Can the wave function of a single free particle in time evolve into two same wave functions (like single free particle) under a potential? What are the conditions under which it can happen?
Let's first decide on your terminology. If you are talking about a wavefunction -- usually that means \psi(\vec{r},t), it depends on just one \vec{r}, hence describes by construction something related to just one particle. |\psi(\vec{r},t)|^2 = the density of probability to find that particle at \vec{r} at time t, etc. -- no room in this description for another particle. What do you imagine could happen to this \psi(\vec{r},t) that it will "evolve into two same wave functions"? Unclear from your question. Same goes about talking about a "free particle" and at the same time about some potential.
As it stands description in terms of the wave function as above is not used for processes in which particles can be created/annihilated. (This \psi(\vec{r},t), or analogous wavefunction for some >1 number of particles, evolves according to Shroedinger equation, which preserves \int |\psi(\vec{{r}_i},t)|^2 d^3{r_i} for a Hermitean hamiltonian; you cannot fit non-conservation of number of particles into this picture). One normally works in the framework of second quantization when processes changing the number of particles are to be considered. Creation operator in that framework does just that -- creates another particle.
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