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Journal of Chongqing Three-gorges University

Uncertain Relation and Energy Level Width of Linear Oscillator in Mesoscopic System

Xu Gang

Published Jan 1, 2007• 0 citations
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Abstract

Applying the theory on quasi-classical particles and the uncertainty relations of Quantum Mechanics,we get the formula of the energy uncertainty of linear oscillator,applying the formula into mesoscopic system,we get the energy level width of linear oscillator and point out the minimum of energy level width is the zero energy,at the same time we exactly obtain the uncertain relation of linear oscillator.

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•Jul 20, 1993•0 citations
Applying the theory on quasi-classical particles and the uncertainty relations of Quantum Mechanics,we get the formula of the energy uncertainty of linear oscillator,applying the formula into mesoscopic system,we get the energy level width of linear oscillator and point out the minimum of energy level width is the zero energy,at the same time we exactly obtain the uncertain relation of linear oscillator.
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 quantum wavefunctions and the corresponding energy levels of a RLC (Resistance-Inductance-Capacity) electric circuit are obtained by using canonical quantization method and unitary transformation from the classical equation of motion. The quantum fluctuations of charge and current in an arbitrary eigenstate of the system have also been given as well as the uncertainty relation. It is showed that even if at 0 K charge and current in the circuit exhibit quantum fluctuations, which originates from fluctuations of zero point vibrations of the system.
The time dependent harmonic oscillator is solved explicitly for quantum mechanics by the operator method with an auxiliary condition as the classical solution. Two classical invariant quantities which determine whether or not the system is bound are derived by the classical equation of motion. We obtain the invariant operator from one classical invariant quantity. Its eigenfunction is related to the solution of Schrodinger equation of the system and its eigenvalue is related to another classical quantity. The wave function is evaluated exactly by the eigenfunction of the invariant operator but it is not the eigenfunction of the Hamiltonian of the system. The uncertainty which calculates with the wave function is not a minimum one. We will confirm that the function which holds minimum uncertainty is a eigenfunction of the Hamiltonian.