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Physical Review A

Minimum uncertainty products via the variational principle

Joydev Lahiri, V Jayaram Menon

Published Nov 1, 1988• 5 citations
Publisher

Abstract

The problem of finding the class of wave packets \ensuremath{\psi} that minimizes the quantum-mechanical uncertainty product ${\ensuremath{\Delta}}_{x}$${\ensuremath{\Delta}}_{p}$ is treated via the constrained variational principle. This leads to a harmonic-oscillator-type wave equation for \ensuremath{\psi} whose explicit solutions are obtained when the potential V has, or does not have, an infinite step. The method can be fruitfully applied to the quantum cosmology of the inflationary universe.

DOI
10.1103/physreva.38.5412

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The problem of finding the class of wave packets \ensuremath{\psi} that minimizes the quantum-mechanical uncertainty product {\ensuremath{\Delta}}_{x}$${\ensuremath{\Delta}}_{p} is treated via the constrained variational principle. This leads to a harmonic-oscillator-type wave equation for \ensuremath{\psi} whose explicit solutions are obtained when the potential V has, or does not have, an infinite step. The method can be fruitfully applied to the quantum cosmology of the inflationary universe.
We derive from a dynamical symmetry property that the linear and nonlinear Schr\"odinger equations with harmonic potential possess an infinite string of shape-preserving coherent wave-packet states with classical motion. Unlike the Schr\"odinger state with $\ensuremath{\Delta}x\ensuremath{\Delta}p=\frac{\ensuremath{\hbar}}{2}$, the uncertainty product can be arbitrarily large for these states showing that classical motion is not necessarily linked with minimum uncertainty. We obtain a generalization of Sudarshan's diagonal coherent-state representation in terms of these states.
The minimization problem of finding the number-phase minimum uncertainty states (MUS) is considered and its solutions are found either numerically or, under some special conditions, analytically. The phase uncertainty measure is based on the Bandilla-Paul dispersion. The problem is treated (i) in a finite-dimensional Hilbert space and (ii) for a countably infinite-dimensional Hilbert space (i.e. the standard quantum harmonic oscillator), with the constraint of a given mean photon number. The MUS relations between the photon number uncertainty and phase uncertainty are presented. Connections to some other minimization problems are discussed.
We present several variations of a proof of the position-momentum uncertainty principle that are based on the calculus of variations and does not rely on the Cauchy–Schwartz inequality. We show that the stationary uncertainty wave functions are the Hermite–Gaussian solutions to the quantum harmonic oscillator problem, that the minimum uncertainty wave function is the Gaussian, and that stationary uncertainty wave functions must be their own Fourier transforms. We also provide a calculus of variations proof of the Cauchy–Schwartz inequality. Finally, we discuss the properties of wave functions that are their own Fourier transforms and provide examples of such functions that may be of interest to teachers of undergraduate and graduate level quantum mechanics courses.
The usual Heisenberg uncertainty relations are generalized to the case of nonsimultaneous measurements of position and momentum. The product of the corresponding uncertainties is bounded from below by a quantity depending on the dynamics of the system (the potential), and which may be lower than ħ/2 and even vanish. The case of two successive measurements of momentum and position is also treated, and one recovers the standard phenomenon of spreading of the wave packet. Whereas the case of an arbitrary potential can only be treated to lowest orders in the lime delay, a complete solution is given for the harmonic oscillator.
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.