I'm continuing[1,2] the study of an infinite square well in the context of quantum mechanics. 

Ultimate goal is to calculate the product $\Delta x\Delta k$, for various eigenstates, that is for various values of number $n$. I have finished with $\Delta x$, but I'm stuck with $\Delta k$.

ClearAll["Global`*"];
(* The length of the well *)
L = 1;

(* The eigenfunctions, n=1,2,3,... *)
u[n_, x_] := If[x &lt;= 0 || x &gt;= L, 0, Sqrt[2/L] Sin[n π x / L]]

(* The Fourier transform of eigenfunctions u[n,x] from the position
   domain onto the momentum domain *)
φ[n_, k_] :=
    Simplify[
        FourierTransform[u[n, x], x, k, FourierParameters -&gt; {0, -1}],
        n ∈ Integers]

(* The probability density function η(n,k) *)
η[n_, k_] := 
    FullSimplify[φ[n, k] \[Conjugate] φ[n, k],
    {n ∈ Integers, k ∈ Reals}]

(* Calculate (Δk)^2 = &lt;k^2&gt; - &lt;k&gt;^2 = &lt;k^2&gt; *)
Integrate[
    k^2 η[n, k], {k, -∞, +∞},
    (* Edited: Was: {n ∈ Integers, n &gt; 0}, but this edit didn't
       fix the problem. *) 
    Assumptions -&gt; n ∈ Integers &amp;&amp; n &gt; 0]


The problem is that Mathematica can't calculate the last integral for any arbitrary $n$, although it can, correctly, calculate its value for hardcoded $n$s. Like $n=1,2,...$.

My question is:
Do you have any idea on how I could calculate it, perhaps by rewriting it a bit, or by using some other trick? In case it helps, the result should be $n^2\pi^2$.

Note: Actually it can be calculated with Cauchy's residue theorem, but I'd like to avoid taking that route, if possible. Though, if it can't be done otherwise, I will post a solution with residual calculation so that this question has an answer.

Mathematica.SE related (to the physical problem) questions:

Is there a more mathematica-y way to label these plots?

Why does FourierTransform converge while same integral manually written does not?


Question: What do I do when a co-author takes too long to give feedback during the peer review process?
Answer: It's inappropriate to submit a paper for publication before all authors agree that it is ready.  Your boss shouldn't have advised you to do this.

First, you should apologize profusely to your coauthor.  Make sure he knows about the deadline, then wait patiently until he is able to give his comments.  If the deadline gets close, you could ask the editor whether you can have more time.  If not, it may be necessary to withdraw your submission and resubmit later.

When you collaborate with coauthors, you have to accept that the final product needs to be something that everyone can agree on.  Yes, this may be inconvenient if one of them is slower, or has different standards than the others, and it can cause you to miss opportunities.  This is the price of collaboration. If it becomes a problem, you should talk to your coauthors and try to work out a solution that's agreeable to all, but you cannot act unilaterally.