Generalization

Abstract

Abstract In this chapter, we generalize the previous path integral expression to a wider class of physical systems. To this end, we first discuss a change of variables in the path integral and the relationship to the operator formalism, and build up the polar coordinate path integral formula. Next we determine those cases where the p and q are confined within some restricted region, known as constrained systems, by applying the Faddeev-Senjanovic path integral formula to a sphere. Further we discuss quantum mechanics on a ring and the Aharonov-Bohm effect as the simplest examples of multiply connected space characterized by unshrinkable closed loops in configuration space. Also we make some comments on quantum mechanics for a more general manifold. In the next section the path integral for spin is de- rived with a newly introduced coherent state that also gives an alternative expression for the harmonic oscillator. Then, finally, cases where the WKB approximation happens to be exact are analysed.