# Measurement and uncertainty principle in QM

The Wikipedia says on the page for the uncertainty principle:

Mathematically, the uncertainty relation between position and momentum arises because the expressions of the wave function in the two corresponding bases are Fourier transforms of one another (i.e., position and momentum are conjugate variables).

Does that mean that position and momentum are just 2 different measurements of the same wave function? I.e., it is the same thing that is being measured, just in two different ways? Meaning, they are not really two different things, but two different views on the same thing?

-

Can I provide a pedestrian answer? When you measure the position of a quantum, you project or force it to commit to a unique position, and from Fourier Analysis, this commitment requires all possible momenta. Think of focusing a quantum wave to single location (ala Dirac Delta), this would require a wave generator to combine all momenta (and therefore no unique momentum). On the other hand, a measured momenta, would hold for the wave throughout all space, and make its position totally arbitrary. Also, when you do measure a quantum system, you do change it as it changes your measurement apparatus, unless you just performed the very same measurement a moment before. A quantum measurement is generally not objective, since your apparatus and the quantum wave are both involved. Measuring is an active process in Quantum Mechanics, changing your lab (its measuring dials) and the quantum wave.

-
Any measurement in physics is in general described by a probability distribution of different outcomes. This distribution depends both on the state of the system being measured and on the measurement apparatus, which are two different things. In quantum mechanics states are described by vectors in Hilbert space $\left|\psi\right>$ (wavefunctions may be seen as their coordinates in some basis), and measurements by Hermitian operators $\hat{A}$ acting on this space (this is the simpliest case, actually the formalizm is a bit more complicated). Probability distribution of measurement outcomes is given by eigenvalues of these operators, and average values of measured quantities by $\hat{\left<A\right>}=\left<\psi\right|\hat{A}\left|\psi\right>$.
Position and momentum measurements are described by two different operators $\hat{x}$ and $\hat{p}$, such that $\hat{x}\hat{p}-\hat{p}\hat{x}=i\hbar$. Their noncommutativity leads to Heisenberg uncertainty relations for variances of corresponding measurements, as described in wikipedia. So the answer is no, they are different things, measured with different apparatus, but if their are done on a system in a given state, their variances turn out to be related.