Thin layer quantization in higher dimensions and codimensions

Abstract

We consider the thin layer quantization with use of only the most elementary notions of differential geometry. We consider this method in higher dimensions and get an explicit formula for quantum poten

Canonical quantization of motion on submanifolds

“…(For Prokhorov's view see [30].) This method gives the same results [2] as the thin layer approach due to a very simple physical reason. The lowest energy level wave functions (in the model with two infinite potential walls) have nodes at x n = ±δ and the bunch at x n = 0: ∂ n χ = 0 or, equivalently,P n Ψ = 0.…”

“…where ∆ LB is the Laplace-Beltrami operator on the surface x n = const. The simplest way [2] to obtain the thin layer limit is to consider the tangent paraboloid of the surface y n = 1…”

“…For example, in the tubular neighbourhood of a curve the normal connection rotates the normal vectors not only with the normal hyperplanes but also around the curve together with its Frenet frame. Of course, on a curve we can exclude this effect by a suitable choice of rotating coordinates [2] as in the previous subsubsection. But for higher dimensional (and codimensional) manifolds the normal bundle can be non-flat and such exclusion would be impossible.…”

“…Da Costa concluded [34] that the thin layer quantization would not work well in this situation. This statement was repeated in [2]. Strictly speaking, this attitude is not right because all the dangerous terms sum up to the angular momenta operators corresponding to rotations in normal sections of the thin tube [8,23,39,40].…”

Some remarks on approximation methods for quantum systems in higher space dimensions

“…For example, the N-dimensional analogy of the hydrogen atom has been investigated extensively over the years (Avery and Herschbach, 1992;Kirchberg et al, 2003;Saelen et al, 2007). In addition, the generalization to higher dimensions is useful in random walks (Mackay et al, 2002), in Casimir effect (Bender et al, 1992), in harmonic oscillator (Oyewumi et al, 2008;Al-Jaber, 2008;Rothos et al, 2009), and in mathematical physics (Bredies,2009;Szmytkowski, 2007;Bouda and Ghabri, 2008;Golovnev, 2006). It is interesting to expose undergraduate students to simple, but illustrative, quantum systems in higher space dimensions.…”