The Role of Unbound Wavefunctions
in Energy Quantization and Quantum Bifurcation
The energy eigenvalues of a confined quantum system are traditionally determined by solving the time-independent Schrödinger equation for square-integrable solutions. The resulting bound solutions give rise to the well-known phenomenon of energy quantization, but the role of unbound solutions, which are not square integrable, is still unknown. In this paper, we release the square-integrable condition and consider a general solution to the Schrödinger equation, which contains both bound and unbound wavefunctions. With the participation of unbound wavefunctions, we derive universal quantization laws from the discrete change of the number of zero of the general wavefunction, and meanwhile we propose a quantum dynamic description of energy quantization, in terms of which a new phenomenon regarding the synchronization between energy quantization and quantum bifurcation is revealed.