How to determine a Limits of Integration of Wave Packet

  • #1

Homework Statement

Consider a force-free particle of mass m described, at an instant of time t = 0, by
the following wave packet:
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 \\
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

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

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 !
  • #2
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
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
Where did you pick the first one from ? Is there an interval where the wavefunction is 1 ?
  • #5
So, should i include first integral ? That was my problem/question ?

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