A quantum particle which is almost at rest but whose position is random! Assume a particle is given by a quantum state which is constructed in such a way that it is equally probable to find it anywhere in an fixed interval $(0,L)$ but has arbitrarily low velocity. The particle is almost at rest but its position is still random!
I will give a construction for such a quantum state. Let 
$$\psi(x) = B(x)e^{ikx},$$ 
where $B(x)$ is a function with smooth cut off which is almost like a square function which is $1$ in $(0,L)$ and zero everywhere else. ( I have introduced $B(x)$ over square function to eliminate arguments about continuity of $\psi(x)$).
The most probable momentum of this particle is $k$ with a manageble uncertainty. If we fix $L$ to be at certain high value, then the uncertainty in momentum is fixed at a low value.
Now the question is, in the above set up we can choose $k$ to be arbitrarily small and with almost same low uncertainty and still have its position equally distributed in an entire interval $(0,L)$.
In essence we have a valid quantum state in which a particle is almost at rest, but whose position is totally random in an interval. Although I am new to quantum mechanics, I find this very intriguing and almost hard to imagine such a real world situation. Is this a possible flaw with in QM? I'd appreciate your comments and explanation of any pitfalls in this argument.
 A: I) A mollifier $B$ times a plane wave $e^{ikx}$ is an example of a smooth wave packet $\psi\in {\cal H}=L^2(\mathbb{R})$. The Heisenberg uncertainty relation is a mathematical consequence of the CCR, and holds in particular for the above mentioned wave packet $\psi$. 
II) Intuitively, the expectation value 
$$\langle \hat{p}\rangle~=~\frac{\langle \psi |\hat{p}|\psi\rangle}{\langle \psi|\psi\rangle}$$ 
of the momentum operator is roughly the bare plane wave momentum $\hbar k$; and the slowly varying mollifier $B$ unavoidably creates a possible small, but strictly non-zero uncertainty/fuzziness 
$${\rm Var}(\hat{p})~=~\frac{\langle \psi |(\hat{p}-\langle \hat{p}\rangle)^2|\psi\rangle}{\langle \psi|\psi\rangle}~>~0$$ 
in the momentum; independently of the value of the plane wave wavenumber $k$. 
III) If the reader is unconvinced by the above arguments, then it is a good educational exercise to replace the mollifier $B$ by a slowly varying Gaussian function, and explicitly calculate how everything works out.   
A: This is a basic consequence of the uncertainty principle which is not only not a possible flaw with QM, it is an experimental fact. You can observe this pretty much directly in Bose-Einstein condensates of ultracold atoms, for instance. Conversely, the confinement pressure is essential to the stability of atoms as you will find out in your QM course. It has been verified in a thousand other ways as well. It even has budgetary consequences: it is the reason why the LHC has to be so big to find things like the Higgs boson which can only been seen at very short distance scales.
