# How to determine a Limits of Integration of Wave Packet

1. Nov 11, 2011

### Zaknife

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

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

3. 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:
$A \ \mathrm{for} \ |x| ≤ a$.
Do you have any ideas ? Thanks in advance !

2. Nov 11, 2011

### dextercioby

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. Nov 11, 2011

### Zaknife

Ok , so far i have:
$\int_{??}^{??} A^{2} dx (?) -\frac{A^{2}}{\epsilon^{2}}\int_{a}^{a+\epsilon} (x-a-\epsilon)^{2} dx +\frac{A^{2}}{\epsilon^{2}}\int_{-a-\epsilon}^{a} (x+a+\epsilon)^{2} dx =1.$ My question is what are the integration limits for the first integral ?

4. Nov 11, 2011

### dextercioby

Where did you pick the first one from ? Is there an interval where the wavefunction is 1 ?

5. Nov 12, 2011

### Zaknife

So, should i include first integral ? That was my problem/question ?