Consider the potential well defined by

V(x)= {0 -L/2<x<L/2}

{Vo x>L/2 or x<L/2}

1. For E>Vo (unbound state, find the solutions to the time-independent Schroedinger equation for a particle incident from the left and traveling to the right(Use boundary conditions to evaluate coefficients A-F)

2. Find expressions for the transmission and reflection as a function of energy.

3. Find the necessary conditions for perfect transmission. Interpret this condition on physical grounds (note that k=2pi/gamma, which reflects the wave vector ,k, to the wavelength).

7.11 Consider two cases involving parallel flow of dry air at V = 1 m/s, T_∞ = 45°C, and atmospheric pressure over an isothermal plate at T_s = 20°C. In the first case, Re_x,c = 5 × 10⁵, while in the second case the flow is tripped to a turbulent state at x = 0 m. At what x‐location are the thermal boundary layer thicknesses of the two cases equal? What are the local heat fluxes at this location for the two cases?

Answer: x=.142; q”_lam = -129 W/m²;  q”_turb = -175 W/m²

Shows a new process in which 0.0100 kg of methane (an ideal gas) is compressed from a pressure of 0.100 MPa and a temperature of 20.0 °C to a pressure of 10.0 MPa in a polytropic process with n = 1.35. Determine the moving boundary work required.

Determine the work required in Example 4.5 if the final pressure of the methane is 0.500 MPa.

If the work required in Example 4.5 is ?5.00 kJ, determine the final temperature and pressure of the methane.

If the gas used in Example 4.5 were air, determine the work required to compress it polytropically from 14.7 psia, 70.0°F to 150.°F with n = 1.33.

1. The refractive index of a transparent material can be determined by measuring the critical angle when the solid is in air. If θ_c= 40.1° what is the index of refraction of the material?

2. A light ray strikes this material (from air) at an angle of 35.5° with respect to the normal of the surface. Calculate the angle of the reflected ray (in degrees).

3. Calculate the angle of the refracted ray (in degrees).

4. Assume now that the light ray exits the material. It strikes the material-air boundary at an angle of 35.5° with respect to the normal. What is the angle of the refracted ray?

A maximal plane graph is a plane graph G = (V, E) with n ≥ 3 vertices such that if we join any two non-adjacent vertices in G, we obtain a non-plane graph.

a) Draw a maximal plane graphs on six vertices.

b) Show that a maximal plane graph on n points has 3n − 6 edges and 2n − 4 faces.

c) A triangulation of an n-gon is a plane graph whose infinite face boundary is a convex n-gon and all of whose other faces are triangles. How many edges does a triangulation of an n-gon have?

Using the wave functions

for the potential energy step, apply the boundary conditions of ψ and

dψ/dx

to find B' and C' in terms of A', for the potential step when particles are incident from the negative x direction. Evaluate the reflection and transmission coefficients

R=

and

T=

.

( k₀=42 and k₁=12)

Boundary-Value Problems in Other Coordinate Systems

1. Solve ∆u = 0 in a disk x^2 + y^2 ≤ 25, where u(5, θ) = 7 sin 3θ − 6 sin 8θ and u is bounded when   r = 0.
2. Solve ∆u = 0 in an annulus 1 ≤ x^2+y^2 ≤ 4, where u(1, θ) = 75 sin θ, u(2, θ) = 60 cos θ.
3. Find the steady-state temperature distribution in a disk of radius 1 if the upper half of the circumference is kept at 100◦ and the lower half is kept at 0◦ .

Quantity take-off problem: A capital improvement project requires the installation of a property line fence along the 250 -ft northern boundary line. The decorative aluminum fence is constructed of posts spaced at 10-ft center to center and an ornate picket infill panel between two posts. Given the material costs below, the cost for the construction of the fence is most nearly:

a. $65,771.25

b. $66,461.60

c. $68,402.10

d. $71, 065.58

Material costs:

Aluminum posts: $645.35 each

Ironworker: $78/hr

Picket infill panel: $1,985.50 each

Placed concrete $498.00/cy

1) Several central banks have used negative official interest rates in recent years, which until recently seemed unimaginable. Indeed, economists used to talk about a zero interest-rate boundary (ZIRB). How is it possible for a central bank to cut its official interest rate below zero?

2) Japan has the world's highest level of government debt. The interest rate (or yield) on 10-year Japanese government debt has been approximately zero for the past three years. What has made this possible? How is it related to quantitative easing?