• Support PF! Buy your school textbooks, materials and every day products Here!

How to determine a Limits of Integration of Wave Packet

  • Thread starter Zaknife
  • Start date
  • #1
12
0

Homework Statement


Consider a force-free particle of mass m described, at an instant of time t = 0, by
the following wave packet:
[itex]
\begin{array}{l}
0 \ \mathrm{for} \ |x| > a + \epsilon \\
A \ \mathrm{for} \ |x| ≤ a \\
-\frac{A}{\epsilon} (x − a − \epsilon) \ \mathrm{for} \ a < x ≤ a + \epsilon \\
\frac{A}{\epsilon}(x + a + \epsilon) \ \mathrm{for} \ − a − \epsilon ≤ x < a \\
\end{array}
[/itex]
where a, ε, and a normalization constant A are all positive numbers. Calculate mean
values and variances of the position and momentum operators [itex] x , x^{2} , \sigma_{x} \ and \ p_{x}[/itex] .



Homework Equations


[itex]
1=\int_{-\infty}^{\infty} |\psi(x,t)|^{2}
[/itex]

The Attempt at a Solution


I want to determine normalization constant A. I don't know what kind of integration limits i should use for the case:
[itex] A \ \mathrm{for} \ |x| ≤ a [/itex].
Do you have any ideas ? Thanks in advance !
 

Answers and Replies

  • #2
dextercioby
Science Advisor
Homework Helper
Insights Author
12,981
540
You have a lot of intervals and need to use the fact that the integral will be a sum of integrals for each interval. So write all the integrals, compute them all the find A. Then compute all other items.
 
  • #3
12
0
Ok , so far i have:
[itex]\int_{??}^{??} A^{2} dx (?) -\frac{A^{2}}{\epsilon^{2}}\int_{a}^{a+\epsilon} (x-a-\epsilon)^{2} dx +\frac{A^{2}}{\epsilon^{2}}\int_{-a-\epsilon}^{a} (x+a+\epsilon)^{2} dx =1.[/itex] My question is what are the integration limits for the first integral ?
 
  • #4
dextercioby
Science Advisor
Homework Helper
Insights Author
12,981
540
Where did you pick the first one from ? Is there an interval where the wavefunction is 1 ?
 
  • #5
12
0
So, should i include first integral ? That was my problem/question ?
 

Related Threads on How to determine a Limits of Integration of Wave Packet

  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
9
Views
1K
  • Last Post
Replies
2
Views
770
  • Last Post
Replies
10
Views
1K
Replies
4
Views
819
  • Last Post
Replies
2
Views
5K
  • Last Post
Replies
1
Views
3K
Replies
4
Views
1K
Replies
2
Views
7K
  • Last Post
Replies
2
Views
5K
Top