Particle in a box model

The above particle in a box model for motion in two dimensions can obviously be extended to three dimensions or to one. 

This simple particle-in-a-box model does not yield orbital energies that relate to ionization energies unless the potential inside the box is specified. Choosing the value of this potential Vo such that Vo + 7t2 h2/2m [ 52/L2] is equal to minus the lowest ionization energy of the 1,3,5,7-nonatetraene radical, gives energy levels (as E = Vo + Jt2 h2/2m [ n2/L2]) which then are approximations to ionization energies. 

Atomic orbitals may be combined to form molecular orbitals. In such orbitals, there is a nonzero probability of finding an electron on any of the atoms that contribute to that molecular orbital. Consider an electron that is confined in a molecular orbital that extends over two adjacent carbon atoms. The electron can move freely between the two atoms. The C-C distance is 139 pm. (a) Using the particle in a box model, calculate the energy required to promote an electron from the n = 1 to n = 2 level assuming that the length of the box is equal to the distance between two carbon atoms, (b) To what wavelength of radiation does this correspond (c) Repeat the calculation for a linear chain of 1000 carbon atoms. 

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 estimate of eq. A.18 can be obtained from the expression hw = h2/Ma (particle-in-a-box model) or from the relation hw - (harmonic oscillator model). Since the frequency... 

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. 

A rough estimate of the optical excitation energy can be obtained from the particle in a box model (18). The Is - 2p transition energy, Eu 2p, is roughly given by... 

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. 

One of the present authors [7] used caterpillar trees to model Wreath product groups while Balasubramanian [8] used a particle in-a box model to represent the same type of permutation ... 

The particle in a box model can be used to illustrate many of the techniques of quantum mechanics in chemistry. It is also of some use in predicting the absorption spectra of delocalised systems such as hexatriene. For the particle in a box model the true ground state energy is given by... 

The formula for the ground state energy for the particle in a box model was met in Chapter 13. In fact there is a series of energy levels given by the general formula... 

A possible wavefunction 4 for the first excited state in the particle in a box model, which we have met on several previous occasions, is... 

These are the energy levels predicted by the particle in a box model for any particle in a one-dimensional box of length a. The energy levels are quantized according to quantum numbers n = 1, 2, 3,... 

The details of several steps in the particle in a box model in this chapter have been omitted. Work out the details of the following steps ... 

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. 

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. 

We see in part (c) that the frequency of light absorbed should be inversely proportional to the length of the chain. Short-chain conjugated molecules absorb in the ultraviolet, whereas longer chain molecules absorb in the visible. This qualitative trend is predicted by the simple particle-in-a-box model. Later chapters detail how these results can be improved. 

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. 

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