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2D vs axial symmetry 2D modeling, Conical Quantum Dot
Posted Aug 1, 2010, 3:07 p.m. EDT 3 Replies
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Hi guys,
I asked the Support, Case: 426303, but got too shallow an answer, and my objections went unnoticed.
It seems the Conical Quantum Dot built-in model (3.5a version of COMSOL) must not produce correct results, if I got it right.
The model is based on 2D geometry, not axial symmetry (2D), while the independent variables are r and z as if it is in the "axial symmetry (2D)" geometry. The independent variables r and z are just names, and COMSOL must have treated them as Cartesian x and y.
But suppose in Cartesian coordinates we have an integral $\int F(x,y,z) dx dy dz$. Its analog in cylindrical coordinates is $\int F(r,\phi,z) r dx d\phi dz$. The extra "r" under the integration sign is just a determinant of the Jacobian of the transformation from Cartesian to cylindrical coordinates. So dealing with cylindrical coordinates all integrals must include unit volume "r". For example, in quantum mechanics, orthogonality of real functions F_1 and F_2 means $\int r F_1 F_2 dr d\phi dz =0 with that extra r.
I'm absolutely sure COMSOL uses some internal integrations to solve a problem, in particular as it must use the weak formulation. Thus for "2D" space it must use the unit volume = 1, and for "axial 2D" the unit volume = r. That's why I doubt the correctness of the Quantum Dot model. Once again, the equation and boundary conditions are fine, but all internal integrations COMSOL uses to solve the problem should have included an extra r.
It can be viewed from different side, the Hamiltonian of the quantum mechanical model is Hermitian only if r is a cylindrical coordinate. Try to use the coordinate r as a Cartesian coordinate x, and you'll have problems with Hermiticity, which might lead to complex eigenenergies.
Guys, am I right?
Another question: can we correct the written model just making a change of a volume parameter like dvol = r instead of dvol =1? But it seems that this parameter cannot be varied...
WBR
Ed
I asked the Support, Case: 426303, but got too shallow an answer, and my objections went unnoticed.
It seems the Conical Quantum Dot built-in model (3.5a version of COMSOL) must not produce correct results, if I got it right.
The model is based on 2D geometry, not axial symmetry (2D), while the independent variables are r and z as if it is in the "axial symmetry (2D)" geometry. The independent variables r and z are just names, and COMSOL must have treated them as Cartesian x and y.
But suppose in Cartesian coordinates we have an integral $\int F(x,y,z) dx dy dz$. Its analog in cylindrical coordinates is $\int F(r,\phi,z) r dx d\phi dz$. The extra "r" under the integration sign is just a determinant of the Jacobian of the transformation from Cartesian to cylindrical coordinates. So dealing with cylindrical coordinates all integrals must include unit volume "r". For example, in quantum mechanics, orthogonality of real functions F_1 and F_2 means $\int r F_1 F_2 dr d\phi dz =0 with that extra r.
I'm absolutely sure COMSOL uses some internal integrations to solve a problem, in particular as it must use the weak formulation. Thus for "2D" space it must use the unit volume = 1, and for "axial 2D" the unit volume = r. That's why I doubt the correctness of the Quantum Dot model. Once again, the equation and boundary conditions are fine, but all internal integrations COMSOL uses to solve the problem should have included an extra r.
It can be viewed from different side, the Hamiltonian of the quantum mechanical model is Hermitian only if r is a cylindrical coordinate. Try to use the coordinate r as a Cartesian coordinate x, and you'll have problems with Hermiticity, which might lead to complex eigenenergies.
Guys, am I right?
Another question: can we correct the written model just making a change of a volume parameter like dvol = r instead of dvol =1? But it seems that this parameter cannot be varied...
WBR
Ed
3 Replies Last Post Aug 2, 2010, 3:43 p.m. EDT
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Posted:
1 decade ago
Hi
I agree with you that COMSOL is using a special integration for the 2DAxi symmetry case, it's also stated briefly in the doc. This is to avoid the singularity at the origine. But so far, my experience is that COMSOLcorrects everything also for the results so that it is transparent for us (an xception in the ealy 4.0 release, corrected since). Now you might have hit a particular poin,t but still he matematicians and physicist behind COMSOL also know there lesson so I also trust them.
You can always recalculate it in 2D and define yourself the axisymmetry coordinates and check out your case (and keep us users here, but also CMSOL support informed).
Fhe "dvol" as you state is, in my knowledge, an internal value of "read only" type, but it would be influence by the way you define your "private" axisymmetric coordinate
--
Good luck
Ivar
I agree with you that COMSOL is using a special integration for the 2DAxi symmetry case, it's also stated briefly in the doc. This is to avoid the singularity at the origine. But so far, my experience is that COMSOLcorrects everything also for the results so that it is transparent for us (an xception in the ealy 4.0 release, corrected since). Now you might have hit a particular poin,t but still he matematicians and physicist behind COMSOL also know there lesson so I also trust them.
You can always recalculate it in 2D and define yourself the axisymmetry coordinates and check out your case (and keep us users here, but also CMSOL support informed).
Fhe "dvol" as you state is, in my knowledge, an internal value of "read only" type, but it would be influence by the way you define your "private" axisymmetric coordinate
--
Good luck
Ivar
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Posted:
1 decade ago
Hi Ivar, all
Thank you for your interest. It turns out that my question is deeper than I thought initially, and probably it would be instructive to model and compare the results in both 2d and axial symmetric 2d spaces. But even if they coincided, this is not a general rule. We were lucky with the Conical Quantum Dot model - the equation had real coefficients so it could produce real eigenvalues. But suppose we are dealing with a matrix Hamiltonian with complex coefficients, the Hamiltonian in cylindrical coordinates has a Hermitian form only when cylindrical coordinates are meant. So, working in Cartesian space it may generate complex eigenvalues, which is wrong.
BTW, there is no problems with the singularity at the origin, as the model can explicitly set zero boundary condition for non-zero angular momentum.
WBR
Ed
Thank you for your interest. It turns out that my question is deeper than I thought initially, and probably it would be instructive to model and compare the results in both 2d and axial symmetric 2d spaces. But even if they coincided, this is not a general rule. We were lucky with the Conical Quantum Dot model - the equation had real coefficients so it could produce real eigenvalues. But suppose we are dealing with a matrix Hamiltonian with complex coefficients, the Hamiltonian in cylindrical coordinates has a Hermitian form only when cylindrical coordinates are meant. So, working in Cartesian space it may generate complex eigenvalues, which is wrong.
BTW, there is no problems with the singularity at the origin, as the model can explicitly set zero boundary condition for non-zero angular momentum.
WBR
Ed
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Posted:
1 decade ago
Hi
when I said "working in 2D" I ment starting COMSOL in standard 2D but then add your own cylindrical transform as user variables. Then you know exactly what is happening, as you control all formulas, and you can compare with the 2D axi result of standard COMSOL values
--
Good luck
Ivar
when I said "working in 2D" I ment starting COMSOL in standard 2D but then add your own cylindrical transform as user variables. Then you know exactly what is happening, as you control all formulas, and you can compare with the 2D axi result of standard COMSOL values
--
Good luck
Ivar