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Journal of Hubei University of Education

Classics Harmonic Oscillator and Quantum Harmonic Oscillator

Ming Liu

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

This essay discusses the Linear Harnonic Oscillator from two different aspects: Classics and Quantum.It also discusses Zero—point energy by using the uncertainty relation.

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In classical physics, for a harmonic oscillator with a certain energy, position and momentum oscillate in time sinusoidally. However, in a Quantum harmonic oscillator, the expectation of position and momentum operators at a given energy level (En) are both zero. So the two oscillators have different dynamics. However, under certain Quantum state, called Coherent state, the Quantum HO exhibits prominent classical behavior. The goal of my thesis is to find analytical solution of this state and discuss certain of its features.
Abstract We consider the time-independent Schrödinger equation for the harmonic oscillator potential and determine its bound states and energy levels in one spatial dimension using two approaches: the algebraic and the analytic method. Using the former, we introduce ladder operators: the annihilation and creation operators, as well as the occupation-number operator, and determine the ground state of the harmonic oscillator to be a Gaussian wave packet, while the latter method yields the general bound state solutions in terms of the Hermite polynomials. We further discuss the zero-point energy and uncertainty relation for the quantum harmonic oscillator and make a comparison to the classical harmonic oscillator. Finally, we examine the three-dimensional harmonic oscillator, which leads us to the description of systems with multiple degrees of freedom via the tensor product
The quantum harmonic oscillator is a quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Coherent state is physically interesting because it is a minimum uncertainty state at which the Heisenberg uncertainty relation reaches its minimum. If a charged particle of charge moves in an electromagnetic field, there is a Lorentz force acting on the particle. The electric and magnetic are invariant under the gauge transformation. In the presence of an external vector potential, the Hamiltonian is changed, so is the probability current operator.
The harmonic oscillator is defined by a quadratic potential. Many important problems in physics reduce to that of a quantum harmonic oscillator because the leading term of a potential near its minimum is quadratic. It is a general and noteworthy feature of quantum systems that the lowest energy is not zero. Calculate the probability that the quantum oscillator is found beyond the classical limit. Agreement with the uncertainty principle may be checked by calculating the expectation values of x2 and p2 and then taking the square root of the product. For large quantum numbers, the solution must approach that of the classical harmonic oscillator.