Evaluate the integral from -∞ to ∞ of the expression exp ip²t 2mħ 4ah² + ipad + fact that the integral from -∞ to ∞ of exp(-a²x² + bx) dx equals exp problem. a 6² 4a² dp. Use the to solve the
Evaluate the integral from -∞ to ∞ of the expression exp ip²t 2mħ 4ah² + ipad + fact that the integral from -∞ to ∞ of exp(-a²x² + bx) dx equals exp problem. a 6² 4a² dp. Use the to solve the
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![**Problem Statement:**
Evaluate the integral from \(-\infty\) to \(\infty\) of the expression:
\[
\exp\left( -\frac{p^2}{4 \alpha \hbar^2} + \frac{ipx}{\hbar} + \frac{ip^2 t}{2 m \hbar} \right) \, dp.
\]
**Hint:** Use the fact that the integral from \(-\infty\) to \(\infty\) of
\[
\exp\left( -a^2 x^2 + bx \right) \, dx
\]
equals
\[
\sqrt{\frac{\pi}{a}} \, \exp\left( \frac{b^2}{4a^2} \right)
\]
to solve the problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0695e04c-6177-4dae-ae8e-62e62853c8ae%2F8ccc9a7e-a05b-4445-93b7-d2c8c4f0d964%2F11c0wc_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Evaluate the integral from \(-\infty\) to \(\infty\) of the expression:
\[
\exp\left( -\frac{p^2}{4 \alpha \hbar^2} + \frac{ipx}{\hbar} + \frac{ip^2 t}{2 m \hbar} \right) \, dp.
\]
**Hint:** Use the fact that the integral from \(-\infty\) to \(\infty\) of
\[
\exp\left( -a^2 x^2 + bx \right) \, dx
\]
equals
\[
\sqrt{\frac{\pi}{a}} \, \exp\left( \frac{b^2}{4a^2} \right)
\]
to solve the problem.
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