Particle-in-a-box model and

Particle-in-a-box models and the qnantnm harmonic oscillator illustrate a number of important features of quantum mechanics. The energy level structure depends on the natnre of the potential in the particle in a box, E n, whereas for the harmonic oscillator, E n. The probability distributions in both cases are different than for the classical analogs. The most probable location for the particle-in-a-box model in its gronnd state is the center of the box, rather than uniform over the box as predicted by classical mechanics. The most probable position for the quantum harmonic oscillator in the ground state is at its equilibrium position, whereas the classical harmonic oscillator is most likely to be fonnd at the two classical turning points. Normalization ensures that the probabilities of finding the particle or the oscillator at all positions add np to one. Finally, for large values of n, the probability distribution looks mnch more classical, in accordance with the correspondence principle. 

Excitation at the HOMO-LUMO gap produces an electron-hole pair, whose energy is modified by the Coulomb interaction between them. In previous tight-binding calculations this Coulomb correction was estimated from the particle in a box model and then artificially added to the tight-binding result [24], The simplicity of the tight-binding model used here... 

The increase in entropy of a substance as its temperature is raised (Eq. 2 and Table 7.2) can also be interpreted in terms of the Boltzmann formula. We shall use the same particle in a box model of a gas, but this reasoning also applies to liquids and solids, even though their energy levels are much more complicated. At low temperatures, the molecules of a gas can occupy only a few of the energy levels, so W is small and the entropy is low. As the temperature is raised, the molecules have access to... 

The energy with n = 1 is 10 eV, and the next allowed one (n = 2) about 30 eV higher. These are typical energy magnitudes for electrons in atoms and molecules. Of course, the particle-in-a-box model is ridiculously oversimplified to apply seriously here (although it has been used to represent it electrons in systems with conjugated double bonds see Problem 5 below). It does, however, confirm our expectation that the quantum theory must be used for problems involving electrons in atoms and molecules. 

Using a particle in a box model with an infinite potential drop at the wall as the boundary condition, and taking into account that the exciton consists of an electron-hole pair, the Schrodinger equation can be solved yielding the energy of the lowest excited state (Brus 1983, 1984 Rossetti et al. 1984), i.e., the lower edge of the conduction band, as... 

The behavior of the polarizability of the n-alkanes may be rationalized with reference to the simple particle-in-a-box model. In this model, the polarizability increases as the fourth power in the length of the box, which implies that the polarizability per unit length (i.e., the differential polarizability) increases as the third power in the length of the box. Obviously, in the n-alkanes, the electrons do not at all behave according to this simple model, but we may still attribute the increasing differential polarizability of the n-alkanes to a more pronounced delocalization of the electrons in the longer chains, as the electrons become more loosely attached to the system. As the chain grows, this effect becomes less important and the differential polarizability becomes constant. 

Energy quantization arises for all systems that are confined by a potential. The one-dimensional particle-in-a-box model shows why quantization 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 observe for systems that contain more than a few hnndred atoms or so. 

We present quantitative, computer-generated plots of the solutions to the particle-in-a-box models in two and three dimensions and use these examples to introduce contour plots and three-dimensional isosurfaces as tools for visual representation of wave functions. We show our students how to obtain physical insight into quantum behavior from these plots without relying on equations. In the succeeding chapters we expect them to use this skill repeatedly to interpret quantitative plots for more complex cases. 

Other things equal, molar entropy increases with molar mass. Heavier species have more vibrational energy levels available to them than lighter ones, so W and S are larger. (See text Fig. 7.13 for use of particle-in-a-box model to understand this better.)... 

If we wish to predict the absorption spectrum of a molecule, we must know the energy levels of the molecule. Sadly, the hydrogen atom is the only real atomic/molecular system for which an analytic solution is known. Luckily for us, for the proper choice of molecule, some of the simpler quantum mechanical models are valid. I guess that means we must select the molecule to fit the theory But our purpose here is to develop a case study, so we ll accept that and apply the one-dimensional particle-in-a-box model to a... 

Many models have been presented to explain quantitatively the dependence of exciton energy on the cluster size [7, 11, 21-30]. This problem was first treated by Efros et al. [7], who considered a simple particle in a box model. This model assumes that the energy band is parabolic in shape, equivalent to the so-called effective mass approximation. The shift in absorption threshold, AE, is dependent upon the value of the cluster radius, R, Bohr radius of the electron, ae ( = h2v.jmce2), and Bohr radius of the hole, ah (= h2e/mhe2). When (1) R ah and R ae, and (2) ah R ac,... 

The important physical properties of simple metals and, in particular, the alkali metals can be understood in terms of a free electron model in which the most weakly bound electrons of the constituent atoms move freely throughout the volume of the metal (231). This is analogous to the free electron model for conjugated systems (365) where the electrons are assumed to be free to move along the bonds throughout the system under a potential field which is, in a first approximation, constant (the particle-in-a-box model). The free electron approach can be improved by replaeing the constant potential with a periodic potential to represent discrete atoms in the chain (365). This corresponds to the nearly free electron model (231) for treating electrons in a metal. 

Apply the particle-in-a-box model to electrons in onedimensional semiconductor quantum wells and to n electrons in conjugated molecules... 

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