Rayleigh’s criterion is used to determine when two objects are barely resolved by a lens of diameter d. The angular separation must be greater than $\theta_{R}$ where $\theta_{R} =1.22 \lambda /d$ . In order to resolve two objects 4000 nm apart at a distance of 20 cm with a lens of diameter 5 cm, what energy (a) photons or (b) electrons should be used? Is this consistent with the uncertainty principle?

A beam of 2.0-keV electrons incident on a crystal is refracted and observed (by transmission) on a screen 35 cm away. The radii of three concentric rings on the screen, all corresponding to first-order diffraction, are 2.1 cm, 2.3 cm, and 3.2 cm. What is the lattice-plane spacing corresponding to each of the three rings?

We can approximate an electron moving in a nano-wire (a small, thin wire) as a one-dimensional infinite square-well potential. Let the wire be 2.0 $\mu$m long. The nanowire is cooled to a temperature of 13 K, and we assume the electron’s average kinetic energy is that of gas molecules at this temperature (=3kT/2). (a) What are the three lowest possible energy levels of the electrons? (b) What is the approximate quantum number of electrons moving in the wire?

Use Equation $\Psi(x, t)=\int \widetilde{A}(k) \cos (k x-\omega t) d k$ with $\tilde{A}(k)=A_{0}$ for the range of $k=k_{0}-\Delta k / 2$ to $k_{0}+\Delta k / 2$, and $\tilde{A}(k)=0$ elsewhere, to determine $\Psi(x, 0)$ , that is, at t=0. Sketch the envelope term, the oscillating term, and $|\Psi(x, 0)|^{2}$. Approximately what is the width $\Delta x$ over the full-width-half-maximum part of $|\Psi(x, 0)|^{2}$?What is the value of $\Delta x \Delta k$

Assume that the total energy E of an electron greatly exceeds its rest energy E0. If a photon has a wavelength equal to the de Broglie wavelength of the electron, what is the photon’s energy? Repeat the same problem assuming $E = 2E_{o}$ for the electron.

An atom in an excited state of 4.7 eV emits a photon and ends up in the ground state. The lifetime of the excited state is $1 \times 10^{-13}$ s. (a) What is the energy uncertainty of the emitted photon? (b) What is the spectral line width (in wavelength) of the photon?

Find the kinetic energy of (a) photons, (b) electrons, (c) neutrons, and (d) particles that have a de Broglie wavelength of 0.13 nm.

Write down the normalized wave functions for the first three energy levels of a particle of mass m in a one-dimensional box of width L. Assume there are equal probabilities of being in each state.

A wave function $\Psi$ is $A(e^{i x}+e^{-i x} )$ in the region $-\pi<x<\pi$ and zero elsewhere. Normalize the wave function and find the probability of the particle being (a) between x =0 and x = $\pi/8$ (b) between x =0 and x = $\pi/4$

What is the expected result when you roll a single die? Specify 40 rolls, and then click on "roll the dice." After the last roll, click on "show mean." Does your expected sum agree with the green mean? What happens to the plotted points as the number of rolls increases? If the picture is not clear, click on "roll the dice" for a new total of 80 rolls. Describe what you see in a sentence. What is the name for this principle?

Will help you prepare for the material covered in the first section of the next chapter. Solve each equation by using rhe cross-products principle to clear fractions from the proportion:

$$\text { If } \frac { a } { b } = \frac { c } { d } , \text { then } a d = b c . ( b \neq 0 \text { and } d \neq 0 )$$

Round to the nearest tenth. Solve for

$$B , 0 < B < 180 ^ { \circ } : \frac { 81 } { \sin 43 ^ { \circ } } = \frac { 62 } { \sin B }$$

Refer to the figure and use the principle of superposition to Find the component of the current $i$ through $R_{3}$ that is due to $V_{S 2}$.

$$\begin{aligned} & V_{S 1}=V_{S 2}=450 \mathrm{~V} \\ & R_{1}=7 \Omega \quad R_{2}=5 \Omega \\ & R_{3}=10 \Omega \quad R_{4}=R_{5}=1 \Omega \end{aligned}$$

The following exercise is a problem that may require permutations, combinations, or the multiplication principle.

According to the Baskin-Robbins Web site, there are 21 "classic flavors" of ice cream.

(a) How many different double-scoop cones can be made if order does not matter (for example, putting chocolate on top of vanilla is equivalent to putting vanilla on top of chocolate)?

(b) How many different triple-scoop cones can be made if order does matter?

Underline any verbs that are used incorrectly. In the space provided, write the correct form (or $C$ if correct) and the number of the $\mathrm{G} / \mathrm{M}$ principle illustrated. When you finish, compare your responses with those provided at the bottom of this page. If your responses differ, study carefully the principles in parentheses.

Everyone except the temporary workers employed during the past year has become eligible for health benefits.