Quantum mechanics particle in a box

PROBLEM 5.2.8. Prove Eq. (5.2.17), assuming that a perfect gas PV = nRT = NfikBT is treated as a fiCE of CB with quantum-mechanical particle-in-a-box energies. 

We solved an equation of this form in Section 8.5, for two alternative sets of boundary conditions. The preceding solution of the Helmholtz equation in Cartesian coordinates is applicable to the Schrddinger equation for the quantum-mechanical particle-in-a-box problem. 

Figure 11.5 In contrast to the classical probability (a) of finding a particle distributed uniformly between x = 0 and L, the quantum mechanical particle in a box (b) has preferred locations, depending on n. The first row shows the wavefunction ip x) and the second row shows the probability distribution for the location of...

It is now necessary to examine the partition function in more detail. The energy states for translation are assumed to be given by the quantum-mechanical picture of a particle in a box. For a one-dimensional box of length a. 

The canonical ensemble is the name given to an ensemble for constant temperature, number of particles and volume. For our purposes Jf can be considered the same as the total energy, (p r ), which equals the sum of the kinetic energy (jT(p )) of the system, which depends upon the momenta of the particles, and the potential energy (T (r )), which depends upon tlie positions. The factor N arises from the indistinguishability of the particles and the factor is required to ensure that the partition function is equal to the quantum mechanical result for a particle in a box. A short discussion of some of the key results of statistical mechanics is provided in Appendix 6.1 and further details can be found in standard textbooks. 

Lx, Lr, and L , so that the volume of the box is LXLVL . The potential energy of the particle inside the box is zero but goes to infinity at the walls. The quantum mechanical solution in the. v direction to this particle in a box problem gives... 

Most students are introduced to quantum mechanics with the study of the famous problem of the particle in a box. While this problem is introduced primarily for pedagogical reasons, it has nevertheless some important applications. In particular, it is the basis for the derivation of the translational partition function for a gas (Section 10.8.1) and is employed as a model for certain problems in solid-state physics. 

The motion of activated complexes within the transition state may be analyzed in terms of classical or quantum mechanics. In terms of classical physics, motion along the reaction coordinate may be analyzed in terms of a onedimensional velocity distribution function. In terms of quantum mechanics, motion along the reaction coordinate within the limits of the transition state corresponds to the traditional quantum mechanical problem involving a particle in a box. 

Drexel undergraduate students in both the lecture and the laboratory of physical chemistry have b n using TKISolver for such calculations as least squares fitting of experimental data, van der Waals gas calculations, and quantum mechanical computations (plotting particle-in-a-box wavefunctions, atomic orbital electron densities, etc.). I use TKISolver in lectures (on a Macintosh with video output to a 25" monitor) to solve simple equations and plot functions of chemical interest. 

The theory assumes that the nuclei stay fixed on their lattice sites surrounded by the inner or core electrons whilst the outer or valence electrons travel freely through the solid. If we ignore the cores then the quantum mechanical description of the outer electrons becomes very simple. Taking just one of these electrons the problem becomes the well-known one of the particle in a box. We start by considering an electron in a one-dimensional solid. 

Atomic structure is fundamental to inorganic chemistry, perhaps more so even than organic chemistry because or the variety or elements and their electron configurations that must be dealt with. It will be assumed that readers will have brought with them from earlier courses some knowledge oT quantum mechanical concepts such as the wave equation, the particle-in-a-box. and atomic spectroscopy. 

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 other cause, the density effect, is especially important at high densities, where molecules are more or less confined to cells formed by their neighbors. In analogy to the well-known quantum mechanical problem of a particle in a box, the translational energies of such molecules are quantized, and this has an effect on the thermodynamic properties. In 1960 Levelt Sengers and Hurst [3] tried to describe the density quantum effect in term of the Lennard-Jones-Devonshire cell model, and in 1980 Hooper and Nordholm proposed a generalized van der Waals theory [4]. The disadvantage of both approaches is that, in the classical limit, they reduce to rather unsatisfactory equations of state. 

This size efiect is generally described by the quantum mechanics of a particle in a box . The electron and hole are limited by potential walls of small dimensions, which leads to a quantization of the energy levels. Therefore, particles of this small size are usually called Q-semiconductors (Q stands for quantized). The effect described above occurs when the size of the small particles comes in the order of the De Broglie wavelength of charge carriers, which is given by... 

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... 

Studies of electron solvation are popular with chemical physicists largely due to the perceived simplicity of the problem. The latter notion rests upon the mental picture of the solvated electron as a single quantum mechanical particle confined in a classical potential well a particle in a box. This picture was first suggested by Ogg in 1946 and subsequently elaborated by Cohen, Rice, Platzmann, Jortner, Castner, and many others. First such models were static, but... 

In contrast to the particle in a box and the harmonic oscillator, the hydrogen atom is a real physical system that can be treated exactly by quantum mechanics. In addition to their inherent significance. Ihe.se solutions suggest prototypes for atomic orbitals used in approximate treatments of complex atoms and molecules. 

The wave functions for a particle in a box illustrate another important principle of quantum mechanics the correspondence principle. We have already stated earlier (and will often repeat) that all successful physical theories must reproduce the explanations and predictions of the theories that preceded them on the length and mass scales for which they were developed. Figure 4.25 shows the probability density for the n = 5, 10, and 20 states of the particle in a box. Notice how the probability becomes essentially uniform across the box, and at m = 20 there is little evidence of quantization. The correspondence principle requires that the results of quantum mechanics reduce to those of classical mechanics for large values of the quantum numbers, in this case, n. 

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. 

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