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Uncertainty Principle and the Zero-Point Energy of the Harmonic Oscillator

R. A. Newing

NatureSep 7, 19352 citations

The Momentum Probability Amplitude and the Equality Form Uncertain Relation of One-dimensional Harmonic Oscillator

Luo Ling-xiao

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.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.

Journal of Xuchang UniversityJan 1, 20040 citations

Energy Levels of Harmonic Oscillator in Classical-Like Formulation of Quantum Mechanics

Vladimir I. Man’ko

Jan 1, 19971 citation

Local States in One-Dimensional Symmetrical Quantum Systems

Jürgen Brickmann

Abstract Local quantum states, which play an important role in quantum dynamical treatments, are expanded analytically with respect to a basis of eigen functions of a symmetrical Hamiltonian ℋ̂(x) = ℋ̂(- x). Exact local states (ELS) in one-dimensional symmetrical quantum systems are therein defined as quantum states which are local eigenstates of the Hamiltonian ℋ̂(x) on one half space ℝ+ or ℝ- and are identically equal to zero on the other half space. Local properties like the projection operator on one half space can be given in terms of ELS-basis, but it is shown that the energy moments 〈(〈ℋ̂ 〉 - 〈ℋ̂)k〉 with respect to the ELS do not converge. Consequently, if one uses the ELS as quasistationary initial states, as has been done recently by some authors [5], the lifetimes of these states cannot be estimated from time energy uncertainty relation using the second energy moment as an energy uncertainty measure. A harmonic oscillator system and a symmetrical double oscillator are treated as examples.Abstract Local quantum states, which play an important role in quantum dynamical treatments, are expanded analytically with respect to a basis of eigen functions of a symmetrical Hamiltonian ℋ̂(x) = ℋ̂(- x). Exact local states (ELS) in one-dimensional symmetrical quantum systems are therein defined as quantum states which are local eigenstates of the Hamiltonian ℋ̂(x) on one half space ℝ+ or ℝ- and are identically equal to zero on the other half space. Local properties like the projection operator on one half space can be given in terms of ELS-basis, but it is shown that the energy moments 〈(〈ℋ̂ 〉 - 〈ℋ̂)k〉 with respect to the ELS do not converge. Consequently, if one uses the ELS as quasistationary initial states, as has been done recently by some authors [5], the lifetimes of these states cannot be estimated from time energy uncertainty relation using the second energy moment as an energy uncertainty measure. A harmonic oscillator system and a symmetrical double oscillator are treated as examples.

Zeitschrift für Naturforschung ADec 1, 19791 citation

Path integral for the harmonic oscillator in one dimension with nonzero minimum position uncertainty

Khireddine Nouicer

Physics Letters AFeb 10, 200617 citations

Damping the zero-point energy of a harmonic oscillator

T. G. Philbin, S. A. R. Horsley

The physics of quantum electromagnetism in an absorbing medium is that of a field of damped harmonic oscillators. Yet until recently the damped harmonic oscillator was not treated with the same kind of formalism used to describe quantum electrodynamics in a arbitrary medium. Here we use the techniques of macroscopic QED, based on the Huttner--Barnett reservoir, to describe the quantum mechanics of a damped oscillator. We calculate the thermal and zero-point energy of the oscillator for a range of damping values from zero to infinity. While both the thermal and zero-point energies decrease with damping, the energy stored in the oscillator at fixed temperature increases with damping, an effect that may be experimentally observable. As the results follow from canonical quantization, the uncertainty principle is valid for all damping levels.The physics of quantum electromagnetism in an absorbing medium is that of a field of damped harmonic oscillators. Yet until recently the damped harmonic oscillator was not treated with the same kind of formalism used to describe quantum electrodynamics in a arbitrary medium. Here we use the techniques of macroscopic QED, based on the Huttner--Barnett reservoir, to describe the quantum mechanics of a damped oscillator. We calculate the thermal and zero-point energy of the oscillator for a range of damping values from zero to infinity. While both the thermal and zero-point energies decrease with damping, the energy stored in the oscillator at fixed temperature increases with damping, an effect that may be experimentally observable. As the results follow from canonical quantization, the uncertainty principle is valid for all damping levels.

arXiv:1304.0977Apr 3, 20134 citations

The energy levels' width and lifetime of one dimensional harmonic oscillator

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The energy levels' width and lifetime of one dimensional harmonic oscillator

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