Schrodinger equation confinement potential with

As an illustration of the application of the time-independent Schrodinger equation to a system with a specific form for F(x), we consider a particle confined to a box with infinitely high sides. The potential energy for such a particle is given by... 

To find the energy levels of an electron in a hydrogen atom, we have to solve the appropriate Schrodinger equation. An electron in an atom is like a particle in a box, in the sense that it is confined within the atom by the pull of the nucleus, so we can expect quantized energy levels. However, the Coulomb potential energy of the electron, % varies with distance, r, from the nucleus ... 

The solution to the Schrodinger equation for a particle confined within a simple harmonic potential well is a set of discrete allowed energy levels with equal intervals of energy between them. It is related to the familiar simple solution for a particle in an infinite square well, with the exception that in the case of the simple harmonic potential, the particle has a non-zero potential energy within the well. The restoring force in a simple harmonic potential well is fcsc, and thus the potential energy V(x) is x/2 kx2 at... 

The confinement of electrons or holes in potential wells leads to the creation of discrete energy levels in the wells, compared with the continuum of states in hulk material quantisation also leads to a major change in the density of states. The energy levels can be calculated by solving the Schrodinger equation for the well-known particle in a box problem. Using the effective mass envelope function approximation (Bastard, 1981 and 1982 Altarelli, 1985 Bastard and Brum, 1986), the electron wavefunction % is then... 

As a specific realization of these ideas, consider the celebrated problem of the one-dimensional particle in a box already introduced earlier in the chapter. Our ambition is to examine this problem from the perspective of the finite element machinery introduced above. The problem is posed as follows. A particle is confined to the region between 0 and a with the proviso that within the well the potential V (x) vanishes. In addition, we assert the boundary condition that the wave function vanishes at the boundaries (i.e. V (0) = rfr (a) = 0). In this case, the Schrodinger equation is... 

To treat the quantum size effect, an electron or hole inside a spherical particle is modeled as a particle in a sphere. " " The electron is considered to be confined to a spherical potential well of radius a with potential energy V(r) = 0 for r < u and V(r) = oo for r > u. Solving the Schrodinger equation yields the wave functions" ... 

Leaute B, MarciUiacy G (1986) On the Schrodinger equation of rotating harmonic, three-dimensional and doubly anharmonie oscillators and a class of confinement potentials in connection with the biconfluent Heun differential equation. J Phys A Math Gen 19 3527-3533... 

The time-independent Schrodinger equation will have at least one solution for any finite value of E. However, if the potential energy function F(r) confines a particle to a definite region of space, only certain values of E are consistent with Bom s restrictions on physically meaningful wavefunctions. All other values of E either make ip go to infinity somewhere or have some other unacceptable property such as a discontinuous first derivative with respect to position. This means that the possible energies for a bound particle are quantized, as illustrated in Fig. 2.1. The solutions to the Schrodinger equation for a free particle are not quantized in this way such a particle can have any energy above the threshold needed to set it free (Fig. 2.1). 

Once a potential surface has been obtained, a variety of dynamics methods can be used to determine reaction cross sections and rate constants. The "exact approach would involve solving the nuclear-motion Schrodinger equation for the appropriate scattering wave function. Because of the complexity associated with doing this, such exact solutions have been confined to the H4-H2 reaction. Much progress in the past year has been reported in the determination of approximate quantum solutions in three dimensions, so some of these applications are described in section II.B. Many more atom-diatom reactions have been treated exactly using one-dimensional models. 

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