefore can be measured simultaneously without error. b C h as in the case of x and px. Then AA AB 1.4 The Copenhagen interpretation 37 Fig. 1.13. Illustration of the Heisenberg uncertainty principle. al x 2 as function of coordinate x. Wave function x can be expanded in the infinite series T x f p Cp exp ipx where p denotes the momentum. Note that each individual function exp ipx is an eigenfunction of momentum and therefore if x exp ipx a measurement of momentum gives exactly p. If however x Ep Cp exp ipx then such a measurement yields a given p with the probability cp 12. Fig. a2 shows Cp 2 as function of p. As one can see a broad range of p large uncertainty of momentum assures a sharp T x 2 distribution small uncertainty of position . Simply the waves exp ipx to obtain a sharp peak of x should exhibit a .