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 R3 is,
mathbfr(xi,zeta)mathbf=left(x(xi,zeta),y(xi,zeta),z(xi,zeta)right).
The covariant derivatives of mathbfr are mathbfrmu=partialmathbfr/partialxmu .
The contravariant derivatives
mathbfrmuequivgmuupsilonmathbfrupsilon
is the generalized inverse of the covariant ones mathbfrmu.
The unit normal vector at point (xi,zeta) is mathbfn=rxitimesmathbfrzeta/sqrtg.
The Hermitian Cartesian momentum mathbfp takes a compact form,
mathbfp=−ihbar(mathbfrmupartialmu+Hmathbfn),
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 Hmathbfn vanish. Then the Cartesian momentum operator reproduces its usual form as, mathbfp=−ihbarnabla.
For a particle moves on the surface of a sphere of radius r, x=rsinthetacosvarphi,texty=rsinthetasinvarphi,textz=rcostheta,
the hermitian operators for Cartesian momenta pi are respectively,
px=−fracihbarr(costhetacosvarphifracpartialpartialtheta−fracsinvarphisinthetafracpartialpartialvarphi−sinthetacosvarphi),
py=−fracihbarr(costhetasinvarphifracpartialpartialtheta+fraccosvarphisinthetafracpartialpartialvarphi−sinthetasinvarphi),
pz=fracihbarr(sinthetafracpartialpartialtheta+costheta).
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.