  A salient feature of the Schr\"{o}dinger equation is that the classical
radial momentum term $p_{r}^{2}$ in polar coordinates is replaced by the
operator $\hat{P}^{\dagger}_{r} \hat{P}_{r}$, where the operator $\hat{P}_{r}$
is not hermitian in general. This fact has important implications for the path
integral and semi-classical approximations. When one defines a formal hermitian
radial momentum operator $\hat{p}_{r}=(1/2)((\frac{\hat{\vec{x}}}{r})
\hat{\vec{p}}+\hat{\vec{p}}(\frac{\hat{\vec{x}}}{r}))$, the relation
$\hat{P}^{\dagger}_{r}
\hat{P}_{r}=\hat{p}_{r}^{2}+\hbar^{2}(d-1)(d-3)/(4r^{2})$ holds in
$d$-dimensional space and this extra potential appears in the path integral
formulated in polar coordinates. The extra potential, which influences the
classical solutions in the semi-classical treatment such as in the analysis of
solitons and collective modes, vanishes for $d=3$ and attractive for $d=2$ and
repulsive for all other cases $d\geq 4$. This extra term induced by the
non-hermitian operator is a purely quantum effect, and it is somewhat analogous
to the quantum anomaly in chiral gauge theory.
