vendredi 28 décembre 2018

The page of Prof Dr. Robert Gilmore

Education

B.S.	Physics	MIT	1962
B.S.	Mathematics	MIT	1962
Ph. D.	Physics	MIT	1967

Contact Info

Physics Department, Drexel University, Philadelphia, PA 19104 USA
Phone: (215) 895-2779; FAX: (215) 895-5934
e-mails: robert.gilmore [at] drexel [dot] edu; bob [at] newton [dot] physics [dot] drexel [dot] edu; bob [at] bach [dot] physics [dot] drexel [dot] edu

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Current Book: The Symmetry of Chaos

New Book: Geometry and Lie Groups

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samedi 22 décembre 2018

A Memoir on the Theory of Matrices

Author(s): Arthur Cayley

Source: Philosophical Transactions of the Royal Society of London, Vol. 148 (1858), pp. 17- 37
Published by: Royal Society

Stable URL: https://www.jstor.org/stable/108649
Accessed: 22-12-2018 10:44 UTC

Received December 10, 1857,-Read January 14, 1858.

The term matrix might be used in a more general sense, but in the present memoir I consider only square and rectangular matrices, and. the term matrix used without quali- fication is to be understood as meaning a square matrix; in this restricted sense,' a set of quantities arranged in the form of a square,

To download the article click on the link below:

https://www.jstor.org/stable/pdf/108649.pdf

THE THEORY OF THE ANHARMONIC OSCILLATOR

BY K. S. VISWANATHAN, F.A.Sc.

(Memoir No. 97 of the Raman Research Institute, Bangalore-6)

Received May 22, 1957

l. INTRODUCTION

ANHARMONICITY of vibration plays an important role in several branches of physics. Its importance was first recognised in acousticsa; the presence of overtones of the fundamental mode of a tuning fork and the alteration of its pitch with intensity changes were successfully accounted for as the effects of the anharmonicity of the oscillator. More recently, the subject has acquired a fresh interest in relation to the subject of molecular spectroscopy. 2 The thermal expansion '~ of crystals also owes its origin to the anharmonic nature of the vibrations inside a crystal lattice. It is the object of the present note to give an exact treatment of the classical problem of the anharmonic oscillator and also to draw attention to certain of its features which by an appeal to the Correspondence Principle lead to the results of wave mechanics, in Section 2, exact expressions have been given for the shift in the mean position of the oscillator from the origin, its frequency and the amplitudes of the different harmonics, in section 4, the eigenvalues of the oscillator are determined by using the W.K.B. method. It is shown that these are the solutions of a transcendental equation involving elliptic functions and that the first correction to any energy level of the system from its harmonic oscillator value is identical with the one obtained from the perturbation theory.

To download the article click on the link below:

https://www.ias.ac.in/article/fulltext/seca/046/03/0203-0217

Classic anharmonic oscillator

In classical mechanics, anharmonicity is the deviation of a system from being a harmonic oscillator. An oscillator that is not oscillating in harmonic motion is known as an anharmonic oscillator where the system can be approximated to a harmonic oscillator and the anharmonicity can be calculated using perturbation theory. If the anharmonicity is large, then other numerical techniques have to be used.

As a result, oscillations with frequencies

2\omega

and

3\omega

etc., where

\omega

is the fundamental frequency of the oscillator, appear. Furthermore, the frequency

\omega

deviates from the frequency

\omega _{0}

of the harmonic oscillations. As a first approximation, the frequency shift

\Delta \omega =\omega -\omega _{0}

is proportional to the square of the oscillation amplitude

A

\Delta \omega \propto A^{2}

In a system of oscillators with natural frequencies

\omega _{\alpha }

\omega _{\beta }

, ... anharmonicity results in additional oscillations with frequencies

\omega _{\alpha }\pm \omega _{\beta }

Anharmonicity also modifies the energy profile of the resonance curve, leading to interesting phenomena such as the foldover effect and superharmonic resonance.

General principle[edit]

2 DOF elastic pendulum exhibiting anharmonic behavior.

Harmonic vs. Anharmonic Oscillators

The "block-on-a-spring" is a classic example of harmonic oscillation. Depending on the block's location, x, it will experience a restoring force toward the middle. The restoring force is proportional to x, so the system exhibits simple harmonic motion.

A pendulum is a simple anharmonic oscillator. Depending on the mass's angular position θ, a restoring force pushes coordinate θ back towards the middle. This oscillator is anharmonic because the restoring force is not proportional to θ, but to sin(θ). Because the linear function y=θapproximates the nonlinear function y=sin(θ) when θ is small, the system can be modeled as a harmonic oscillator for small oscillations.

An oscillator is a physical system characterized by periodic motion, such as a pendulum, tuning fork, or vibrating diatomic molecule. Mathematically speaking, the essential feature of an oscillator is that for some coordinate x of the system, a force whose magnitude depends on x will push x away from extremes values and back toward some central value x₀, causing x to oscillate between extremes. For example, x may represent the angular position of a pendulum. When x is too positive or too negative, gravity pushes it back towards its lowest point.

In harmonic oscillators, the restoring force is proportional in magnitude (and opposite in direction) to the displacement of x from its natural position x₀. The resulting differential equation implies that x must oscillate sinusoidally over time, with a period of oscillation that is inherent to the system. x may oscillate with any amplitude, but will always have the same period.

Anharmonic oscillators, however, are characterized by the nonlinear dependence of the restorative force on the displacement x. Consequently, the anharmonic oscillator's period of oscillation may depend on its amplitude of oscillation.

As a result of the nonlinearity of anharmonic oscillators, the vibration frequency can change, depending upon the system's displacement. These changes in the vibration frequency result in energy being coupled from the fundamental vibration frequency to other frequencies through a process known as parametric coupling.^{[clarification needed]}

Treating the nonlinear restorative force as a function F(x-x₀) of the displacement of x from its natural position, we may replace F by its linear approximation F₁=F'(0)*(x-x₀) at zero displacement. The approximating function F₁ is linear, so it will describe simple harmonic motion. Further, this function F₁ is accurate when x-x₀ is small. For this reason, anharmonic motion can be approximated as harmonic motion as long as the oscillations are small.

Examples in physics[edit]

There are many systems throughout the physical world that can be modeled as anharmonic oscillators in addition to the nonlinear mass-spring system. For example, an atom, which consists of a positively charged nucleus surrounded by a negatively charged electronic cloud, experiences a displacement between the center of mass of the nucleus and the electronic cloud when an electric field is present. The amount of that displacement, called the electric dipole moment, is related linearly to the applied field for small fields, but as the magnitude of the field is increased, the field-dipole moment relationship becomes nonlinear, just as in the mechanical system.

Further examples of anharmonic oscillators include the large-angle pendulum; nonequilibrium semiconductors that possess a large hot carrier population, which exhibit nonlinear behaviors of various types related to the effective mass of the carriers; and ionospheric plasmas, which also exhibit nonlinear behavior based on the anharmonicity of the plasma. In fact, virtually all oscillators become anharmonic when their pump amplitude increases beyond some threshold, and as a result it is necessary to use nonlinear equations of motion to describe their behavior.

Anharmonicity plays a role in lattice and molecular vibrations, in quantum oscillations ^[1], and in acoustics. The atoms in a molecule or a solid vibrate about their equilibrium positions. When these vibrations have small amplitudes they can be described by harmonic oscillators. However, when the vibrational amplitudes are large, for example at high temperatures, anharmonicity becomes important. An example of the effects of anharmonicity is the thermal expansion of solids, which is usually studied within the quasi-harmonic approximation. Studying vibrating anharmonic systems using quantum mechanics is a computationally demanding task because anharmonicity not only makes the potential experienced by each oscillator more complicated, but also introduces coupling between the oscillators. It is possible to use first-principles methods such as density-functional theory to map the anharmonic potential experienced by the atoms in both molecules^[2] and solids.^[3] Accurate anharmonic vibrational energies can then be obtained by solving the anharmonic vibrational equations for the atoms within a mean-field theory. Finally, it is possible to use Møller–Plesset perturbation theory to go beyond the mean-field formalism.

Potential energy from period of oscillations[edit]

Let us consider a potential well

U(x)

. Assuming that the curve

U=U(x)

is symmetric about the

U

-axis, the shape of the curve can be implicitly determined from the period

T(E)

of the oscillations of particles with energy

E

according to the formula:^{[citation needed]}

U(x)={\frac {1}{2\pi {\sqrt {2m}}}}\int _{0}^{U}{\frac {T(E)\,dE}{\sqrt {U(x)-E}}}

Conversely the oscillation period may be derived ^[4]

T={\sqrt {2m}}\int _{x_{-}}^{x_{+}}{\frac {dx}{\sqrt {E-U}}}

Reference: https://en.wikipedia.org/wiki/Anharmonicity

The Quantum Harmonic Oscillator

C. David Sherrill
School of Chemistry and Biochemistry
Georgia Institute of Technology

Introduction

The harmonic oscillator is extremely useful in chemistry as a model for the vibrational motion in a diatomic molecule. Polyatomic molecules can be modeled by coupled harmonic oscillators. The atoms are viewed as point masses which are connected by bonds which act (approximately) like springs obeying Hooke's law. In these notes we will review the classical harmonic oscillator problem and then discuss the quantum harmonic oscillator.

Reference: http://vergil.chemistry.gatech.edu/notes/ho/ho.html

Classical harmonic oscillator

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:

{\vec {F}}=-k{\vec {x}},

where k is a positive constant.

If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).

If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:

Oscillate with a frequency lower than in the undamped case, and an amplitude decreasing with time (underdamped oscillator).
Decay to the equilibrium position, without oscillations (overdamped oscillator).

The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called critically damped.

If an external time-dependent force is present, the harmonic oscillator is described as a driven oscillator.

Mechanical examples include pendulums (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.

Simple harmonic oscillator[edit]

Mass-spring harmonic oscillator

Simple harmonic motion

A simple harmonic oscillator is an oscillator that is neither driven nor damped. It consists of a mass m, which experiences a single force F, which pulls the mass in the direction of the point x = 0 and depends only on the mass's position x and a constant k. Balance of forces (Newton's second law) for the system is

F=ma=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=m{\ddot {x}}=-kx.

Solving this differential equation, we find that the motion is described by the function

x(t)=A\cos(\omega t+\varphi ),

where

\omega ={\sqrt {\frac {k}{m}}}={\frac {2\pi }{T}}.

The motion is periodic, repeating itself in a sinusoidal fashion with constant amplitude A. In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its period T, the time for a single oscillation or its frequency f = 1/T, the number of cycles per unit time. The position at a given time t also depends on the phase φ, which determines the starting point on the sine wave. The period and frequency are determined by the size of the mass m and the force constant k, while the amplitude and phase are determined by the starting position and velocity.

The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position, but with shifted phases. The velocity is maximal for zero displacement, while the acceleration is in the direction opposite to the displacement.

The potential energy stored in a simple harmonic oscillator at position x is