Nano-Mechanical Oscillator and Basic Dynamics: Part I

Duncan G. Steel

Published Apr 30, 2021• 0 citations

Abstract

Abstract This discussion introduces the student to the reality, in quantum technology, that analysis of any problem necessarily begins with the Hamiltonian representing the system. The quantum Hamiltonian represents the total energy of the system, the sum of kinetic energy plus potential energy, written in canonical coordinates and conjugate momenta, and where these variables become time independent quantum operators. The nature of the potential energy for the nano-vibrator, following Hooke’s law, serves to localize the particle. The relevance of the nano-vibrator Hamiltonian—sometimes called the harmonic oscillator Hamiltonian—is perhaps one of the most important Hamiltonians in quantum systems. Not only can it be extended to cover things like phonons in solids, vibrations in molecules, and the behavior of bosons, but it is also the basis for leading to the concept of a photon, the quantum radiation field, and the quantum vacuum. This chapter provides the basic introduction for vibration of a particle or a nano-rod and looks at the wave-like behavior that emerges from the solution to the time independent Schrödinger equation. When we include the time evolution, we can observe dynamical behavior and begin to examine the meaning of quantum measurement.

DOI: 10.1093/oso/9780192895073.003.0002

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