Franz Schwabl Quantum Mechanics 4th Edit · Problem 3.2

The Uncertainty Relation

Physics — Quantum Mechanics · coherent-states

Problem

(a) Show that coherent states can be written in the form

(3.37'')

. Hint: Use (2.47). (b) Demonstrate that

(\varphi_\alpha, \varphi_\beta) = \mathrm{e}^{-\frac{1}{2}(|\alpha|^2 + |\beta|^2) + \alpha^* \beta}

holds. (c) Prove the relation

\int \frac{d^2\alpha}{\pi} \varphi_\alpha(x) \varphi^*_\alpha(x') = \delta(x - x') \ ,

where

d^2\alpha = (\mathrm{Re}\, \alpha)(\mathrm{Im}\, \alpha) \ .