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The gaussian wave packet. A free particle has the initial wave function $$ \Psi(x, 0)=A e^{-a x^{2}} $$ where $A$ and $a$ are constants ( $a$ is real and positive). (a) Normalize $\Psi(x, 0)$. (b) Find $\Psi(x, t) .$ Hint: Integrals of the form $$ \int_{-\infty}^{+\infty} e^{-\left(a x^{2}+b x\right)} d x $$ can be handled by "completing the square": Let $y \equiv \sqrt{a}[x+(b / 2 a)]$, and note that $\left(a x^{2}+b x\right)=y^{2}-\left(b^{2} / 4 a\right)$. Answer: $$ \Psi(x, t)=\left(\frac{2 a}{\pi}\right)^{1 / 4} \frac{e^{-a x^{2} / 11+(2 i \hbar a t / m)}}{\sqrt{1+(2 i \hbar a t / m)}} $$ (c) Find $|\Psi(x, t)|^{2}$. Express your answer in terms of the quantity $$ w \equiv \sqrt{\frac{a}{1+(2 \hbar a t / m)^{2}}} $$ Sketch $|\Psi|^{2}$ (as a function of $x$ ) at $t=0$, and again for some very large $t$. Qualitatively, what happens to $|\Psi|^{2}$, as time goes on? (d) Find $\langle x\rangle,\langle p\rangle,\left\langle x^{2}\right\rangle,\left\langle p^{2}\right\rangle, \sigma_{x}$, and $\sigma_{p} .$ Partial answer: $\left\langle p^{2}\right\rangle=a \hbar^{2}$, but it may take some algebra to reduce it to this simple form. (e) Does the uncertainty principle hold? At what time $t$ does the system come closest to the uncertainty limit?

          The gaussian wave packet. A free particle has the initial wave function
$$
\Psi(x, 0)=A e^{-a x^{2}}
$$
where $A$ and $a$ are constants ( $a$ is real and positive).
(a) Normalize $\Psi(x, 0)$.
(b) Find $\Psi(x, t) .$ Hint: Integrals of the form
$$
\int_{-\infty}^{+\infty} e^{-\left(a x^{2}+b x\right)} d x
$$
can be handled by "completing the square": Let $y \equiv \sqrt{a}[x+(b / 2 a)]$, and note that $\left(a x^{2}+b x\right)=y^{2}-\left(b^{2} / 4 a\right)$. Answer:
$$
\Psi(x, t)=\left(\frac{2 a}{\pi}\right)^{1 / 4} \frac{e^{-a x^{2} / 11+(2 i \hbar a t / m)}}{\sqrt{1+(2 i \hbar a t / m)}}
$$
(c) Find $|\Psi(x, t)|^{2}$. Express your answer in terms of the quantity
$$
w \equiv \sqrt{\frac{a}{1+(2 \hbar a t / m)^{2}}}
$$
Sketch $|\Psi|^{2}$ (as a function of $x$ ) at $t=0$, and again for some very large $t$. Qualitatively, what happens to $|\Psi|^{2}$, as time goes on?
(d) Find $\langle x\rangle,\langle p\rangle,\left\langle x^{2}\right\rangle,\left\langle p^{2}\right\rangle, \sigma_{x}$, and $\sigma_{p} .$ Partial answer: $\left\langle p^{2}\right\rangle=a \hbar^{2}$, but it
may take some algebra to reduce it to this simple form.
(e) Does the uncertainty principle hold? At what time $t$ does the system come closest to the uncertainty limit?

Added by F-Tima S.

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