Chapter 2

Application of Quantum Mechanics to Nanomaterial Structures

2.1 Introduction

Quantum mechanics can provide precise answers to many properties of nanomaterials, but it provides only the average value of individual measurements made on a given dynamical system in a certain initial state. One of the fundamental differences between classical mechanics and quantum theory is that quantum mechanics does not measure all variables with specific accuracy at the same time, while classical mechanics can do so. Another difference is that, classically, the effects of the disturbances due to the measurements can be exactly allowed in predicting the future behavior of the system, whereas quantum mechanically, the exact effects of the disturbances accompanying any measurements are inherently unknown. An example is that the measurement of the position of a particle introduces an unpredictable uncertainty regarding its momentum.

The development of quantum mechanics is made possible due to the failure of classical mechanics in explaining many phenomena. For example, the blackbody radiation from incandescent hot bodies was actually the first step in developing quantum mechanics. In 1901, Planck described the spectral intensity of blackbody radiation by assuming that oscillators at equilibrium with radiation can have certain discrete energy En, given by

where ω0 is the oscillator frequency ...