Uncertainty Principle and the Infinite Well

For the infinite square well in one-dimension the wavefunctions have the form Acos(kx) where k is the wavenumber which is proportional to momentum. Now due to H.U.P. if Δx is fixed as the infinite well size we can't know the exact momentum. I presume this is because the wavefunction exists as a superposition of all the wavefunctions with k quantized so the momentum is not precisely known ?

If I am correct so far ; what happens when the wavefunction is observed ? Surely it collapses to one of its eigenstates with k and hence momentum now precisely known which violates the H.U.P. as Δx is fixed ?



Thanks