Energy levels particle in a box

The particle-in-a-box energy levels can be used to predict the qualitative behavior of an electron trapped in a spherical cavity of radius r. The relevant equation from Section 1.7 is now... 

We know the particle-in-a-box energy levels are very close together when the dimension L of the... 

The more sophisticated—and more general—way of finding the energy levels of a particle in a box is to use calculus to solve the Schrodinger equation. First, we note that the potential energy of the particle is zero everywhere inside the box so V(x) = 0, and the equation that we have to solve is... 

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. We can therefore expect the electron s wavefunctions to obey certain boundary conditions, like the constraints we encountered when fitting a wave between the walls of a container. As we saw for a particle in a box, these constraints result in the quantization of energy and the existence of discrete energy levels. Even at this early stage, we can expect the electron to be confined to certain energies, just as spectroscopy requires. 

An ideal gas consists of a large number of molecules that occupy the energy levels characteristic of a particle in a box. For simplicity, we consider a one-dimensional box (Fig. 7.9a), but the same considerations apply to a real three-dimensional container of any shape. At T = 0, only the lowest energy level is occupied so W = 1 and the entropy is zero. There is no disorder, because we know which state each molecule occupies. 

FIGURE 7.9 The energy levels of a particle in a box (a) become closer together as the width of the box is increased, (b) As a result, the number of levels accessible to the particles in the box increases, and the entropy of the system increases accordingly. Die range of thermally accessible levels is shown by the tinted band. The change from part (a) to part (b) is a model of the isothermal expansion of an ideal gas. The total energy of the particles is the same in each case. 

The time-dependent Schrddinger equation (2.30) for the particle in a box has an infinite set of solutions tpn(x) given by equation (2.40). The first four wave functions tpn(x) for = 1, 2, 3, and 4 and their corresponding probability densities ip (x) are shown in Figure 2.2. The wave function ipiix) corresponding to the lowest energy level Ei is called the ground state. The other wave functions are called excited states. 

A very crude model to calculate the increase in bandgap energy is the effective-mass particle-in-a-box approximation. Assuming parabolic bands and infinitely high barriers the lowest conduction band (CB) level of a quantum wire with a square cross-section of side length w is shifted by AEC compared to the value Ec of the bulk crystal [Lei, Ho3] ... 

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... 

We shall need to know how to evaluate these separated partition functions. The translational energy levels can be derived from the quantum mechanical solution for a particle in a box they are so closely spaced that the partition function can be evaluated in closed form by integration, and has the value... 

The three-dimensional particle in a box corresponds to the real life problem of gas molecules in a container, and is also sometimes used as a first approximation for the free conduction electrons in a metal. As we found for one dimension (Section 2.3), the allowed energy levels are extremely closely spaced in macroscopically sized boxes. For many purposes they can be regarded as a continuum, with no discernible energy gaps. Nevertheless, there are problems, for example in the theory of metals and in the calculation of thermodynamic properties of gases, where it is essential to take note of the existence of discrete quantized levels, rather than a true continuum. 

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