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Physical Review A

Position-momentum uncertainty products for exactly solvable potentials

Michael Martin Nieto

Published Sep 1, 1979• 30 citations
Publisher

Abstract

The author shows how the position-momentum uncertainty products may be obtained analytically for the eigenstates of the exactly solvable symmetric P"oschl-Teller, symmetric Rosen-Morse, and one-dimensional Morse potentials.
DOI
10.1103/physreva.20.700

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arXiv:quant-ph/0608054•Aug 7, 2006•0 citations
We obtain the exact, normalized, closed-form eigenfunctions for the one-dimensional Morse oscillator, as well as the raising and lowering operators. We next review a new method for obtaining the coherent states for arbitrary potentials, it being based on the classical motion of a particle. We apply this method to the Morse oscillator. After demonstrating that in the appropriate limit our results reduce to those for the harmonic oscillator, we obtain analytic insights into published numerical results on uncertainty products for Morse-oscillator wave packets.
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.
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.
The moments are calculated for some levels of Poschl-Teller potentials. Using isospectral Hamiltonians, the calculations are exte nded to deformed wave functions and their properties are graphically demonstrated. Two Hamiltonians are said to be strictly isospectral, if they have exactly same eigenvalue s pectrum and S-matrix. The wave functions and their dependent quantities are differ ent but related. This property is utilized to obtain the uncertainty in position space. Due to the presence of a deformation parameter, the uncertainty in position space is rea rranged.