An ion of mass sits in a trap with angular frequency
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Quantum dynamics and its formulations
The Structure of Quantum Mechanics, Ch Coherent states, Ehrenfest's theorem and the classical limit
Problem
An ion of mass m=2.85×10−25kg sits in a trap with angular frequency ω=2π×1.10MHz. Its motional state is a coherent state ∣α⟩ with α=2−i (a dimensionless complex number fixing the mean position and momentum, in units where x^=ℏ/2mω(a^+a^†)).
((a)) Using a^∣α⟩=α∣α⟩, compute Δx and Δp for ∣α⟩, and show that neither depends on α itself.
((b)) Check the generalised uncertainty relation from Chapter 5, ΔxΔp≥ℏ/2, against your numbers and state whether ∣α⟩ saturates it.