Transformations of the schrödinger equation with discrete energy spectrum

Sergey Dubovichenko

Zhenjun Ding

On the condition that the electric charge takes discrete value,the finite differenc Schrodinger equation ofthemesoscopic RLC circuit is achieved.With a unitary transformation,the Schrodinger equation is transformed into the standard Mathieu equation in the generalized momentum presentation.The energy spectrum and the wave functions of the system are obtained.The average of currents and square of the current are calculated by using the WKM method.The results show the existence of the current.

Yinghua Ji

The quantization of a mesoscopic transmission line in accordance with the discreteness of electric charges is proposed under the Born–Von-Karmann periodic boundary condition. The finite-difference Schrödinger equation of the circuit has been obtained in charge representation, and the Schrödinger equation is transformed into Mathieu equation in current representation. The wavefunction and energy spectrum can be solved by adopting the canonical transformation and unitary transformation method. Our results indicate that in the transmission line, quantum fluctuation of currents, which also have distributed properties, are related to both the circuit parameters and the positions and the mode of signals.

Yu. N. Demkov, N. V. Semenova

Rafael de la Madrid

It is shown that the natural framework for the solutions of any Schrödinger equation whose spectrum has a continuous part is the rigged Hilbert space (RHS) rather than just the Hilbert space. The difficulties of using only the Hilbert space to handle unbounded Schrödinger Hamiltonians whose spectrum has a continuous part are disclosed. Those difficulties are overcome by using an appropriate RHS. The RHS is able to associate an eigenket with each energy in the spectrum of the Hamiltonian, regardless of whether the energy belongs to the discrete or to the continuous part of the spectrum. The collection of eigenkets corresponding to both discrete and continuous spectra forms a basis system that can be used to expand any physical wavefunction. Thus the RHS treats discrete energies (discrete spectrum) and scattering energies (continuous spectrum) on the same footing.

G. Levin, Wesley Jones, Kamil Walczak +1 more