HW05

ASSIGNED:     FRI 22 FEB
DUE:                 FRI 01 MAR

In all problems the Mathematica output is part of your grade and must be turned in with your paper-and-pencil calculations.

For the Mathematica calculations, such as in the Integrate, Simplify, FullSimplify commands, you need to specify that the constants such as a, σ, λ, etc., are real numbers greater than zero or else it will assume they are complex numbers and produce crazy looking answers. To do this read the "tutorial/UsingAssumptions" which you will find in the Mathematica Help / Documentation Center.

1. GRIFFITHS PROBLEM 2.26, PAGE 77.  Do the analytic proof by hand first. Then try to get Mathematica to reproduce the result using the Integrate command. What happens? Then realized that the integral is the Fourier transform of the function "1" multiplied by 1/Sqrt[2*Pi] and use the Mathematica FourierTransform command instead to arrive at the result of Eq.2.144. (Look up FourierTransform command in the Mathematica help.)

2. GRIFFITHS PROBLEM 2.29, PAGE 82. Use Mathematica to solve the equation graphically as shown in notes. EyebalL the graphical solutions to guess all the roots for z0=8 (as for the even solutions) then use (as in notes) the Mathematica FindRoot command to find all of the roots to at least three significant digits and express the quantized energies in terms of these. Compare the EXACT quantized energies for the odd solutions to the FSW to the EXACT quantized energies for the odd solutions to the ISW as done in the notes.

3. GRIFFITHS PROBLEM 2.32, PAGE 83. Take a deep breath and solve this by hand. Then use the Solve command to solve the four equations and four unknowns simultaneously and check your hand calculation. Hint: See notes.

4. GRIFFITHS PROBLEM 2.33, PAGE 83. Hint: I did all of the cases in the notes except for the case E=Vo so just go through the proofs again so you understand them. Plot the transmission coefficient T[ε, νo] for each of the three cases as done in the notes. Hint: For the case E=Vo you can get the correct result from the solution to 0<E<Vo (formula given in the problem and in notes) by taking the  limit E->Vo carefully. Try expanding the Sin[2*a*Sqrt[Vo-E]]^2 in a Taylor series around Vo–E=0 and then cancel stuff out to find T^(–1).

5. GRIFFITHS PROBLEM 2.34, PAGE 83. You will have to solve two equations and two unknows for E<Vo and again for E>Vo to get r and t. Do it by hand then use the Solve command in Mathematica to check your work. For parts (a), (b), and (c) use Mathematica to plot T and R as functions of E for reasonable choices of Vo.

6. GRIFFITHS PROBLEM 2.36, PAGE 85. Hint: Look at the notes. Plot the first three solutions on top of V[x] in Mathematica.


Erwin Schrödinger
 


"I don't like [quantum mechanics], and I'm sorry I ever had anything to do with it."

--- Erwin Schrödinger (1887–1961)