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My research group finds/proposes a fundamental operator in quantum mechanics, the Cartesian momentum as I called (I think for mathematician the ref. 2007 is sufficient). However, I do not know whether it is self-adjoint or not (we are all physicists). If a mathematician can give a definite answer to it for even simple surfaces such as cylindrical and spherical, he has then a nice paper.

The standard representation of the curved smooth surface $M$ embedded\ in $ R^{3}$ is,

$\mathbf{r}(\xi ,\zeta )\mathbf{=}\left( x(\xi ,\zeta ),y(\xi ,\zeta ),z(\xi ,\zeta )\right)$.

The covariant derivatives of $\mathbf{r}$ are $\mathbf{r}_{\mu }=\partial \mathbf{r}/ \partial x^{\mu }$ .

The contravariant derivatives

$\mathbf{r}^{\mu }\equiv g^{\mu \upsilon }\mathbf{r}_{\upsilon }$

is the generalized inverse of the covariant ones $\mathbf{r}_{\mu }$.

The unit normal vector at point $(\xi ,\zeta )$ is $\mathbf{n=r}^{\xi } \times \mathbf{r}^{\zeta }/ \sqrt{g}$.

The Hermitian Cartesian momentum $\mathbf{p}$ takes a compact form,

$\mathbf{p=}-i\hbar (\mathbf{r}^{\mu }\partial _{\mu }+H\mathbf{n),}$

where $H$ is the mean curvature of the surface. When the motion is constraint-free or in a flat plane, i.e., when $H=0$, the constraint induced terms $H\mathbf{n}$ vanish. Then the Cartesian momentum operator reproduces its usual form as, $\mathbf{p=}-i\hbar \nabla $.

For a particle moves on the surface of a sphere of radius $r$, $ x=r\sin \theta \cos \varphi ,\text{ }y=r\sin \theta \sin \varphi ,\text{ }z=r\cos \theta$,

the hermitian operators for Cartesian momenta $p_{i}$ are respectively,

$p_{x} =-\frac{i\hbar }{r}(\cos \theta \cos \varphi \frac{\partial }{\partial \theta }-\frac{\sin \varphi }{\sin \theta }\frac{\partial }{\partial \varphi }-\sin \theta \cos \varphi ), $

$p_{y} =-\frac{i\hbar }{r}(\cos \theta \sin \varphi \frac{\partial }{\partial \theta }+\frac{\cos \varphi }{\sin \theta }\frac{\partial }{\partial \varphi }-\sin \theta \sin \varphi ), $

$p_{z} =\frac{i\hbar }{r}(\sin \theta \frac{\partial }{\partial \theta }+\cos \theta ).$

On the spherical surface, the complete set of the spherical harmonics defines the Hilbert space.


Refs.

2003, Liu Q H and Liu T G, Int. Quantum Hamiltonian for the Rigid Rotator, J. Theoret. Phys. 42(2003)2877.

2004, Liu Q H, Hou J X, Xiao Y P and Li L X, Quantum Motion on 2D Surface of Nonspherical Topology, Int. J. Theoret. Phys. 43(2004)1011.

2005, Xiao Y P, Lai M M, Hou J X, Chen X W and Liu Q H, A Secondary Operator Ordering Problem for a Charged Rigid Planar Rotator in Uniform Magnetic Field, Comm. Theoret. Phys. 44(2005)49.

2006a, Lai M M, Wang X, Xiao Y P and Liu Q H, Gauge Transformation and Constraint Induced Operator Ordering for Charged Rigid Planar Rotator in Uniform Magnetic Field, Comm. Theoret. Phys. 46(2006) 843.

2006b, Wang X, Xiao Y P, Liu T G, Lai M M and Rao, Quantum Motion on 2D Surfaces of Spherical Topology, Int. J. Theoret. Phys. 45(2006)2509.

2006c, Liu Q H, Universality of Operator Ordering in Kinetic Energy Operator for Particles Moving on two Dimensional Surfaces, Int. J. Theoret. Phys. 45(2006)2167.

2007, Liu Q H., Tong C L., Lai M M., Constraint-induced mean curvature dependence of Cartesian momentum operators J. Phys. A 40(2007)4161.

2010, Zhu X M, Xu M and Liu Q H, Wave packets on spherical surface viewed from expectation values of Cartesian variables, Int. J. Geom. Meth. Mod. Phys., 7(2010)411-423.

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1 Answer 1

If H is bounded and the surface is geodesically complete, then I bet the operator is self-adjoint. The proofs of such theorems rest on an approximation of 1 by smooth compactly supported functions with bounded derivatives, constructed from the Riemannian distance function. Should be quite simple.

I have no library access currently. Do you have a link to 2007?

The trick should be found in one of the following (but I couldn't check).

M.P. Gaffney, A Special Stokes's Theorem for Complete Riemannian Manifolds, Ann. of Math. (2) 60 (1954), 140 145.

P.R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Funct. Anal. 12 (1973), 401 414.

R.S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983), 48 79.

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@ Martin Gisser: iopscience.iop.org/1751-8121/40/15/007 or arxiv.org/abs/0910.0769 –  QHLIU Sep 12 '10 at 3:22

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