Determination of energy levels, probabilities, and expectation values of particles in the three-dimensional box at quantum numbers ≤5 (original) (raw)
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183 philosophical implications of quantum mechanics and develop a new way of thinking about nature on the nanometer-length scale. This was undoubtedly one of the most signiicant shifts in the history of science. The key new concepts developed in quantum mechanics include the quantiza-tion of energy, a probabilistic description of particle motion, wave–particle duality, and indeterminacy. These ideas appear foreign to us because they are inconsistent with our experience of the macroscopic world. Nonetheless, we have accepted their validity because they provide the most comprehensive account of the behavior of matter and radiation and because the agreement between theory and the results of all experiments conducted to date has been impressively accurate. Energy quantization arises for all systems whose motions are connned by a potential well. The one-dimensional particle-in-a-box model shows why quantiza-tion only becomes apparent on the atomic scale. Because the energy level spacing is inversely proportional to the mass and to the square of the length of the box, quantum effects become too small to be observed for systems that contain more than a few hundred atoms. Wave–particle duality accounts for the probabilistic nature of quantum mechanics and for indeterminacy. Once we accept that particles can behave as waves, we can form analogies with classical electromagnetic wave theory to describe the motion of particles. For example, the probability of locating the particle at a particular location is the square of the amplitude of its wave function. Zero-point energy is a consequence of the Heisenberg indeterminacy relation; all particles bound in potential wells have nite energy even at the absolute zero of temperature. Particle-in-a-box models illustrate a number of important features of quantum mechanics. The energy-level structure depends on the nature of the potential, E n n 2 , for the particle in a one-dimensional box, so the separation between energy levels increases as n increases. The probability density distribution is different from that for the analogous classical system. The most probable location for the particle-in-a-box model in its ground state is the center of the box, rather than uniformly over the box as predicted by classical mechanics. Normalization ensures that the probability of nding the particle at some position in the box, summed over all possible positions, adds up to 1. Finally, for large values of n, the probability distribution looks much more classical, in accordance with the correspondence principle. Different kinds of energy level patterns arise from different potential energy functions, for example the hydrogen atom (See Section 5.1) and the harmonic oscil-lator (See Section 20.3). These concepts and principles are completely general; they can be applied to explain the behavior of any system of interest. In the next two chapters, we use quantum mechanics to explain atomic and molecular structure, respectively. It is important to have a rm grasp of these principles because they are the basis for our comprehensive discussion of chemical bonding in Chapter 6.
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The problems of a free classical particle inside a one-dimensional box: (i) with impenetrable walls and (ii) with penetrable walls, were considered. For each problem, the classical amplitude and mechanical frequency of the τ -th harmonic of the motion of the particle were identified from the Fourier series of the position function. After using the Bohr-Sommerfeld-Wilson quantization rule, the respective quantized amplitudes and frequencies (i.e., as a function of the quantum label n) were obtained. Finally, the classical-quantum results were compared to those obtained from modern quantum mechanics, and a clear correspondence was observed in the limit of n ≫ τ .
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This article offers the calculations and visualization of calculation results for quantum system made with use of the software package of MATLAB. The electron is in an infinitely deep square well with width of l. The solution of the Schrodinger equation allows to obtain the energy levels of a particle in the well and corresponding eigen-functions. The program for calculation and visualization of results in the MATLAB language is developed. Results are presented in the form of graphs of wave functions and probability of finding of an electron at various distances from walls of a potential well by drawing the in one picture. For the best perception and understanding the behavior of the electron in the well the graphs of the wave functions and probability of finding electron at various distances from walls of the potential well are presented separately for each quantum state. The probabilities of the electron location in different regions of the well and energies at discrete states are calculated for the case when the electron moves along x-axis, i.e. for one-dimensional motion. It is shown that the total probability of finding electron in the well is defined by summation of probabilities of the electron location in different regions of the well and equals one.