A Modern Approach to Quantum Mechanics · Problem 7.17

Path Integrals — John S. Townsend

Physics — Quantum Mechanics · coherent-states-classical-limit

Problem

We can safely say that the coherent state is as close to purely classical oscillatory motion as we are going to get in quantum mechanics. An interesting limiting case is worthy of mention. Show that as ℏ→0\hbar \to 0 the probability density ∣⟨x∣0⟩∣2=mωπℏ e−mωx2/ℏ|\langle x|0\rangle|^2 = \sqrt{\frac{m\omega}{\pi\hbar}}\,e^{-m\omega x^2/\hbar} for the ground state becomes a Dirac delta function (see Appendix C). What happens to the momentum and position uncertainties for the coherent state in this limit? Is your result in accord with the uncertainty relation Δx Δpx=ℏ/2\Delta x\,\Delta p_x = \hbar/2? Explain.

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