The Quantum Mechanics Solver 22

The Quantum Mechanics Solver 22 uniquely illustrates the application of quantum mechanical concepts to various fields of modern physics. It aims at encouraging the reader to apply quantum mechanics to research problems in fields such as molecular physics, condensed matter physics or laser physics. Advanced undergraduates and graduate students will find a rich and challenging source of material for further exploration. This book consists of a series of problems concerning present-day experimental or theoretical questions on quantum mechanics | Solutions 213 P-1 t 1 otherwise there would exist n and n such that - 2 n n. d Since Jn wt 0 for wt 2 n n only sites for which m 1 nwt 2 are reached at time t. For large wt the amplitude is the same for all sites of the same parity Pm t - -cos2 wt m 1 - . nwt 2 4 We notice in particular that Pm t Pm 1 t 2 nwt is independent of m and varies slowly with t. e The probability wave becomes delocalized very quickly on the chain wt a few n . The edges of the region where the probability is non zero propagate in opposite directions with the velocity v nwa 2. This is comparable with what we have found in for a wave packet near q n 2. The experimental data displayed in this chapter were obtained by B. Dorner et al. Z. Phys. B 72 487 1988 . 21 A Quantum Box In recent years it has become possible to devise quantum boxes also called quantum dots of nanometric dimensions inside which the conduction electrons of a solid are confined at low temperatures. The ensuing possibility to control the energy levels of such devices leads to very interesting applications in micro-electronics and opto-electronics. A quantum box is made of a material A on which another material B is deposited. A set of quantum boxes is shown on Fig. . The dots of In As material B are deposited on a substate of GaAs material A . Fig. . Picture of a set of quantum boxes obtained with a tunneling microscope. The side of the square is 1 pm long and the vertical scale will be studied below 216 21 A Quantum Box In this chapter we are interested in the motion of an electron in a twodimensional box. We note q the electric charge of the electron and we neglect spin effects. We shall assume that in a solid the dynamics of an electron is described by the usual Schrodinger equation where i the mass of the electron is replaced by an effective mass p ii the atoms of the materials A and B create an effective potential V x y which is slowly varying on the atomic scale. Results on the One-Dimensional .

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