physicsA free particle has the initial wave function $\Psi(x, 0)=A e^{-a x^{2}}$, where A and a are (real and positive) constants. 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 $\left(a x^{2}+b x\right)=y^{2}-\left(b^{2} / 4 a\right)$. (c) Find $|\Psi(x, t)|^{2}$. Express your answer in terms of the quantity $w \equiv \sqrt{a /\left[1+(2 \hbar a t / m)^{2}\right]}$. 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 $\sigma_{p}$. (e) Does the uncertainty principle hold? At what time t does the system come closest to the uncertainty limit? 6th Edition•ISBN: 9781260475678Janice Gorzynski Smith2,029 solutions

3rd Edition•ISBN: 9781119316152 (6 more)David Klein3,099 solutions

2nd Edition•ISBN: 9781439047910 (9 more)Lawrence S. Brown, Thomas A. Holme945 solutions

3rd Edition•ISBN: 9780471254249 (1 more)Octave Levenspiel228 solutions