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Journal of Tianshui Normal University

The Variation Method Leading to the Uncertain Relation of One-dimensional Harmonic Oscillator

LI Ru-heng

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

Based on the general form of the uncertainty relation, using the variation method, the paper here deduces the equality form uncertainty relation which with quantum number of one- dimensional.Introducing such an easy and clear matter to the beginners,will effectively deepening their comprehension on the uncertainty relation, and avoid forming the wrong idea that the uncertainty relation could only express into inequality form.

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Results in Physics•Jan 1, 2016•7 citations
Based on the general form of the uncertainty relation, using the variation method, the paper here deduces the equality form uncertainty relation which with quantum number of one- dimensional.Introducing such an easy and clear matter to the beginners,will effectively deepening their comprehension on the uncertainty relation, and avoid forming the wrong idea that the uncertainty relation could only express into inequality form.
In this paper,the uncertainty in positon,the momentum probability amplitude and momentum probability density's recurrence formulas,the uncertainty in momentum and the uncertain relation of the equality form momentum-position of one-dimensional harmonic oscillator were deduced.This paper also pointed out that the lower limit of one-dimensional harmonic oscillator's ΔpΔx is exactly the lower limit of /2 which given by the general form of the uncertain relation,however the lower limit of one-dimensional infinite well's ΔpΔx is a little bigger than /2,so the lower limit of a quantum system's ΔpΔx is needn't equal to /2.
The concept of harmonic oscillator particularly one dimensional (1-D) is mentioned in literature repeatedly and is explained in more complex manner by using various methods. This creates difficulties in understanding the description of the concept for new learners. The purpose of this article was to enlighten different methods to formulate harmonic oscillator in improving knowledge about detailed steps to derive eigen energy values in more comprehensible manner for the beginners. The energy values are derived by using classical method, quantum mechanically, Schrodinger time independent equation, perturbation technique, variation method, WKB approximation etc. A coherent way of derivation of eigen values using various approaches makes this article as unique.
The Heisenberg uncertainty relation, ΔxΔpx ⩾ ℏ/2 (ℏ ≡ h/2π), is well known. The purpose of this paper is to study a linkage between macroscopic thermodynamic quantities and the uncertainty relation. To simplify matters, we consider one-dimensional harmonic oscillator model. The calculations are carried out by two steps. First, the thermal average of the uncertainty relation is obtained for canonical system in thermal equilibrium at temperature T, which is denoted by (Δx)T(Δpx)T. Second, the thermodynamic quantities are expressed analytically as a function of (Δx)T(Δpx)T within the harmonic oscillator model. Finally, a connection between the energy fluctuations and the uncertainty relation is clarified for the Einstein’s heat capacity model. The analysis is made on the basis of fluctuation theory. The uncertainty relation is valid not only in quantum state but also at finite temperature.