HW01

ASSIGNED:     FRI 18 JAN
DUE:                  FRI 25 JAN

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 1.3, PAGE 12: First carry out the integrals by hand using lookup tables and then do them again using the Integrate function in Mathematica. Using the Plot function in Mathematica, plot the function ρ(x) as a function of  x for the three values σ = 0.1, 1.0, 10.0 (with a = 1) with all three curves in a single plot. Then plot the function ρ(x) as a function of  x for the three values a = –1.0, 0.0, +1.0 (with σ = 1.0) with all three curves again in a single plot.

2. GRIFFITHS PROBLEM 1.5, PAGE 14: 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 part (c) produce three graphs in a single plot for the three values σ = 0.1, 1.0, 10.0. Then calculate, using Mathematica, the three probabilites that the particle is found outside the range 〈x〉±σ  for these three different values of σ using the Integrate function. Hint: To handle the |x|, break the integral up into two regions, x>0 and x<0. To integrate a function f(x) from 0 to infinity you just type:
Integrate[f[x], {x, 0, Infinity}] into Mathematica.

3. GRIFFITHS PROBLEM 1.7, PAGE 18. This can only be done by hand so no Mathematica.

4. GRIFFITHS PROBLEM 1.9, PAGE 20. This can only be done by hand so no Mathematica.

5. GRIFFITHS PROBLEM 1.14, PAGE 21. This can only be done by hand so no Mathematica. Add an additional part (c): The isotope carbon-14 is used in radiocarbon dating to estimate the age of carbon-bearing objects (such as human fossils and trees) up to about 60,000 years old. Carbon-14 radioactively decays into nitrogen-14 (as well as an electron and a neutrino) with a half-life of  η = 5,730 years.  That is if you start with 100 carbon-14 atoms at time zero, after 5,730 years you will have (on average) 50 carbon-14 atoms left. By measuring the ratio of carbon-14 to nitrogen-14 in an object, you can then estimate its age. The half life is defined to be the time it takes for an initial sample of N atoms to decay to N/2 atoms and it is independent of N. Using this information, derive a formula for the lifetime τ of a single carbon-14 atom in terms of the given half life η for a collection of N carbon-14 atoms, and then use η = 5,730 years to estimate τ  (in years) for carbon-14 (to 4 significant digits). Using this value you find for τ, compute P(t) evaluated at the half life time of t = η = 5,730 and then explain the result. Hint!