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: How can I do some free design work?
Answer: I started out much like you my friend, I had a lot of talent and skills and no real knowledge of how to put these things to use, obviously I had a little bit of learning to do so what I did in my spare time was contact companies in my local area that I was interested in, and either did some work on my own, and brought it in to ask what they think, or id reach out to them and simply ask them if they needed any work done. This will go a long with in both establishing a client base, or atleast a group of people who are familiar with you and who can provide your name to someone they know needing the services in the future and it will also teach you a lot about how to work with clients, and probably the best part of all, you get to meet new people in your community and lend a helping hand!