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Black-body Radiation, Einstein and Planck's Law
Published in Caio Lima Firme, Quantum Mechanics, 2022
A harmonic oscillator is a system that is displaced from its equilibrium position experiencing a restoring force proportional to its displacement having sinusoidal oscillations about the equilibrium point. Simple mechanics examples are pendulums and masses connected to springs. The light has two sinusoidal oscillating fields (electric and magnetic fields) and it is a two-coupled harmonic oscillators. In Section 4 of the Chapter four we presented the solution for the classical harmonic oscillator. As we observed in Chapter five, it is probable that the Lorentz’s theory (an electron is harmonically bound to the nucleus of the atom, following the equation of motion of a harmonic oscillator) used to elucidate the Zeeman effect was the basis for Planck’s theory on the thermal black body.
Applications of the Formalism-II
Published in Shabnam Siddiqui, Quantum Mechanics, 2018
A harmonic oscillator is a system that exhibits simple harmonic motion or periodic motion, a motion that repeats itself after equal intervals of time. Such a motion is caused by a restoring force, which is proportional to displacement of the oscillator and acts in a direction opposite to the displacement. Many systems such as spring, simple pendulum, vibrating string and molecular vibration can be approximated as simple harmonic oscillators. In classical physics, such systems are well understood. Therefore, the most natural question to ask is, what is a quantum mechanical simple harmonic motion? What are its properties? Its study is key to understanding the vibration of individual atoms in molecules and crystals. It is also very important for understanding particle properties of an electromagnetic wave. In this chapter, first we solve the Schrodinger equation for a harmonic oscillator using two methods, analytical method and algebraic method. Later, we discuss the quantum properties of the harmonic oscillator.
Linear modal theory and damping
Published in Ihor Raynovskyy, Alexander Timokha, Sloshing in Upright Circular Containers, 2020
Ihor Raynovskyy, Alexander Timokha
Linear vibrating systems whose amplitude decreases over time are frequently called damped harmonic oscillators. Since nearly all physical systems involve considerations such as friction, air resistance, pollution (non-ideal nature/system), intermolecular forces, etc., where energy in the system is lost to heat or sound, accounting for the damping is important. Examples of the damped harmonic oscillators include any real oscillatory system like the clock pendulum, liquid sloshing in a cup of coffee or guitar string: after shaking the cup or guitar string vibrating, their amplitude slows down and stops over time.
Design and analysis of computer experiment via dimensional analysis
Published in Quality Engineering, 2018
Weijie Shen, Dennis K. J. Lin, Chia-Jung Chang
In the classic mechanics, a damped harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F which is proportional to the displacement x, and a frictional force f, proportional to the velocity v. Harmonic oscillators are very important in physics because any object in stable equilibrium acts like a harmonic oscillator for small vibrations. Extensive researches have been done associated with damped harmonic oscillator in mechanical engineering, control engineering, structural engineering, electrical engineering, among others. Reference of computer-aided experiment on damped harmonic oscillator could be found in McInerney (1985).