A wave function of a particle with mass m is given by

$$\psi(x) = \left\{ \begin{array}{ll} A ~ \cos{(\alpha x)} , & - \dfrac{\pi}{2\alpha} \leq x \leq \dfrac{\pi}{2\alpha} \\ 0 , & \text{Otherwise.} \end{array}\right.$$

where $\alpha = 1.00 ~ \times 10^{10} ~ \mathrm{m}^{-1}$. (a) Find the normalization constant. (b) Find the probability that the particle can be found on the interval $0 \leq x \leq 0.5 ~ \times 10^{-10} ~ \mathrm{m}$. (c) Find the particle’s average position. (d) Find its average momentum. (e) Find its average kinetic energy $-0.5 ~ \times 10^{-10} \mathrm{m} \leq x \leq 0.5 ~ \times 10^{-10} ~ \mathrm{m}$.

Evaluate the following integral

$$\int \frac{x}{x-2} d x$$

Find the normalization constant A for the following wave function of a particle in a cubical box

$$\psi =A \sin \left(\frac{\pi x}{L}\right) \sin \left(\frac{\pi y}{L}\right) \sin \left(\frac{\pi z}{L}\right)$$

A 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?

A particle with mass m moving along the x-axis and its quantum state is represented by the following wave function: $$ \Psi(x,t) = \left\{ \begin{array}{ll} 0 , & x<0 \\ A ~ x e^{-\alpha x }~ e^{iEt/\hbar} , & x \( \geq \) 0 \end{array}\right. $$ where $\alpha = 2.0 ~ \times 10^{10} ~ \mathrm{m}^{-1}$ (a) Find the normalization constant. (b) Find the probability that the particle can be found on the interval $0 \leq x \leq L$ (c) Find the expectation value of position. (d) Find the expectation value of kinetic energy.

The wave function for a quantum particle is given by $\psi(x)=A x$ between x = 0 and x = 1.00, and $\psi(x)=0$ elsewhere. Find (a) the value of the normalization constant A, (b) the probability that the particle will be found between x = 0.300 and x = 0.400, and (c) the expectation value of the particle’s position.

Why are three quantum numbers needed to describe the state of a one-electron atom (ignoring spin)?

Calculate $|\Psi(x, t)|^2$ for the function $\Psi(x, t)=\psi(x) \sin \omega t$, where $\omega$ is a real constant.

An electron is trapped in a one-dimensional infinite well of width 250 pm and is in its ground state. What are the (a) longest, (b) second longest, and (c) third longest wavelengths of light that can excite the electron from the ground state via a single photon absorption?

Find the real solutions, if any, of each equation. Use the quadratic formula. $\frac{2 x}{x-3}+\frac{1}{x}=4$

find all eigenvalues and eigenvectors of the given matrix. ⎝11/9−2/98/9−2/92/910/98/910/95⎪

What is the 3rd person singular imperfect conjugation of the verb 'regresar' ?

The selection sort algorithm sorts the sequence $s_1,...,s_n$ in nondecreasing order by first finding the smallest item, say $s_i$, and placing it first by swapping $s_1$ and $s_i$. It then finds the smallest item in $s_2,...,s_n$, again say $s_i$, and places it second by swapping $s_2$ and $s_i$. It continues until the sequence is sorted. Write selection sort in pseudo code.

What is the de Broglie wavelength of an electron traveling at $1.35 \times 10^{5} \mathrm{m} / \mathrm{s} ?$