10. Continuous Variables (V)

In this lecture, I show how the ``minimum uncertainty” state can be derived and show that this is the cohearent state for momentum and position uncertainty. Then show that there’s no such an uncertainty between angular momentum and angle operators because the mathematical assumptions in deriving uncertainty relations do not hold in this case. Then I show other difficulties arised by continuous spectra in commutation relations. Finally, I show that continuous operators do not have eigenvectors at all! but since we’ve defined measurements outcomes as eigenvectors, this is a serious difficulty that must be resolved. I’ve introduced the method which ``truncates’’ the Hilbert space in order to approximamte the actual infinite-dimensional Hilbert space by a finite-dimensional one (which doesn’t have such difficulties). But it will turn out that trunction method (which is notably successful in chemical physics), has unfortunate results and cannot be considered as an appropriate way to deal with continuous variables.