Tips for problem set 8
-For Question 1, there are many practice questions in chapter 40 of the text. Here is another simple practice question and solution.
-Question
2 is closely related to question 3 from tutorial 10, so you may want to
look at the solutions for that (or look at the notes on complex
numbers).
A key point is that when the wavefunction is complex, the probabilty density is the magnitude squared of the complex number. So if Ψ(x) = a + i b
then:
|Ψ(x) |2= a2 + b2
-For Questions 3 and 4, there are some integrals you
have to do. I don't really care how you do them, so feel free to use
integration tables or software like Wolfram Alpha. However, you
will probably still need to simplify your initial integrals in
order to be able to use a table or computer. For example, if we had an
integral like:
integral ( 1/(1+ 4 a2 h2 p2) dp)
from
p=p0 to p=infinity. Then as a first step, it's usually good to change
variables to get rid of the constants and h inside. So we could define
x = 2 a h p. We also have dp = dx/(2 a h), so the integral becomes
1/(2 a h) integral(1/(1+ x2) dx)
The
integral is now from x = 2 a h p0 to x = infinity. In this form, it
will be no problem to use a table or software to do the integral.
For part 4b, remember that A(p) is interpreted as the wavefunction for momentum.