A free Gaussian state from Chapter 6 has and at

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Observables, commutators and the uncertainty relations

Problem

A free Gaussian state from Chapter 6 has

\Delta x_0 = s

and

\Delta p = \hbar/2s

t = 0

, so it saturates the bound proved in this chapter. Using Chapter 6's spreading law,

\Delta x(t) = \Delta x_0\sqrt{1 + (\hbar t/2m\Delta x_0^2)^2}

, and its doubling time

t_2 = 2\sqrt3\,ms^2/\hbar

, find

\Delta x\,\Delta p

t = t_2

and check it still satisfies the generalised uncertainty principle.