Franz Schwabl Quantum Mechanics 4th Edit · Problem 2.1

One-Dimensional Problems

Physics — Quantum Mechanics · gaussian-integrals-fourier-transform

Problem

(a) Show that, for complex

\alpha

with

\text{Re}\,\alpha > 0

\int_{-\infty}^{+\infty} dx\, e^{-\alpha x^2} = \sqrt{\pi/\alpha}

holds. (b) Compute

\int_{-\infty}^{+\infty} d^3k\, e^{i\boldsymbol{k}\cdot\boldsymbol{x}}\, e^{-k^2\alpha^2}\,.