I got this integral in one quantum mechanics problem: $$\int_{-\infty}^{\infty} e^{-i\cdot p_y y/\hbar}e^{-\beta y^2}dy$$ and the solution was apparently $\sqrt{\frac{\pi}{\beta}} e^{-\frac{p_y^2}{4\hbar^2 \beta}}$. Like how? I can see that $\sqrt{\frac{\pi}{\beta}}$ factor since it is a solution to the integral $\int_{-\infty}^{\infty} e^{-\beta y^2}dy$ but what about the rest? It just doesn't seem like a solution to be done with integration by parts, does it?
2026-04-02 09:57:09.1775123829
Complex integral from quantum mechanics
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