Uncertainty relations for angular momentum eigenstates in two and three spatial dimensions

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 Delta r and Delta p of a two-dimensional wave packet in position and momentum space is bounded from below by Delta r Delta p >=(h) over bar(vertical bar m vertical bar+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 Delta r Delta p >=(h) over bar (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. (C) 2011 American Association of Physics Teachers. [DOI: 10.1119/1.3534840]