Particle in a 1-D box and the correspondence principle Consider the particle in a 1-d box, we know very  well the solutions of it. I'd like to see how the correspondence principle will work out in this case, if we consider position probability density function (pdf) of the particle.
In the classical case, under the absence of any potential, its fair enough to assume the position pdf $P_{classic}$ of the particle to be constant inside the box and zero outside of it.
In the quantum case we know it is $$P_{quantum}(x) = \frac{2}{L} \sin^2(\frac{n\pi x}{L})$$ inside the box and zero outside.
A picture's worth a thousand word so here I grab a picture from a university website (actually google) for which I am thankful to the authors.

From the picture we can see that when we consider large enough quantum number $n$, we see that the $P_{Quantum}$ is still jiggling and no matter how large $n$ we consider, it keeps jiggling and doesn't really bother to converge to $P_{classic}$, but the consolation is, they match in an average sense.
Ideally, for such a fundamental problem like this, I would expect the function $P_{quantum}(x)$ to converge to $P_{classic}(x)$, pointwise. Am I asking for too much? There are infinite number of types of $P_{quantum}$ matching $P_{classic}$ in an average sense, so that would mean there can be infinite number of quantum mechanical theories which obey correspondence principle in average matching, which I think is not ery appealing property of a theory.
My question is, what can we do to make the pdf converge pointwise to the classical pdf?
 A: It is worth noting that while the quantum PDF still exhibits rapid fluxuations, they get to be very high frequency, and when you go to measure the distribution with a physical instrument you introduce a resolution.
At some point that resolution gets to be much larger then the wavelength of the fluxuations and you in fact detect the classical distribution.
A: 
My question is, what can we do to make the pdf converge pointwise to
  the classical pdf

Answer : Nothing, if you accept Quantum principles.
You may see below an other example with the harmonic oscillator, with $N= 50$, and you see the same behaviour you are worrying about (classical behaviour in blue, quantum behaviour in red):
The fundamental reason is that Quantum Mechanics is about correlation amplitudes and probability amplitudes, so  classical statistical concepts  like correlations probabilities or probabilities is a "baby" view of the physic world. So, you cannot avoid " jiggling". The simplest approach is that probabilities amplitudes, or correlation amplitudes, for bound states, are  "naturally" oscillating, this been traduced in probabilities as oscillations between zero and "maximum" values, while, for great quantum numbers, a "mean"  looks like a classical probability.

