Write solutions to the Schrodinger equation appropriate in the various regions, impose required continuity conditions, and obtain the energy quantization condition: $\sqrt{E} \cot \left(\frac{\sqrt{2 m E}}{\hbar} L\right)=-\sqrt{U_{0}-E}$

What does it mean to say that an electron occupies discrete energy levels in an atom?

A particle with mass m and total energy E tunnels through a square barrier of height $U_0$ and width $L$. When the transmission coefficient is not much less than unity, it is given by

$$T=\left[1+\frac{\left(U_0 \sinh \kappa L\right)^2}{4 E\left(U_0-E\right)}\right]^{-1}$$

where $\sinh \kappa L=\left(e^{\kappa L}-e^{-\kappa L}\right) / 2$ is the hyperbolic sine of $\kappa L$.

(a) Show that if $\kappa L \gg 1$, this expression for $T$ approaches

$T=G e^{-2 \kappa L}$ where $G=16 \frac{E}{U_0}\left(1-\frac{E}{U_0}\right)$ and $\kappa=\frac{\sqrt{2 m\left(U_0-E\right)}}{\hbar}$

(b) Explain why the restriction $\kappa L \gg 1$ in part (a) implies either that the barrier is relatively wide or that the energy $E$ is relatively low compared to $U_0$.

(c) Show that as the particle's incident kinetic energy $E$ approaches the barrier height $U_0$, $T$ approaches $\left[1+(k L / 2)^2\right]^{-1}$, where $k=\sqrt{2 m E} / \hbar$ is the wave number of the incident particle.

A particle is in a box with infinitely rigid walls. The walls are at x=-L/2 and x=+L/2. How does the set of energy levels you found in the mentioned part compare to the energy levels given by the mentioned equation?

Show that for the case of a particle of energy $E$ incident on a step barrier $U_0<E$, the wave numbers $k_1$ and $k_2$ are related by

$$\frac{k_2}{k_1}=r=\sqrt{1-\frac{U_0}{E}}$$

A particle of energy E approaches a step barrier of height $U_0$.Find the ratio $E / U_0$ so that the reflection coefficient is $\frac{1}{2}$.

A particle of energy $E=5 \mathrm{eV}$ approaches an energy barrier of height $U=8 \mathrm{eV}$. Quantum mechanically there is a finite probability that the particle tunnels through the barrier. If the barrier height is slowly decreased, the probability that the particle will reflect from the barrier will be a) decrease. b) increase. c) not change.

Give an example of how a manager can decrease variable costs while increasing fixed costs.

Draw an energy level diagram for a nonrelativistic particle confined inside a three-dimensional cube-shaped box, showing all states with energies below $15 \cdot\left(h^{2} / 8 m L^{2}\right).$ Be sure to show each linearly independent state separately, to indicate the degeneracy of each energy level. Does the average number of states per unit energy increase or decrease as E increases?

From the energy-level diagram in figure , how much energy would be required to ionize a hydrogen atom when it is in its lowest energy level?

Under what conditions can a plane wall be treated as a semi-infinite medium?

De Broglie developed initially for photons, assuming that they had a small but finite mass. His assumption was that RF waves with $\lambda=30 \mathrm{~m}$ traveled at a speed of at least 99 percent of that of visible light with $\lambda=500 \mathrm{~nm}$. Beginning with the relativistic expression $h f=\gamma m c^2$, verify de Broglie's calculation that the upper limit of the rest mass of a photon is $10^{-44} \mathrm{~g}$. (Hint: Find an expression for $\mathrm{V} / \mathrm{c}$ in terms of $h f$ and $m c^2$ and then let $m c^2 \ll h f ;\left(\gamma=1 / \sqrt{1-v^2 / c^2}\right.$.)

The fin efficiency is defined as the ratio of the actual heat transfer from the fin to(a) The heat transfer from the same fin with an adiabatic tip (b) The heat transfer from an equivalent fin which is infinitely long (c) The heat transfer from the same fin if the temperature along the entire length of the fin is the same as the base temperature (d) The heat transfer through the base area of the same fin (e) None of the above

Oil at $15^\circ C$ is to be heated by saturated steam at 1 atm in a double-pipe heat exchanger to a temperature of $25^\circ C.$ The inner and outer diameters of the annular space are 3 cm and 5 cm, respectively, and oil enters with a mean velocity of 0.8 m/s. The inner tube may be assumed to be isothermal at $100^\circ C,$ and the outer tube is well insulated. Assuming fully developed flow for oil, determine the tube length required to heat the oil to the indicated temperature. In reality, will you need a shorter or longer tube? Explain.