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Journal of Xuchang University

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

Luo Ling-xiao

Published Jan 1, 2004• 0 citations
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Abstract

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.

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Position and momentum uncertainties of the normal and inverted harmonic oscillators under the minimal length uncertainty relation

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We analyze the position and momentum uncertainties of the energy eigenstates of the harmonic oscillator in the context of a deformed quantum mechanics, namely, that in which the commutator between the position and momentum operators is given by $[\stackrel{^}{x},\stackrel{^}{p}]=i\ensuremath{\hbar}(1+\ensuremath{\beta}{\stackrel{^}{p}}^{2})$. This deformed commutation relation leads to the minimal length uncertainty relation $\ensuremath{\Delta}x\ensuremath{\ge}(\ensuremath{\hbar}/2)(1/\ensuremath{\Delta}p+\ensuremath{\beta}\ensuremath{\Delta}p)$, which implies that $\ensuremath{\Delta}x\ensuremath{\sim}1/\ensuremath{\Delta}p$ at small $\ensuremath{\Delta}p$ while $\ensuremath{\Delta}x\ensuremath{\sim}\ensuremath{\Delta}p$ at large $\ensuremath{\Delta}p$. We find that the uncertainties of the energy eigenstates of the normal harmonic oscillator ($m>0$), derived in L. N. Chang, D. Minic, N. Okamura, and T. Takeuchi, Phys. Rev. D 65, 125027 (2002), only populate the $\ensuremath{\Delta}x\ensuremath{\sim}1/\ensuremath{\Delta}p$ branch. The other branch, $\ensuremath{\Delta}x\ensuremath{\sim}\ensuremath{\Delta}p$, is found to be populated by the energy eigenstates of the ``inverted'' harmonic oscillator ($m<0$). The Hilbert space in the inverted case admits an infinite ladder of positive energy eigenstates provided that $\ensuremath{\Delta}{x}_{\mathrm{min}}=\ensuremath{\hbar}\sqrt{\ensuremath{\beta}}>\sqrt{2}[{\ensuremath{\hbar}}^{2}/k|m|{]}^{1/4}$. Correspondence with the classical limit is also discussed.

arXiv:1109.2680•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Nov 18, 2011•38 citations
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.
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.
We analyze the position and momentum uncertainties of the energy eigenstates of the harmonic oscillator in the context of a deformed quantum mechanics, namely, that in which the commutator between the position and momentum operators is given by $[\stackrel{^}{x},\stackrel{^}{p}]=i\ensuremath{\hbar}(1+\ensuremath{\beta}{\stackrel{^}{p}}^{2})$. This deformed commutation relation leads to the minimal length uncertainty relation $\ensuremath{\Delta}x\ensuremath{\ge}(\ensuremath{\hbar}/2)(1/\ensuremath{\Delta}p+\ensuremath{\beta}\ensuremath{\Delta}p)$, which implies that $\ensuremath{\Delta}x\ensuremath{\sim}1/\ensuremath{\Delta}p$ at small $\ensuremath{\Delta}p$ while $\ensuremath{\Delta}x\ensuremath{\sim}\ensuremath{\Delta}p$ at large $\ensuremath{\Delta}p$. We find that the uncertainties of the energy eigenstates of the normal harmonic oscillator (m>0m>0m>0), derived in L. N. Chang, D. Minic, N. Okamura, and T. Takeuchi, Phys. Rev. D 65, 125027 (2002), only populate the $\ensuremath{\Delta}x\ensuremath{\sim}1/\ensuremath{\Delta}p$ branch. The other branch, $\ensuremath{\Delta}x\ensuremath{\sim}\ensuremath{\Delta}p$, is found to be populated by the energy eigenstates of the ``inverted'' harmonic oscillator (m<0m<0m<0). The Hilbert space in the inverted case admits an infinite ladder of positive energy eigenstates provided that $\ensuremath{\Delta}{x}_{\mathrm{min}}=\ensuremath{\hbar}\sqrt{\ensuremath{\beta}}>\sqrt{2}[{\ensuremath{\hbar}}^{2}/k|m|{]}^{1/4}$. Correspondence with the classical limit is also discussed.