HW03

ASSIGNED:     FRI 01 FEB
DUE:                  FRI 15 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 2.11, PAGE 50: First carry out the integrals by hand using lookup tables and then do them again using the Integrate function and derivative function D in Mathematica. Hint: You may find some of the formulas in CH2.3.1 useful.

2. GRIFFITHS PROBLEM 2.13, PAGE 50: Again carry out the calculations first by hand using paper and pencil and an integral lookup table and then again using the Integrate function in Mathematica. For parts (b) and (c) use the Manipulate and Animate functions to produce animated plots of the wave function Ψ(x,t) and the probability density |Ψ(x,t)|2 and the time-dependent expectation value <x(t)> as shown in EX02.2.NB. Hints: ψ0 and ψ1 are from EQ02.59 and EQ02.62. For part (c): <p>, figure out who the heck Peter Lorre is and then use EQ01.33.

3. GRIFFITHS PROBLEM 2.15, PAGE 57.  Hint: Skip the math table and use Mathematica to carry out the integral, which is the area under the curve in the "wings" of 0(x)|2 outside of the potential well V(x).

4. GRIFFITHS PROBLEM 2.16, PAGE 57. Again carry out the calculations first by hand using paper and pencil and algebra and then write a Mathematica code to do the same.

5. GRIFFITHS PROBLEM 2.17, PAGE 57. Do it by hand and then again in Mathematica. Hint: Take a look-see at CH02.3.NB.

7. A Quantum SHO is in the initial state Ψ[x,0]=f[x]=B*Sin[x]*Exp[–x^2/2] = f[x], where we have taken the unit of length l=√(ħ/mω)=1.

(a) Normalize this state.

(b) Plot the probability density |Ψ[x,0]|2 on top of the scaled potential V[x]=x^2/2.

(c) Construct the first 5 coefficients cn in the expansion of Ψ[x,0] in terms of the SHO stationary states ψn(x). Which of these are zero and why???

(d) Construct expressions for the solution to the TDSE, Ψ(x,t) and Ψ*(x,t), and then the associated probability distribution |Ψ[x,t]|2 for the first FIVE coefficients cn .

(e) Produce an animated plot of |Ψ[x,t]|2 on top of V(x) with these first five coefficients. Be careful to redfine a new fuction ρplot5[x_, t_] using the new output of ρ[x, t, 5, 5] by just copying and pasting. There is a point x (not infinity) where the probability of finding the particle is always exactly zero. What value of x is it and why?

Hint: Don't try doing this by hand but instead take a look at EX2.3.2.NB.

"I turn with terror and horror from this lamentable scourge of continuous functions with no derivatives."

--- Charles Hermite (1822–1901)