On the properties of wave functions

In standard textbooks on preliminary quantum mechanics, it is usually stated that, any physical wave function must be single-valued, which is surely all the time true, as long as the wave function is interpreted as the probability amplitude. Besides, it is usually also stated that, the wave functions (together with its first derivatives) have to be continuous. Obviously, the single-valued-ness does not warrant any continuity. Here I just want to emphasize that, the former property is a direct sequel-a of the Born interpretation, whereas the latter is never a must. Actually, the latter is model dependent: different Hamiltonian can lead to different matching conditions that may not necessarily demand the wave function itself or its derivative be continuous. Indeed, in the conventional p^2/2m case (free from singular potentials), from the Schrodinger equation, H wavefunction=E wave function, directly follows the continuity of the first derivative of the physical wave functions, from which follows the continuity of wave functions themselves. On the other hand, for Dirac-type Hamiltonian that is linear in p (also free from singular potentials), one can only derive from the corresponding Schrodinger equation the continuity of the wave functions, and none can be imposed upon their derivatives. Even more, in the case of singular potentials, for the Dirac (conventional) case, the wave functions (the first derivative) must be discontinuous to satisfy the Schrodinger equation.

Definitely, the above discussions apply to any kind of wave equations, such as Maxwell equations. In summary: (1) single-valued-ness is a must; (2) continuity is not.