Schrodinger equation for the nonrelativistic particle constrained on a hypersurface in a curved space

Angela Foerster, H.O. Girotti, P.S. Kuhn

Akira Shimizu, Tadashi Inamoto, T. Miyazaki

Takahiro Homma

Takahiro Homma, Tadashi Inamoto, Tadashi Miyazaki

Dirac-bracket quantization of the nonrelativistic particle whose motion is constrained on the hypersurface f(x)=const embedded in a general curved space is discussed. The noncanonical nature of commutation relations makes it difficult to obtain the coordinate representation. Then the system with the derivative-type constraint df(x)/dt=0 is alternatively quantized, treating carefully the operator-ordering problem. In this system it is shown that there exist no constraints on coordinates and momenta and that one can thus have a straightforward representation, which leads, in turn, to the representation and the Schr\"odinger equation in the former system.

Qiao Yu Li, Z. Li, X. Wang +1 more

There is a belief that the Ehrenfest theorem holds true universally. We demonstrate that for a classically nonrelativistic particle constrained on an $N-1$ ($N\geq 2$) curved hypersurface embedded in $N$ flat space, the theorem breaks down.

Ding-Kun Lian, Liangdong Hu, Quanhui Liu

In classical mechanics, a nonrelativistic particle constrained on an N − 1 curved hypersurface embedded in N flat space experiences the centripetal force only. In quantum mechanics, the situation is totally different for the presence of the geometric potential. We demonstrate that the motion of the quantum particle is ”driven” by not only the centripetal force, but also a curvature induced force proportional to the Laplacian of the mean curvature, which is fundamental in the interface physics, causing curvature driven interface evolution.