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edited Sep 5 2010 at 8:02 |
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 RaoJ\U{ff0c}Quantum , Quantum Motion on 2D Surfaces of Spherical Topology, Int. J. Theoret. Phys. 45(2006)2509.
2006c, Liu Q HInt. , 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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edited Sep 4 2010 at 11:08 |
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edited Sep 4 2010 at 9:41 |
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 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 ), $
$p_{z} =\frac{i\hbar }{r}(\sin \theta \frac{\partial }{\partial \theta }+\cos \theta ).$
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 J\U{ff0c}Quantum Motion on 2D Surfaces of Spherical Topology, Int. J. Theoret. Phys. 45(2006)2509.
2006c, Liu Q H Int. Universality of Operator Ordering in Kinetic Energy Operator for Particles Moving on two Dimensional Surfaces, 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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edited Sep 4 2010 at 9:35 |
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 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 $\mathbf{r}_{\mu }=\partial \mathbf{r}/ \partial x^{\mu }$ ,
and then the metric tensor $g_{\mu \upsilon }$ is easily formed as$ g_{\mu \upsilon }\equiv \mathbf{r}{\mu }\cdot \mathbf{r}{\upsilon }$.
The unit normal vector at point $(\xi ,\zeta )$ is
$\mathbf{n=r}^{\xi } \times \mathbf{r}^{\zeta }/ \sqrt{g}$. .
The contravariant derivatives
$\mathbf{r}^{\mu }\equiv g^{\mu \upsilon }\mathbf{r}_{\upsilon }$
is the generalized inverse of the covariant ones $\mathbf{r}_{\mu }$ for we have $.
The unit normal vector at point $\mathbf{r}^{\mu }(\xi ,\zeta )$ . is $\mathbf{r}_{\upsilon \mathbf{n=r}^{\xi } $
$=g^{\mu \alpha }\mathbf{r}{\alpha }$ . $\mathbf{r}{\upsilon }$
$=g^{\mu times \alpha mathbf{r}^{\zeta }g_{\alpha / \upsilon }$
$=\delta _{\upsilon }^{\mu }$. 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 ), $
$p_{z} =\frac{i\hbar }{r}(\sin \theta \frac{\partial }{\partial \theta }+\cos \theta ).$
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 J\U{ff0c}Quantum Motion on 2D Surfaces of Spherical Topology, Int. J. Theoret. Phys. 45(2006)2509.
2006c, Liu Q H Int. Universality of Operator Ordering in Kinetic Energy Operator for Particles Moving on two Dimensional Surfaces, 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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edited Sep 4 2010 at 9:30 |
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 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
$\mathbf{r}_{\mu }=\partial \mathbf{r}/ \partial x^{\mu }$ ,
and then the metric tensor $g_{\mu \upsilon }$ is easily formed as
$g_{\mu \upsilon }\equiv \mathbf{r}{\mu }\cdot \mathbf{r}{\upsilon }$.
The unit normal vector at point $(\xi ,\zeta )$ is
$\mathbf{n=r}^{\xi } \times \times \mathbf{r}^{\zeta }/\sqrt{g}$.
/ \sqrt{g}$.
The contravariant derivatives
$\mathbf{r}^{\mu }\equiv g^{\mu \upsilon }\mathbf{r}{\upsilon \mathbf{r}_{\upsilon }$
is the generalized inverse of the covariant ones $\mathbf{r}{\mu \mathbf{r}_{\mu }$ for we have
$\mathbf{r}^{\mu }\cdot \mathbf{r}{\upsilon $ . $\mathbf{r}_{\upsilon }$
$=g^{\mu \alpha }\mathbf{r}{\alpha }\cdot \mathbf{r}$ . $\mathbf{r}{\upsilon }$
$=g^{\mu \alpha }g{\alpha g_{\alpha \upsilon }$
$=\delta _{\upsilon }^{\mu }$.
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 ), $
$p_{z} =\frac{i\hbar }{r}(\sin \theta \frac{\partial }{\partial \theta }+\cos \theta ).$
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 J\U{ff0c}Quantum Motion on 2D Surfaces of Spherical Topology, Int. J. Theoret. Phys. 45(2006)2509.
2006c, Liu Q H Int. Universality of Operator Ordering in Kinetic Energy Operator for Particles Moving on two Dimensional Surfaces, 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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edited Sep 4 2010 at 9:25 |
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 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
$\mathbf{r}{\mu \mathbf{r}_{\mu }$ $=$ $\partial =\partial \mathbf{r}/\partial x^{\mu}$mathbf{r}/ \partial x^{\mu }$ ,
and then the metric tensor $g{\mu g_{\mu \upsilon }$ is easily formed as
$g_{\mu \upsilon }\equiv \mathbf{r}{\mu }\cdot \mathbf{r}{\upsilon }$.
The unit normal vector at point $(\xi ,\zeta )$ is
$\mathbf{n=r}^{\xi }\times \mathbf{r}^{\zeta }/\sqrt{g}$.
The contravariant derivatives
$\mathbf{r}^{\mu }\equiv g^{\mu \upsilon }\mathbf{r}{\upsilon }$
is the generalized inverse of the covariant ones
$\mathbf{r}{\mu }$ for we have $\mathbf{r}^{\mu }\cdot \mathbf{r}{\upsilon }=g^{\mu \alpha }\mathbf{r}{\alpha }\cdot \mathbf{r}{\upsilon }=g^{\mu \alpha }g{\alpha \upsilon }=\delta _{\upsilon }^{\mu }$.
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 ), $
$p_{z} =\frac{i\hbar }{r}(\sin \theta \frac{\partial }{\partial \theta }+\cos \theta ).$
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 J\U{ff0c}Quantum Motion on 2D Surfaces of Spherical Topology, Int. J. Theoret. Phys. 45(2006)2509.
2006c, Liu Q H Int. Universality of Operator Ordering in Kinetic Energy Operator for Particles Moving on two Dimensional Surfaces, 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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edited Sep 4 2010 at 9:18 |
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 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 $ $=$ $\partial \mathbf{r}/\partial x^{\mu}$,
and then the metric tensor $g{\mu \upsilon }$ is easily formed as
$g_{\mu \upsilon }\equiv \mathbf{r}{\mu }\cdot \mathbf{r}{\upsilon }$.
The unit normal vector at point $(\xi ,\zeta )$ is
$\mathbf{n=r}^{\xi }\times \mathbf{r}^{\zeta }/\sqrt{g}$.
The contravariant derivatives
$\mathbf{r}^{\mu }\equiv g^{\mu \upsilon }\mathbf{r}{\upsilon }$
is the generalized inverse of the covariant ones
$\mathbf{r}{\mu }$ for we have $\mathbf{r}^{\mu }\cdot \mathbf{r}{\upsilon }=g^{\mu \alpha }\mathbf{r}{\alpha }\cdot \mathbf{r}{\upsilon }=g^{\mu \alpha }g{\alpha \upsilon }=\delta _{\upsilon }^{\mu }$.
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 ), $
$p_{z} =\frac{i\hbar }{r}(\sin \theta \frac{\partial }{\partial \theta }+\cos \theta ).$
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 J\U{ff0c}Quantum Motion on 2D Surfaces of Spherical Topology, Int. J. Theoret. Phys. 45(2006)2509.
2006c, Liu Q H Int. Universality of Operator Ordering in Kinetic Energy Operator for Particles Moving on two Dimensional Surfaces, 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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Are these operators defined on 2D surfaces self-adjoint?
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 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 $\mathbf{r}{\mu }=\partial \mathbf{r}/\partial x^{\mu}$, and then the metric tensor $g{\mu \upsilon }$ is easily formed as $g_{\mu \upsilon }\equiv \mathbf{r}{\mu }\cdot \mathbf{r}{\upsilon }$. The unit normal vector at point $(\xi ,\zeta )$ is $\mathbf{n=r}^{\xi }\times \mathbf{r}^{\zeta }/\sqrt{g}$. The contravariant derivatives $\mathbf{r}^{\mu }\equiv g^{\mu \upsilon }\mathbf{r}{\upsilon }$ is the generalized inverse of the covariant ones $\mathbf{r}{\mu }$ for we have $\mathbf{r}^{\mu }\cdot \mathbf{r}{\upsilon }=g^{\mu \alpha }\mathbf{r}{\alpha }\cdot \mathbf{r}{\upsilon }=g^{\mu \alpha }g{\alpha \upsilon }=\delta _{\upsilon }^{\mu }$. 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 ), $
$p_{z} =\frac{i\hbar }{r}(\sin \theta \frac{\partial }{\partial \theta }+\cos \theta ).$
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 J\U{ff0c}Quantum Motion on 2D Surfaces of Spherical Topology, Int. J. Theoret. Phys. 45(2006)2509.
2006c, Liu Q H Int. Universality of Operator Ordering in Kinetic Energy Operator for Particles Moving on two Dimensional Surfaces, 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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