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CRC Press eBooks

Quantum Harmonic Oscillator

James W. Rohlf

Published Oct 18, 2023• 1 citation
Publisher

Abstract

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.

DOI
10.1201/9781003395515-7

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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.
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 first application of quantum ideas to a system with a nonuniform potential energy is to the harmonic oscillator: a pendulum, or a mass on a spring. The result is a quantization of energy with equally spaced energy levels. The wave function is localized in the potential well but can do quantum tunneling beyond the turning points into classically forbidden regions. A wave function can be a negative quantity, but the related position probability is always positive. The wave function shows the particle is not in any single place; in a sense it is in many places simultaneously. The fact that the lowest energy quantum state does not vanish is an example of zero-point energy. States of very high energy show a strong analogy to classical oscillator behavior, an example of the Bohr correspondence principle.
Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.
A single closed-form expression is obtained for the probability amplitude ψ(x,t) for a harmonic oscillator whose initial state ψ(x,0) has a minimum value for the uncertainty product of position x and momentum p, Δx Δp=ℏ/2. This form describes both coherent and squeezed states. Oscillations in time for uncertainties in position and momentum, characteristic of squeezed states, are shown to follow directly from the derived form of ψ(x,t) and from Heisenberg equations of motion for operators x̂ and p̂. Expressions for average photon number and its uncertainty are obtained from calculations of average oscillator energy and fluctuations about this energy.