Quantum mechanics with constraints

Heisenberg equation for a nonrelativistic particle on a hypersurface: From the centripetal force to a curvature induced force

“…In fact, the situation is far more complicated than what is anticipated. This is because in quantum mechanics for motion on the hypersurface, there is a curvature induced potential [2][3][4][5] that has no classical correspondence, and we can by no mean assume that same form of the Ehrenfest theorem for the time derivative of mean value of the momentum applies.…”

“…For a spheroid x 2 + y 2 /a 2 + z 2 /b 2 = 1, ∇ 2 M reaches its maximum − b 2 − a 2 b/a 6 at the top (x, y, z) = (0, 0, b) for the prolate spheroid and it reaches its maximum b 2 − a 2 b 2 + 3a 2 / 2ab 6 at the equator for the oblate spheroid, and for spherical surface, χ g = 0. For a torus with R being the distance from the center of the tube to the center of the torus and r (≺ R) being the radius of the circular tube, ∇ 2 M = R(r + R sin θ)/(2r 2 (R + r sin θ) 3 ) reaches its maximum R(R r)/(2r 2 (R r) 3 ) when sin θ = 1 which lines out the circumference of the inside circle (with radius R r) of the torus.…”

Curvature induced out-of-plane spin accumulation in Rashba quantum waveguides

“…For example, the q 1 coordinate can be the x coordinate along the top half of a not necessarily circular cross section of a uniform, infinitely long cylinder, and q 2 the dimension along its length. In such systems where one or more of the dimensions are curved, there is an additional energy contribution commonly known as the da Costa geometric confinement potential [17,18] which can be physically interpreted as the energy contribution from the 'force' confining the charged carriers to stay on the curved surface. This term has the mathematical form of −κ 2 /(8m) where κ is the radius of curvature.…”

Quantum Mechanics on Surfaces

“…definitely does since curved surfaces in ordinary space can generally not be completely charted with Cartesian coordinates. The most generally accepted adaption of the canonical quantization procedure to curved surfaces and linear structures was developed in (5)(6)(7). What follows is a brief account of this adaptation.…”