[Sona]

The wave function for particle in a one dimensional box isnot the eigenfunction of momentum operator p_x
But it is an eigenfunction of p_x²
what is the significance of this result?
is it that the wavefunction cannot give the absolute value of momentum buy it gives only the average value which is agreeing with the Heisenberg's Uncertainity principle.

[Enthalpy]

One use of an operator is to "measure" (...in a mathematical treatment) a quantity. Here p_x applied to ψ gives a momentum along x.

Now, Heisenberg. If a particle is trapped somewhere, do you have some (imprecise) information about that particle? Is it compatible with an other information about that particle? With arbitrary precision?

And when ψ is an eigenfunction of an operator, how well is the quantity (here the momentum) measured by this operator defined?

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"The wave function for particle in a one dimensional box is an eigenfunction of p_x²"

I doubt that in the general case. Some hypothesis may be missing there, for instance "stationary wave function", "fundamental state" or a similar one.

Do you know a quantity that resembles p²? How precisely is it defined under the previous kind of assumptions?