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Finding position-momentum uncertainty product for two special eigenstates

Zafar Ahmed, Indresh Yadav

Published Mar 12, 2014• 0 citations
OpenAlex

Abstract

We point out two interesting features of position-momentum uncertainty product: $U=\Delta x \Delta p$. We show that two special (non-differentiable) eigenstates of the Schr{o}dinger operator with the Dirac Delta potential $[V(x)=-V_0 \delta(x)],V_0>0$, also satisfy the Heisenberg's uncertainty principle by yielding $U> \frac{\hbar}{2}$. One of these eigenstates is a zero-energy and zero-curvature bound state.

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•Jan 19, 2021•0 citations
We point out two interesting features of position-momentum uncertainty product: U=ΔxΔpU=\Delta x \Delta pU=ΔxΔp. We show that two special (non-differentiable) eigenstates of the Schr{o}dinger operator with the Dirac Delta potential [V(x)=−V0δ(x)],V0>0[V(x)=-V_0 \delta(x)],V_0>0[V(x)=−V0​δ(x)],V0​>0, also satisfy the Heisenberg's uncertainty principle by yielding U>ℏ2U> \frac{\hbar}{2}U>2ℏ​. One of these eigenstates is a zero-energy and zero-curvature bound state.
We point out two interesting features of position-momentum uncertainty product: U=ΔxΔpU=Δx ΔpU=ΔxΔp. We show that two special (non-differentiable) eigenstates of the Schr{ö}dinger operator with the Dirac Delta potential [V(x)=−V0δ(x)],V0>0[V(x)=-V_0 δ(x)],V_0>0[V(x)=−V0​δ(x)],V0​>0, also satisfy the Heisenberg's uncertainty principle by yielding U>ℏ2U> \frac{\hbar}{2}U>2ℏ​. One of these eigenstates is a zero-energy and zero-curvature bound state.
The Heisenberg–Robertson (HR) and Robertson–Schrödinger (RS) uncertainty relations are studied. Analyzing the RS uncertainty relation we found, that there can exist a large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of noncommuting observables, A and B, is zero, and which differ from those described in the literature for the HR uncertainty relation. These states are not eigenstates of either the observable A or B. The correlation function for these observables in such states is equal to zero. We have also shown that the so-called “sum uncertainty relations” also do not provide any information about lower bounds on the standard deviations calculated for these states. We additionally show that the uncertainty principle in its most general form has two faces: One is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations is an upper bound on the modulus of the correlation function of a pair of the noncommuting observables in the state under consideration.
Analyzing general uncertainty relations one can find that there can exist such pairs of non-commuting observables A and B and such vectors that the lower bound for the product of standard deviations ΔA and ΔB calculated for these vectors is zero: ΔA·ΔB≥0. Here we discuss examples of such cases and some other inconsistencies which can be found performing a rigorous analysis of the uncertainty relations in some special cases. As an illustration of such cases matrices (2×2) and (3×3) and the position–momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in PT–symmetric quantum theory and the problems associated with it are also studied.
I reexamine Heisenberg’s uncertainty relation for two- and three-dimensional wave packets with fixed angular momentum quantum numbers m or ℓ. A simple proof shows that the product of the average extent Δr and Δp of a two-dimensional wave packet in position and momentum space is bounded from below by ΔrΔp≥ℏ(|m|+1). The minimum uncertainty is attained by modified Gaussian wave packets that are special eigenstates of the two-dimensional isotropic harmonic oscillator, which include the ground states of electrons in a uniform magnetic field. Similarly, the inequality ΔrΔp≥ℏ(ℓ+3/2) holds for three-dimensional wave packets with fixed total angular momentum ℓ and the equality holds for a Gaussian radial profile. I also discuss some applications of these uncertainty relations.
Recently, S\'anchez-Moreno et al. [New J. Phys. 8, 330 (2006)] have shown that the three-dimensional Heisenberg uncertainty relation ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}\ensuremath{\ge}3\ensuremath{\hbar}/2$ might be significantly sharpened if the relevant quantum state describes the particle in a central potential. I extend that result to the case of states which are not the eigenstates of the square of the angular momentum operator. I derive a lower bound for ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}$ which involves the mean value and the variance of the square of the angular momentum operator.
In (1), an approximate method of obtaining the ground state energy is presented by mapping a Schrodinger equation with a given potential V(x) into one with delta function potential with an appropriate weight. Various examples are presented in both (1) and (2). It should be noted that although this approach yields the exact ground state energy for the oscillator it does not yield a correct answer for a particle in a box with infinite potentials at the wall. In this note, we try to investigate a possible physical argument for a mapping into a delta function and argue it is linked to the Heisenberg uncertainty principle. We also suggest that the weighting used in (1) i.e. U1= Integral (x1,x2) V(x)dx where (x1,x2) are classical turning points, should be multiplied by a constant b which ultimately cancels in the expression for the ground state energy.