Dear Kwant developers,
I've found in other threads in the mailing list that the units of current is for example (unit of charge)/(hbar/unit of energy) (https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg01100.html). Also, the local density of states has units of energy/volume (https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg00169.html?).
My question is, what is the units of the output of the density operator? Is it energy/volume as well? I ask this because intuitively I view it as the square of the wavefunction, but it gives me values larger than 1 for each site when there is only 1 mode involved (see attached picture) ?so it is not just the probability of findinge the electron at that site because this should be maximum 1. I have also noticed that the values I get in the colorbar depend on the value of my hopping (e.g. case of graphene), but overall I'm not so sure of the units.
Thank you again for your help.
Kind regards,
Marc
Dear Marc,
There is no problem in a density of probability being larger than one. In fact, this doesn’t prevent it from being normalizable. Example: the probability of the transmission T in random cavities P(T)=1/(2 sqrt(T)) for T in [0, 1]. This function is normalizable despite the fact that it diverges near T=0.
Now, for the wavefunction, you should remember that for infinite systems (open systems) the wavefucntion is not normalizable in the usual way. In fact, we say that it is normalizable in the sens of Dirac-distributions. (delta(k-k'))
The unit of your density is therefore 1/a with 'a' the latice constant. The hopping 't' and the parameter 'a' are related, that is way, changing the hopping gives you a different result. I hope this helps. Adel
On Mon, Oct 22, 2018 at 9:40 AM Marc Vila marc.vila@icn2.cat wrote:
Dear Kwant developers,
I've found in other threads in the mailing list that the units of current is for example (unit of charge)/(hbar/unit of energy) ( https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg01100.html). Also, the local density of states has units of energy/volume ( https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg00169.html ).
My question is, what is the units of the output of the density operator? Is it energy/volume as well? I ask this because intuitively I view it as the square of the wavefunction, but it gives me values larger than 1 for each site when there is only 1 mode involved (see attached picture) so it is not just the probability of findinge the electron at that site because this should be maximum 1. I have also noticed that the values I get in the colorbar depend on the value of my hopping (e.g. case of graphene), but overall I'm not so sure of the units.
Thank you again for your help.
Kind regards,
Marc
Dear Adel,
Thank you for your quick answer. It has certainly been helpful.
Bests,
Marc
------------------------ Marc Vila Tusell La Caixa - Severo Ochoa PhD in the Theoretical and Computational Nanoscience Group Catalan Institute of Nanoscience and Nanotechnology (ICN2) Barcelona Institute of Science and Technology (BIST)
Additional information:
http://icn2.cat/en/theoretical-and-computational-nanoscience-group
https://www.researchgate.net/profile/Marc_Vila_Tusell
https://www.becarioslacaixa.net/marc-vila-tusell-BI00042?nav=true
https://orcid.org/0000-0001-9118-421X
________________________________ From: Abbout Adel abbout.adel@gmail.com Sent: Monday, October 22, 2018 10:40 AM To: Marc Vila Cc: kwant-discuss Subject: Re: [Kwant] Units of density
Dear Marc,
There is no problem in a density of probability being larger than one. In fact, this doesn't prevent it from being normalizable. Example: the probability of the transmission T in random cavities P(T)=1/(2 sqrt(T)) for T in [0, 1]. This function is normalizable despite the fact that it diverges near T=0.
Now, for the wavefunction, you should remember that for infinite systems (open systems) the wavefucntion is not normalizable in the usual way. In fact, we say that it is normalizable in the sens of Dirac-distributions. (delta(k-k'))
The unit of your density is therefore 1/a with 'a' the latice constant. The hopping 't' and the parameter 'a' are related, that is way, changing the hopping gives you a different result. I hope this helps. Adel
On Mon, Oct 22, 2018 at 9:40 AM Marc Vila <marc.vila@icn2.catmailto:marc.vila@icn2.cat> wrote:
Dear Kwant developers,
I've found in other threads in the mailing list that the units of current is for example (unit of charge)/(hbar/unit of energy) (https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg01100.html). Also, the local density of states has units of energy/volume (https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg00169.html?).
My question is, what is the units of the output of the density operator? Is it energy/volume as well? I ask this because intuitively I view it as the square of the wavefunction, but it gives me values larger than 1 for each site when there is only 1 mode involved (see attached picture) ?so it is not just the probability of findinge the electron at that site because this should be maximum 1. I have also noticed that the values I get in the colorbar depend on the value of my hopping (e.g. case of graphene), but overall I'm not so sure of the units.
Thank you again for your help.
Kind regards,
Marc
-- Abbout Adel
Hi again,
......In my previous message, I was referring to the case of 1d. the unit is (1/a). In 2 and 3 dimensions it will be (1/V), where V is the unit volume. Adel
On Mon, Oct 22, 2018 at 9:40 AM Marc Vila marc.vila@icn2.cat wrote:
Dear Kwant developers,
I've found in other threads in the mailing list that the units of current is for example (unit of charge)/(hbar/unit of energy) ( https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg01100.html). Also, the local density of states has units of energy/volume ( https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg00169.html ).
My question is, what is the units of the output of the density operator? Is it energy/volume as well? I ask this because intuitively I view it as the square of the wavefunction, but it gives me values larger than 1 for each site when there is only 1 mode involved (see attached picture) so it is not just the probability of findinge the electron at that site because this should be maximum 1. I have also noticed that the values I get in the colorbar depend on the value of my hopping (e.g. case of graphene), but overall I'm not so sure of the units.
Thank you again for your help.
Kind regards,
Marc
Hi,
I've found in other threads in the mailing list that the units of current is for example (unit of charge)/(hbar/unit of energy) (https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg01100.html). Also, the local density of states has units of energy/volume (https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg00169.html%E...).
The units of local density of states is rather "per energy per volume" (this is what is written in the linked message) not "energy per volume". Though
My question is, what is the units of the output of the density operator? Is it energy/volume as well?
It is "per energy per volume", as is the local density of states.
If you sum the output of a kwant.operator.Density for all scattering states at a given energy and divide the result by 2pi, it will be identical (up to numerical precision) to the output of kwant.ldos. I've attached a script that illustrates this.
I ask this because intuitively I view it as the square of the wavefunction, but it gives me values larger than 1 for each site when there is only 1 mode involved (see attached picture) so it is not just the probability of findinge the electron at that site because this should be maximum 1. I have also noticed that the values I get in the colorbar depend on the value of my hopping (e.g. case of graphene), but overall I'm not so sure of the units.
The scattering wavefunctions are not normalized over the scattering region, so if you sum the absolute square of the wavefunction you will not obtain 1. The lead modes are normalized such that they carry unit current, and the scattering wavefunctions are thus normalized in a way that is commensurate with this normalization of the lead modes.
In the attached script I also show that the norm of the scattering wavefunction over the scattering region is not 1.
Happy Kwanting,
Joe
?Hi Joseph,
Thank you a lot for the fast and detailed response and the example. Now I understand much better.
Kind regards,
Marc
------------------------ Marc Vila Tusell La Caixa - Severo Ochoa PhD in the Theoretical and Computational Nanoscience Group Catalan Institute of Nanoscience and Nanotechnology (ICN2) Barcelona Institute of Science and Technology (BIST)
Additional information:
http://icn2.cat/en/theoretical-and-computational-nanoscience-group
https://www.researchgate.net/profile/Marc_Vila_Tusell
https://www.becarioslacaixa.net/marc-vila-tusell-BI00042?nav=true
https://orcid.org/0000-0001-9118-421X
________________________________ From: Joseph Weston joseph.weston08@gmail.com Sent: Monday, October 22, 2018 11:03 AM To: Marc Vila; kwant-discuss@kwant-project.org Subject: Re: [Kwant] Units of density
Hi,
I've found in other threads in the mailing list that the units of current is for example (unit of charge)/(hbar/unit of energy) (https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg01100.html). Also, the local density of states has units of energy/volume (https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg00169.html?https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg00169.html%E2%80%8B).
The units of local density of states is rather "per energy per volume" (this is what is written in the linked message) not "energy per volume". Though
My question is, what is the units of the output of the density operator? Is it energy/volume as well?
It is "per energy per volume", as is the local density of states.
If you sum the output of a kwant.operator.Density for all scattering states at a given energy and divide the result by 2pi, it will be identical (up to numerical precision) to the output of kwant.ldos. I've attached a script that illustrates this.
I ask this because intuitively I view it as the square of the wavefunction, but it gives me values larger than 1 for each site when there is only 1 mode involved (see attached picture) ?so it is not just the probability of findinge the electron at that site because this should be maximum 1. I have also noticed that the values I get in the colorbar depend on the value of my hopping (e.g. case of graphene), but overall I'm not so sure of the units.
The scattering wavefunctions are not normalized over the scattering region, so if you sum the absolute square of the wavefunction you will not obtain 1. The lead modes are normalized such that they carry unit current, and the scattering wavefunctions are thus normalized in a way that is commensurate with this normalization of the lead modes.
In the attached script I also show that the norm of the scattering wavefunction over the scattering region is not 1.
Happy Kwanting,
Joe
Hi Joseph,
Playing with my code (graphene lattice), I see that if I double my hopping parameter, the density in my colourplot is reduced by 2, which makes sense according to the units. However I'm not finding any changes when varying either the scattering region size or the lattice parameter. I thought that when we said volume (or area in my case) it was related to the unit cell volume. Is it or is for example the area of the scattering region or something else?
Thanks again for your time.
Marc
------------------------
Marc Vila Tusell La Caixa - Severo Ochoa PhD in the Theoretical and Computational Nanoscience Group Catalan Institute of Nanoscience and Nanotechnology (ICN2) Barcelona Institute of Science and Technology (BIST)
Additional information:
http://icn2.cat/en/theoretical-and-computational-nanoscience-group
https://www.researchgate.net/profile/Marc_Vila_Tusell
https://www.becarioslacaixa.net/marc-vila-tusell-BI00042?nav=true
https://orcid.org/0000-0001-9118-421X
________________________________ From: Joseph Weston joseph.weston08@gmail.com Sent: Monday, October 22, 2018 11:03 AM To: Marc Vila; kwant-discuss@kwant-project.org Subject: Re: [Kwant] Units of density
Hi,
I've found in other threads in the mailing list that the units of current is for example (unit of charge)/(hbar/unit of energy) (https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg01100.html). Also, the local density of states has units of energy/volume (https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg00169.html?https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg00169.html%E2%80%8B).
The units of local density of states is rather "per energy per volume" (this is what is written in the linked message) not "energy per volume". Though
My question is, what is the units of the output of the density operator? Is it energy/volume as well?
It is "per energy per volume", as is the local density of states.
If you sum the output of a kwant.operator.Density for all scattering states at a given energy and divide the result by 2pi, it will be identical (up to numerical precision) to the output of kwant.ldos. I've attached a script that illustrates this.
I ask this because intuitively I view it as the square of the wavefunction, but it gives me values larger than 1 for each site when there is only 1 mode involved (see attached picture) ?so it is not just the probability of findinge the electron at that site because this should be maximum 1. I have also noticed that the values I get in the colorbar depend on the value of my hopping (e.g. case of graphene), but overall I'm not so sure of the units.
The scattering wavefunctions are not normalized over the scattering region, so if you sum the absolute square of the wavefunction you will not obtain 1. The lead modes are normalized such that they carry unit current, and the scattering wavefunctions are thus normalized in a way that is commensurate with this normalization of the lead modes.
In the attached script I also show that the norm of the scattering wavefunction over the scattering region is not 1.
Happy Kwanting,
Joe
participants (3)
-
Abbout Adel -
Joseph Weston -
Marc Vila