Some counterexamples in quantum mechanics

By constructing specific examples it is shown that the generally held views about (a) the connection between symmetry and degeneracy (b) the degeneracy of the ground state (c) the number of nodes of the wavefunction (d) invariances and conservation laws and (e) the relation between quantum and classical degeneracies are not always true. In addition it is observed that the separability of the wavefunction in more than one coordinate system is not a necessary condition for the presence of the accidental degeneracy. Other assumption's regarding the nonuniqueness of the ordering of operators, quantizable classical system, determination of the wavefunction and the minimal coupling rule for the motion of a charged particle in an external electromagnetic field are all questioned.