Internal Structure of the Heisenberg and Robertson-Schrödinger Uncertainty Relations

It is known that the Heisenberg and Robertson-Schrodinger uncertainty relations can be replaced by sharper uncertainty relations in which the “classical” (depending on the gradient of the phase of the wave function) and “quantum” (depending on the gradient of the envelope of the wave function) parts of the variances 〈(Δx)2〉 and 〈(Δp)2〉 are separated. In this paper, three types of uncertainty relations for a different number of classical parts (2, 1 or 0) with different time behaviour of their left-hand and right-hand sides are discussed. For the Gaussian wave packet and two classical parts, the left-hand side of the corresponding relations increases for t→∞ as t 2 and is much larger than ħ 2/4. For one classical part, the left-hand side of the corresponding relation goes to the right-hand side equal to ħ 2/4. For no classical part, both the right-hand and left-hand sides of the corresponding relation go quickly to zero. Therefore, the well-known limitations following from the usual uncertainty relations can be overcome in the corresponding measurements.