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Morgan & Claypool Publishers eBooks

Translation and vibration

Ajit J. Thakkar

Published Jun 1, 2014• 0 citations
Publisher

Abstract

Mathjax On | Off Abstract The postulates of quantum mechanics can be understood by considering a simple one-dimensional model of the translational motion of a particle in a wire.
DOI
10.1088/978-1-627-05416-4ch3

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The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.
This chapter considers the quantum mechanics of translation and vibration. Both types of motion can be solved in certain cases, and both are important not only in their own right but also because they form a basis for the description of more complicated types of motion. It mentions translational motion, which has the advantage of introducing in a simple way many of the striking features of quantum mechanics. The chapter describes certain features of wavefunctions that are common to all the problems. The combination of features of wavefunctions with the solution of the Schrödinger equation results in one of the most characteristic features of quantum mechanics, the quantization of energy.
Abstract The quantum equilibrium distribution, ϱQm, of an arbitrary number, m, of momentum or displacement components is determined for atoms that are part of a polyatomic molecule or a solid. This is shown to be a multidimensional Gaussian distribution. Two cases are considered: (1) the motion of the system as a whole is given, (2) it is in itself determined by the statistical equilibrium conditions. In the first case we obtain distributions for the vibrational momenta and displacements and in the second for the total momenta, including the momenta of vibrational, translational and rotational motions. The distributions for momenta and displacements of one particle and for the maximum number of linearly independent components of momenta and displacements of all particles of the system are considered as particular cases. It is shown that the averaging of any function, Fm, depending on an arbitrary number, m, of components of displacements or momenta of particles, over the canonical ensemble is reduced to the integration of this function weighted by ϱQm over all its arguments between infinite limits.