HW04

ASSIGNED:     FRI 15 FEB
DUE:                  FRI 22 FEB

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 command, 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 2.18, PAGE 66.

2. GRIFFITHS PROBLEM 2.19, PAGE 66.

3. GRIFFITHS PROBLEM 2.20, PAGE 66.

4. GRIFFITHS PROBLEM 2.21, PAGE 67. For (b) show that normalizing ψ(x) automatically normalizes φ(k); that is show that ∫ |φ(k)|2 dk (from –Infinity to +Infinity) is one. (This is always true for Fourier transform pairs!) For parts (c) and (d) even Mathematica cannot do the integral in closed form, even in the limit a->0 and a->Infinity so your argument will look like that in EX2.6. For part (d) use Mathematica to plot |ψ(x)|2 and  |φ(k)|2 for a small, intermediate, and large value of a. Show that when when |ψ(x)|2 is broad and shallow then |φ(k)|2 is narrow and tall, and then vice versa. Estimate the width of both and show that the Heisenberg uncertianty principle holds in all three cases: ∆x*∆p≥hbar/2. Recall that ∆p=hbar*∆k.

5. GRIFFITHS PROBLEM 2.22, PAGE 67. Do it by hand using lookup tables and then again in Mathematica. For part (b) you will need to first find φ(k) so also use Mathematica to plot |ψ(x)|2 and  |φ(k)|2 for a small, intermediate, and large value of a. Show that when when |ψ(x)|2 is broad and shallow then |φ(k)|2 is narrow and tall, and then vice versa. Estimate the width of both and show that the Heisenberg uncertianty principle holds in all three cases: ∆x*∆p≥hbar/2. Recall that ∆p=hbar*∆k. For part (c) plot the probablity density |Ψ(x,t)|2 as an animated plot as a function of time for a small, intermediate, and large value of a. Use rescaled time units of τ := 2*hbar/m*t as shown in class.

6. GRIFFITHS PROBLEM 2.24, PAGE 77. Hint: Take a look-see at BARTON.CH01.DIRAC-DELTA-FCN.

"Physical reality is a ray in Hilbert Space!"

--- Paul Adrien Maurice Dirac (1902–1984)