> we can measure the wave function at any given time We can't. Nobody knows how to measure a wavefunction. Instead, what you measure is one of the eigenvalue

> we can measure the wave function at any given time We can't. Nobody knows how to measure a wavefunction. Instead, what you measure is one of the eigenvalues of a linear operator that acts on a wavefunction (which particular eigenvalue? You can't always predict this; hence the randomness in QM). For each "observable" you want to measure (position, momentum, spin), there is a specific linear operator associated with it. For instance, in position-space, the position operator is just the identity operator. If you haven't taken linear algebra, I can clarify this further.