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Ideas of Quantum Chemistry P6

Ideas of Quantum Chemistry P6 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. | 16 1. The Magic of Quantum Mechanics moving along a single coordinate axis x the mathematical foundations of quantum mechanics are given in Appendix B on p. 895 . Postulate I on the quantum mechanical state wave function The state of the system is described by the wave function x t which depends on the coordinate of particle x at time t. Wave functions in general are complex functions of real variables. The symbol x t denotes the complex conjugate of x t . The quantity p x t x t x t dx x t 2dx 1.1 gives the probability that at time t the x coordinate of the particle lies in the small interval x x dx Fig. 1.3.a . The probability of the particle being in the interval a b on the x axis is given by Fig. 1.3.b fb x t 2dx. statistical The probabilistic interpretation of the wave function was proposed by Max interpretation Born.26 By analogy with the formula mass density x volume the quantity probability x t T x t is called the probability density that a particle at time t has posi- density tion x. In order to treat the quantity p x t as a probability at any instant t the wave normalization function must satisfy the normalization condition i x t x t dx 1. 1.2 TO Fig. 1.3. A particle moves along the x axis and is in the state described by the wave function x t . Fig. a shows how the probability of finding particle in an infinitesimally small section of the length dx at xo at time t to is calculated. Fig. b shows how to calculate the probability of finding the particle at t tg in a section a b . 26M. Born Zeitschrift für Physik 37 1926 863. 1.2 Postulates 17 All this may be generalized for more complex situations. For example in threedimensional space the wave function of a single particle depends on position r x y z and time r t and the normalization condition takes the form i dx i dy i dz x y z t x y z t i r t r t dV TO TO TO I r t r t d3r 1. 1.3 When integrating over whole space for simplicity the last two integrals are given without the integration limits but they are .