Schrödinger Equation of a Particle on a Rotating Curved Surface

Abstract

We derive the Schrödinger equation of a particle constrained to move on a rotating curved surface S. Using the thin-layer quantization scheme to confine the particle on S, and with a proper choice of gauge transformation for the wave function, we obtain the well-known geometric potential Vg and an additive Coriolis-induced geometric potential in the co-rotational curvilinear coordinates. This novel effective potential, which is included in the surface Schrödinger equation and is coupled with the mean curvature of S, contains an imaginary part in the general case which gives rise to a non-Hermitian surface Hamiltonian. We find that the non-Hermitian term vanishes when S is a minimal surface or a revolution surface which is axially symmetric around the rolling axis.

Exact Solutions to Schrödinger Equation for a Charged Particle on a Torus in Uniform Electric and Magnetic Fields

“…We wrote the equation as a Sturm-Liouville problem and then used a Liouville-Green transformation, which allowed us to identify the angular differential equation for ψ(θ) as a Hill's differential equation-for the magnetic field case. We solve this Hill's equation, (13), exactly using a technique similar to that presented by Whittaker and Watson [32]. The energy eigenvalues present an interesting feature for the case where there is no magnetic field: each pair of successive levels n-for instance, 4th and 5th, 6th and 7th, and so forth-becomes closer as n increases.…”

“…Inspired by recent developments in nanodevices [1][2][3][4], there is an increasing effort to study Schrödinger [5][6][7][8], Pauli [9], and Dirac [10] equations constrained to surfaces and curves [11][12][13]. The first problem is to obtain such equations considering the correct curvature and other geometrical effects.…”

“…The outline for our paper is the following: in section 2, we solve the equation for a scalar charged particle constrained to move along the surface of a torus in the presence of a uniform magnetic field and then in the presence of an electric field. In the first case, we must deal with the Hill's equation-see (13)-and in the second with a differential equation having not just cosines but also sines. In section 3, we discuss four other external potentials that could be studied using the same approach of section 2.…”

Electrostatically interacting Wannier qubits in curved space

“…Furthermore quantum machine learning can mimic any stochastic finite state machine by usage of tight-binding model as it was shown in [26]. Using the concept of reconfigurable q-graph of quantum dot [22] we can simulate the behaviour of quantum particle in curved space what is the subject of future work [31], [34], [33]. Another future direction is the investigation of holonomic quantum computation [29], [30] with qubits constructed from curvy semiconductor nanowires described in this work.…”

Digraph states and their neural network representations

“…Solutions within the standard framework of quantum mechanics generally require the knowledge of the full quantum many-body wave function. Thus, the problem becomes how to solve the many-body Schrödinger equation [2][3][4] of the system with a large dimension. This is just the so-called quantum manybody problem (QMBP) [5][6][7] in quantum physics, which becomes a hot topic in high energy physics and condensed matter physics.…”