Feynman, in his lecture notes, argues convincingly that you will understand the guts of Quantum Mechanics [QM] as best you can by looking at the two-slit experiment whose Hilbert space is two-dimensional. Or you could look at the Stern-Gerlach experiment, where it is three-dimensional. I recommend you stick to these cases, drop $L^2$, and see carefully what all the concepts boil down to there in terms of the Kahler geometry of projective space. Feynman does this, in physics language, in his 1957 paper with Vernon titled `Geometrical‘Geometrical Representation of the Schrodinger Equation for Solving Maser Problems'Problems’: they solve the needed Schrodinger equation by drawing circles on the two-sphere = $CP^1$.
The ‘Berry Phase’ is a post-Feynman idea that is essentially the curvature of the
canonical connection for the canonical line bundle over Berry Phase' is a post-Feynman idea that is essentially the curvature of the canonical connection for the canonical line bundle over $CP^n$. For a dictionary from standard QM to Kahler geometry and connections you could look at Heisenberg and Isoholonomic inequalites' which you can download from$CP^n$. For a dictionary
http://count.ucsc.edu/~rmont/papers/list.html by goingfrom standard QM to the year 1990 thereKahler geometry and connections you could look at ‘Heisenberg and isoholonomic inequalites’.
