tailieunhanh - Ideas of Quantum Chemistry P21

Ideas of Quantum Chemistry P21 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. | 166 4. Exact Solutions - Our Beacons vibrational quantum number where h is the Planck constant v 0 1 2 . is the vibrational quantum number the variable is proportional to the displacement x TOr I km 1 Pk a y t2 v 2ïïVm frequency Hermite polynomials is the frequency of the classical vibration of a particle of mass m and a force constant k Hv represent the Hermite polynomials31 of degree v defined as32 Hv è -1 v exp dv exp -g2 1 d v and Nv is the normalization constant Nv a 2 Vfv . The harmonic oscillator finger print it has an infinite number of energy levels all non-degenerate with constant separation equal to hv. Note that the oscillator energy is never equal to zero. Fig. shows what the wave functions for the one-dimensional harmonic oscillator look like. Fig. also shows the plots for a two-dimensional harmonic oscillator we obtain the solution by a simple separation of variables the wave function is a product of the two wave functions for the harmonic oscillators with x and y variables respectively . The harmonic oscillator is one of the most important and beautiful models in physics. When almost nothing is known except that the particles are held by some Fig. . Some of the wave functions 1 v for a one-dimensional oscillator. The number of nodes increases with the oscillation quantum number v. 31Charles Hermite was French mathematician 1822-1901 professor at the Sorbonne. The Hermite polynomials were defined half a century earlier by Pierre Laplace. 32Ho 1 H1 2 H2 4f2 - 2 etc. The harmonic oscillator 167 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 Fig. . A graphic representation of the 2D harmonic oscillator wave isolines . The background colour corresponds to zero. Figs. a-i show the wave functions labelled by a pair of oscillation quantum numbers vi U2 . The higher the energy the larger the number of node planes. A reader acquainted with the wave functions of the hydrogen atom will easily recognize a striking resemblance between these figures and