For the harmonic oscillator potential energy $U = \frac{1}{2}{kx^2},$ the ground-state wave function is $\psi(x)=A e^{-(\sqrt{m \kappa} / 2 \hbar) x^{2}}$ and its energy is $\frac{1}{2} \hbar \sqrt{\kappa / m}$

(a) Find the classical turning points for a particle with this energy.

(b) The Schrodinger equation says that $\psi(x)$ and its second derivative should be of the opposite sign when E > U and of the same sign when E< U. These two regions are divided by the classical turning points. Verify the relationship between $\psi(x)$ and its second derivative for the ground-state oscillator wave function.

A particle in the harmonic oscillator potential starts out in the state $\psi(x,0)=A[3\psi_0(x)+4\psi_1(x)].$ (a) Find A. (b) Construct $\psi(x,t)\text{ and }|\psi(x,t)|^2.$ (c) Find $\left \langle x \right \rangle$ and $\left \langle p \right \rangle.$ Don’t get too excited if they oscillate at the classical frequency; what would it have been had I specified instead of check that Ehrenfest’s theorem holds for this wave function. (d) If you measured the energy of this particle, what values might you get, and with what probabilities?

The Rich and Creamy Edibles Factory manufactures and distributes chocolate products. It purchases cocoa beans and processes them into two intermediate products: chocolate-powder liquor base and milk-chocolate liquor base. These two intermediate products become separately identifiable at a single splitoff point. Every 600 pounds of cocoa beans yields 20 gallons of chocolate-powder liquor base and 60 gallons of milk-chocolate liquor base. The chocolate-powder liquor base is further processed into chocolate powder. Every 20 gallons of chocolate-powder liquor base yield 680 pounds of chocolate powder. The milk-chocolate liquor base is further processed into milk chocolate. Every 60 gallons of milk-chocolate liquor base yield 1,100 pounds of milk chocolate. Production and sales data for August 2017 are as follows (assume no beginning inventory): Cocoa beans processed, 27,600 pounds, Costs of processing cocoa beans to split off point (including purchase of beans), $70,000.$

$$\begin{matrix} & \text{Production} & \text{Sales} & \text{Selling Price} & \text{Separable Processing Costs}\\ \text{Chocolate powder} & \text{31,280 pounds} & \text{6,800 pounds} & \text{\$8 per pound} & \text{\$46,035}\\ \text{Milk chocolate} & \text{50,600 pounds} & \text{14,400 pounds} & \text{\$9 per pound} & \text{\$55,085}\\ \end{matrix}$$

$Rich and Creamy Edibles Factory fully processes both of its intermediate products into chocolate powder or milk chocolate. There is an active market for these intermediate products. In August 2017, Rich and Creamy Edibles Factory could have sold the chocolate-powder liquor base for$21 a gallon and milk-chocolate liquor base for $28 a gallon. 1. Calculate how the joint costs of$70,000 would be allocated between chocolate powder and milk chocolate under the following methods: a. Sales value at splitoff, b. Physical measure (gallons), c. NRV, d. Constant gross-margin percentage NRV. 2. What are the gross-margin percentage of chocolate powder and milk chocolate under each of the methods in requirement 1? 3. Could Rich and Creamy Edibles Factory have increased its operating income by a change in its decision to fully process both of its intermediate products? Show your computations.

Given that the particle's total energy E is 0, show that the potential energy is $U(x)=\frac{\hbar^{2}}{2 m b^{4}} x^{2}-\frac{3 \hbar^{2}}{2 m b^{2}}$

A particle is bound by a potential energy of the form

$$U(x)=\left\{\begin{array}{cl}{0} & {|x|<\frac{1}{2} a} \\ {\infty} & {|x|>\frac{1}{2} a}\end{array}\right.$$

This differs from the infinite well in being symmetric about x = 0, which implies that the probability densities must also be symmetric. Noting that either sine or cosine would fit this requirement and could be made continuous with the zero wave function outside the well, determine the allowed energies and corresponding normalized wave functions for this potential well.

Joyce Brown has run Medical Maids, a specialty cleaning service for medical and dental offices, for the past 10 years. Her static budget and actual results for April 2017 are shown below. Joyce has one employee who has been with her for all 10 years that she has been in business. In addition, at any given time she also employs two other less-experienced workers. It usually takes each employee 2 hours to clean an office, regardless of his or her experience. Brown pays her experienced employee $30 per office and the other two employees$15 per office. There were no wage increases in April. Medical Maids Actual and Budgeted Income Statements for the MOnth Ended April 30, 2017.

$$\begin{matrix} \text{ } & \text{Budget } & \text{Actual}\\ \text{Offices cleaned} & \text{140} & \text{160}\\ \text{Revenue} & \text{\$ 26.600} & \text{\$ 36.000}\\ \text{Variable costs:}\\ \text{Costs of supplies} & \text{630} & \text{680}\\ \text{Labor} & \text{3.360} & \text{4.200}\\ \text{Total variable costs} & \text{3.990} & \text{4.880}\\ \text{Contribution margin} & \text{22.610} & \text{31.120}\\ \text{Fixed costs} & \text{4.900} & \text{4.900}\\ \text{Operating income} & \text{\$ 17.710} & \text{\$ 26.220}\\ \end{matrix}$$

1. How many offices, on average, did Brown budget for each employee? How many offices did each employee actually clean? 2. Prepare a flexible budget for April 2017. 3. Compute the sales prices variance and the labor efficiency variance for each labor type. 4. What information, in addition to that provided in the income statements, would you want Brown to gather, if you wanted to improve operational efficiency?

Consider the following system:

$$\left[\begin{array}{ll}{1} & {0} \\ {0} & {4}\end{array}\right] \ddot{\mathbf{x}}(t)+\left[\begin{array}{rr}{3} & {-1} \\ {-1} & {1}\end{array}\right] \mathbf{x}(t)=\mathbf{0}$$

where M is in kg and K is in N/m. (a) Calculate the eigenvalues of the system. (b) Calculate the eigenvectors and normalize them.

The Fourier field equation in cylindrical coordinates is $\frac{\partial T}{\partial t}=\alpha\left(\frac{\partial^{2} T}{\partial r^{2}}+\frac{1}{r} \frac{\partial T}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} T}{\partial \theta^{2}}-\frac{\partial^{2} T}{\partial z^{2}}\right)$. a. What form does this equation reduce to for the case of steady-state, radial heat transfer? b. Given the boundary conditions. Generate an expression for the heat flow rate, qr, using the result from part (b).

$$\begin{array}{ll}{T=T_{i}} & {\text { at } \quad r=r_{i}} \\ {T=T_{o}} & {\text { at } \quad r=r_{o}}\end{array}$$

Westover Motors is a small car dealership. On average, it sells a car for $32,000, which it purchases from the manufacturer for$28,000. Each month, Westover Motors pays $53,700 in rent and utilities and$69,000 for salespeople’s salaries. In addition to their salaries, salespeople are paid a commission of $400 for each car they sell. Westover Motors also spends$10,500 each month for local advertisements. Its tax rate is 40%. Required: 1. How many cars must Westover Motors sell each month to break even? 2. Westover Motors has a target monthly net income of $69,120. What is its target monthly operating income? How many cars must be sold each month to reach the target monthly net income of$69,120?

A particle is subject to a potential energy that has an essentially infinitely high wall al the origin, like the infinite well, but for positive values of x is of the form U(x)=-b/x, where b is a constant, (a) Sketch this potential energy, (b) How much energy could a classical particle have and still be bound by such a potential energy? (c) Add to your sketch a plot of £ for a bound particle and indicate the outer classical turning point (the inner being the origin), (d) Assuming that a quantum-mechanical description is in order, sketch a plausible ground-state wave function, making sure that your function's second derivative is of the proper sign when U(x) is less than E and when it is greater.

Consider two concentric spheres forming an enclosure with diameters of 12 cm and 18 cm and surface temperatures 300 K and 500 K, respectively. Assuming that the surfaces are black, the net radiation exchange between the two spheres is(a) 21 W(b) 140W(c) 160 W(d) 1275 W(e) 3084 W

Tred-America, Inc., manufactures tires for large auto companies. It uses standard costing and allocates variable and fixed manufacturing overhead based on machine-hours. For each independent scenario given, indicate whether each of the manufacturing variances will be favorable or unfavorable or, in case of insufficient information, indicate “CBD” (cannot be determined). $$ \begin{matrix} \text{Scenario} & \text{Variable Overhead Spending Variance} & \text{Variable Overhead Efficiency Variance} & \text{Fixed Overhead Spending Variance} & \text{Fixed Overhead Production-Volume Variance}\\ \text{Production output is 8% more than budgeted, and actual fixed manufacturing overhead costs are 7% less than budgeted} & \text{ } & \text{ } & \text{ } & \text{ }\\ \text{Production output is 11% more than budgeted; actual machine-hours are 5% less than budgeted} & \text{ } & \text{ } & \text{ } & \text{ }\\ \text{Production output is 15\\% less than budgeted} & \text{ } & \text{ } & \text{ } & \text{ }\\ \text{Actual machine-hours are 18\\% greater than flexible-budget machine-hours} & \text{ } & \text{ } & \text{ } & \text{ }\\ \text{Relative to the flexible budget, actual machine-hours are 10% greater, and actual variable manufacturing overhead costs are 15% less} & \text{ } & \text{ } & \text{ } & \text{ }\\ \end{matrix} $$

A gas mixture in a tank at 600 R and 20 psia consists of 1 lbm of $CO_2$ and 3 lbm of $CH_4.$ Determine the volume of the tank and the partial pressure of each gas.

“I'm going to focus on the customers of my business and leave cost-allocation issues to my accountant.” Do you agree with this comment by a division president? Explain.