see http://front.math.ucdavis.edu/0608.4038
For me at the moment this is a bookmark .. I hope there will be discussion of the concepts involved either now or later..
-C2.
Can't open pdf
Opens in WindowsXP .. I don't know what version of pdf reader I have* .. generally I find it best not to enquire about these things until they stop working.
The link should have been to the arxiv page http://arxiv.org/abs/quant-ph/0608038 (as yor_on has already pointed out)
( which currently links to the same file - so no help
)-C2.
* Adobe doo da 8.0
I had (initially) intended to promote (to the best of my ability) a very conventional analysis at the same time as other analyses continue within the forum.
In an attempt to solve the differential equation 'without help' I looked at the wiki article of Fourier Transforms. The current article (22/03/2008) may be unstable - so grab quotes
Edit.. here:- http://en.wikipedia.org/wiki/Fourier_transform
The equality is achieved for the Gaussian function listed above, which shows that the Gaussian function is maximally concentrated in "time-frequency".
The most famous practical application of this property is found in quantum mechanics. Following from the axioms of quantum mechanics, the momentum and position wave functions are Fourier transform pairs to within a factor of h/2π and are normalized to unity. The above expression then becomes a statement of the Heisenberg uncertainty principle.
I'm operating beyond my ability here .. but it looks very interesting.
-C2.
In an attempt to solve the differential equation 'without help' I looked at the wiki article of Fourier Transforms. The current article (22/03/2008) may be unstable - so grab quotes
Edit.. here:- http://en.wikipedia.org/wiki/Fourier_transform
QUOTE (wiki+)
The equality is achieved for the Gaussian function listed above, which shows that the Gaussian function is maximally concentrated in "time-frequency".
The most famous practical application of this property is found in quantum mechanics. Following from the axioms of quantum mechanics, the momentum and position wave functions are Fourier transform pairs to within a factor of h/2π and are normalized to unity. The above expression then becomes a statement of the Heisenberg uncertainty principle.
I'm operating beyond my ability here .. but it looks very interesting.
-C2.
Hi Confused2, yor_on, DavidD et al,
I think you are on to something there Confused2. As a preliminary investigation it is very interesting indeed. It certainly coincides with my point of view and with Feynman's Many Path Concepts. I think Cramer's Interpretational Hypothesis is also supported by this approach. Using the Fourier Process it may be extended to any number of dimensions simply "inverting" in phase at "nodes". One may ask why the authors say this is a "controversial problem", it seems to be one in which "everyone got it wrong" till now? The controversy seems to be around why this "blind spot" in the treatment persists considering the benefits of the approach?
http://uk.arxiv.org/abs/quant-ph/0608038
On the question of access to the documents... I find no specific difficulties with reading this reference. I am using only Adobe Reader 5.1. I have mentioned previously and elsewhere that Adobe Reader 8.0 has some unacknowledged problems reading certain fonts used in certain Scientific Papers... some of them I have made reference to in these threads. This fault renders them unable to be read. I am sure that this fault may be rectified in the future but for now perhaps one could drop back to an earlier version of Acrobat Reader or use another "supplementary" freeware product like Foxit Pdf Reader (be sure that when you run it you do not allow it to associate with the pdf extension to specifically open pdf files using only Foxit).
Foxit Pdf Reader
I would suggest the Foxit program does have a few problems because it does not allow universal text copying of portions of some certain files (but works for most) in other respects is very workable. Saving the file locally and opening in Adobe Reader may also help. Another point is do not bother with the Foxit Pdf Creator if you need to... use something like Pdf995. On the positive side it is a very small download and has most reader functions. The "image" can be partially read using OCR software found in the latest versions of MS Office 2003 or later.
I think you are on to something there Confused2. As a preliminary investigation it is very interesting indeed. It certainly coincides with my point of view and with Feynman's Many Path Concepts. I think Cramer's Interpretational Hypothesis is also supported by this approach. Using the Fourier Process it may be extended to any number of dimensions simply "inverting" in phase at "nodes". One may ask why the authors say this is a "controversial problem", it seems to be one in which "everyone got it wrong" till now? The controversy seems to be around why this "blind spot" in the treatment persists considering the benefits of the approach?
http://uk.arxiv.org/abs/quant-ph/0608038
On the question of access to the documents... I find no specific difficulties with reading this reference. I am using only Adobe Reader 5.1. I have mentioned previously and elsewhere that Adobe Reader 8.0 has some unacknowledged problems reading certain fonts used in certain Scientific Papers... some of them I have made reference to in these threads. This fault renders them unable to be read. I am sure that this fault may be rectified in the future but for now perhaps one could drop back to an earlier version of Acrobat Reader or use another "supplementary" freeware product like Foxit Pdf Reader (be sure that when you run it you do not allow it to associate with the pdf extension to specifically open pdf files using only Foxit).
Foxit Pdf Reader
I would suggest the Foxit program does have a few problems because it does not allow universal text copying of portions of some certain files (but works for most) in other respects is very workable. Saving the file locally and opening in Adobe Reader may also help. Another point is do not bother with the Foxit Pdf Creator if you need to... use something like Pdf995. On the positive side it is a very small download and has most reader functions. The "image" can be partially read using OCR software found in the latest versions of MS Office 2003 or later.
QUOTE (Wiki+)
The equality is achieved for the Gaussian function listed above, which shows that the Gaussian function is maximally concentrated in "time-frequency".
The most famous practical application of this property is found in quantum mechanics. Following from the axioms of quantum mechanics, the momentum and position wave functions are Fourier transform pairs to within a factor of h/2π and are normalized to unity. The above expression then becomes a statement of the Heisenberg uncertainty principle.
http://en.wikipedia.org/wiki/Fourier_transform
The most famous practical application of this property is found in quantum mechanics. Following from the axioms of quantum mechanics, the momentum and position wave functions are Fourier transform pairs to within a factor of h/2π and are normalized to unity. The above expression then becomes a statement of the Heisenberg uncertainty principle.
http://en.wikipedia.org/wiki/Fourier_transform
The "reciprocal dual" 201 and 202 is most important since every other function may be derived from it as a series of delta functions.
http://en.wikipedia.org/wiki/Sinc_function
The Sinc function is similar to the "normalized" Airy function so useful in optics. It is also the solution to Schrödinger's equation for a particle confined within a triangular potential well. Some aspects of this type of treatment we have discussed previously. Would you like to discuss this paper further Confused2? I certainly would.
Three Experiments in One
Cheers
http://en.wikipedia.org/wiki/Sinc_function
The Sinc function is similar to the "normalized" Airy function so useful in optics. It is also the solution to Schrödinger's equation for a particle confined within a triangular potential well. Some aspects of this type of treatment we have discussed previously. Would you like to discuss this paper further Confused2? I certainly would.
Three Experiments in One
Cheers
Hi Confused2, yor_on, DavidD et al,
Using your document I was able to search for other interesting material and I found this...
All comments (I suspect...) welcome...
Cheers
Using your document I was able to search for other interesting material and I found this...
QUOTE
Can degenerate bound states occur in one dimensional quantum mechanics?
Authors: Sayan Kar, Rajesh R. Parwani
(Submitted on 8 Jun 2007 (v1), last revised 17 Sep 2007 (this version, v2))
Abstract: We point out that bound states, degenerate in energy but differing in parity, may form in one dimensional quantum systems even if the potential is non-singular in any finite domain. Such potentials are necessarily unbounded from below at infinity and occur in several different contexts, such as in the study of localised states in brane-world scenarios. We describe how to construct large classes of such potentials and give explicit analytic expressions for the degenerate bound states. Some of these bound states occur above the potential maximum while some are below. Various unusual features of the bound states are described and after highlighting those that are ansatz independent, we suggest that it might be possible to observe such parity-paired degenerate bound states in specific mesoscopic systems.
Comments: 10 pages, 2 figures, to appear in Europhysics Letters
Subjects: Quantum Physics (quant-ph); Mesoscopic Systems and Quantum Hall Effect (cond-mat.mes-hall); High Energy Physics - Theory (hep-th)
Cite as: arXiv:0706.1135v2 [quant-ph]
http://uk.arxiv.org/abs/quant-ph/0608038
This also suggests a similar line of investigation.Authors: Sayan Kar, Rajesh R. Parwani
(Submitted on 8 Jun 2007 (v1), last revised 17 Sep 2007 (this version, v2))
Abstract: We point out that bound states, degenerate in energy but differing in parity, may form in one dimensional quantum systems even if the potential is non-singular in any finite domain. Such potentials are necessarily unbounded from below at infinity and occur in several different contexts, such as in the study of localised states in brane-world scenarios. We describe how to construct large classes of such potentials and give explicit analytic expressions for the degenerate bound states. Some of these bound states occur above the potential maximum while some are below. Various unusual features of the bound states are described and after highlighting those that are ansatz independent, we suggest that it might be possible to observe such parity-paired degenerate bound states in specific mesoscopic systems.
Comments: 10 pages, 2 figures, to appear in Europhysics Letters
Subjects: Quantum Physics (quant-ph); Mesoscopic Systems and Quantum Hall Effect (cond-mat.mes-hall); High Energy Physics - Theory (hep-th)
Cite as: arXiv:0706.1135v2 [quant-ph]
http://uk.arxiv.org/abs/quant-ph/0608038
All comments (I suspect...) welcome...
Cheers
Hi Good Elf et al,
This isn't going as I intended. Instead of simply following an exemplary solution of the Schrodinger equation and deciding what it all means afterwards .. we don't seem to have got that far. I'm not sure to what extent the FT is part of the method and to what extent it is part of the solution.
-C2
This isn't going as I intended. Instead of simply following an exemplary solution of the Schrodinger equation and deciding what it all means afterwards .. we don't seem to have got that far. I'm not sure to what extent the FT is part of the method and to what extent it is part of the solution.
-C2
Hi C2,
I think this was not a good choice of problem for what you wanted to do with it. Probably you saw the title "One dimensinal Hydrogen atom" and thought that it would be a nice simplified "test case" of Schrodinger's equation. That's what I would have thought, too, but after reading the paper, it seems that there are mathematical difficulties in 1 dimension that do not occur in 3 dimensions. The 1 dimensional H atom is actually HARDER to analyze than the 3d one.
The authors of the paper trace this difficulty to the fact that the potential energy is negative infinity at the origin. In 3d, the origin is simply a point, and you can consider a wave function that is defined everywhere except the origin, and things work just fine. On the other hand, in 1d, the origin cuts the 1d space into two separate pieces, so you get two disconnected parts of the wave function, with no clear way to connect them. It's as if you made a jigsaw puzzle with square pieces: ANY piece might connect to any other piece. So the authors have to fall back on physical interpretation, i.e. what PICTURE do the pieces make, in order to match the left and right sides up correctly. That brings up a whole lot of issues that aren't elementary, such as domains of Hermiticity (which the authors refer to as self-adjointness). That takes you into functional analysis territory, which is somewhere you do NOT want to go when you're just coming to grips with solving Schrodinger's equation.
The 2d H atom would probably be easier, but be warned that the solution will involve the SAME mathematical technique (separation of variables) that is used to solve the 3d H atom. These techniques are related to Fourier analysis, in the sense that you consider how to build up your function out of standardized pieces. But in the 2d case, these pieces are Bessel functions in the radial direction (if I recall correctly -- someone tell me if I'm wrong, please), and imaginary exponentials in the angular direction (using polar coordinates of course; Cartesian coordinates are ridiculous for problems with circular symmetry). Bessel functions are rather like the trig functions, but less familiar to most people, which makes them seem mysterious at first. Again, this is probably not what you were looking for.
The 3d H atom solution will involve separation of variables in spherical coordinates, and will produce solutions combining a radial part (which looks like a polynomial * a decreasing exponential), a latitude angle part (which looks like sin(n*theta) and a longitude angle part (which looks like an imaginary exponential).
You will learn more about mathematics than about physics by studying these systems, but it is very useful mathematics. Still, if what you want is to see clearly what Schrodinger's equation says and how it works, without getting too deep into the math, your best choices are probably the "particle in a box," "step potential," "delta potential," or "harmonic oscillator." The first one has the easiest math and is a good representation of how an electron behaves inside a rectangular chunk of metal. The second one demonstrates how particles can scatter off a boundary. The third shows how a particle deals with a very narrow, deep energy hole, essentially a "sticky spot." The last is a physical system that shows up everywhere in lots of different contexts, basically because whenever a smooth function has a minimum, then near the miminum, the function USUALLY looks like a parabola, and when the potential energy looks like a parabola, you get a harmonic oscillator.
If any of those sound interesting, I'll try to direct you toward an outline of solving the Schrodinger equation for one of them. From there, you might work up to the H atom, although as I mentioned, these other functions have physical meanings of their own, and are not just abstract stuff to try out the equation on, so there is some physical merit in studying them for their own sakes, not just as a stepping stone to further things.
Hope that helps!
--Stuart Anderson
I think this was not a good choice of problem for what you wanted to do with it. Probably you saw the title "One dimensinal Hydrogen atom" and thought that it would be a nice simplified "test case" of Schrodinger's equation. That's what I would have thought, too, but after reading the paper, it seems that there are mathematical difficulties in 1 dimension that do not occur in 3 dimensions. The 1 dimensional H atom is actually HARDER to analyze than the 3d one.
The authors of the paper trace this difficulty to the fact that the potential energy is negative infinity at the origin. In 3d, the origin is simply a point, and you can consider a wave function that is defined everywhere except the origin, and things work just fine. On the other hand, in 1d, the origin cuts the 1d space into two separate pieces, so you get two disconnected parts of the wave function, with no clear way to connect them. It's as if you made a jigsaw puzzle with square pieces: ANY piece might connect to any other piece. So the authors have to fall back on physical interpretation, i.e. what PICTURE do the pieces make, in order to match the left and right sides up correctly. That brings up a whole lot of issues that aren't elementary, such as domains of Hermiticity (which the authors refer to as self-adjointness). That takes you into functional analysis territory, which is somewhere you do NOT want to go when you're just coming to grips with solving Schrodinger's equation.
The 2d H atom would probably be easier, but be warned that the solution will involve the SAME mathematical technique (separation of variables) that is used to solve the 3d H atom. These techniques are related to Fourier analysis, in the sense that you consider how to build up your function out of standardized pieces. But in the 2d case, these pieces are Bessel functions in the radial direction (if I recall correctly -- someone tell me if I'm wrong, please), and imaginary exponentials in the angular direction (using polar coordinates of course; Cartesian coordinates are ridiculous for problems with circular symmetry). Bessel functions are rather like the trig functions, but less familiar to most people, which makes them seem mysterious at first. Again, this is probably not what you were looking for.
The 3d H atom solution will involve separation of variables in spherical coordinates, and will produce solutions combining a radial part (which looks like a polynomial * a decreasing exponential), a latitude angle part (which looks like sin(n*theta) and a longitude angle part (which looks like an imaginary exponential).
You will learn more about mathematics than about physics by studying these systems, but it is very useful mathematics. Still, if what you want is to see clearly what Schrodinger's equation says and how it works, without getting too deep into the math, your best choices are probably the "particle in a box," "step potential," "delta potential," or "harmonic oscillator." The first one has the easiest math and is a good representation of how an electron behaves inside a rectangular chunk of metal. The second one demonstrates how particles can scatter off a boundary. The third shows how a particle deals with a very narrow, deep energy hole, essentially a "sticky spot." The last is a physical system that shows up everywhere in lots of different contexts, basically because whenever a smooth function has a minimum, then near the miminum, the function USUALLY looks like a parabola, and when the potential energy looks like a parabola, you get a harmonic oscillator.
If any of those sound interesting, I'll try to direct you toward an outline of solving the Schrodinger equation for one of them. From there, you might work up to the H atom, although as I mentioned, these other functions have physical meanings of their own, and are not just abstract stuff to try out the equation on, so there is some physical merit in studying them for their own sakes, not just as a stepping stone to further things.
Hope that helps!
--Stuart Anderson
Hi Mr Homm,
I hope I may be forgiven for paddling out out to sea and waiting to be rescued..
".. how an electron behaves inside a rectangular chunk of metal" (particle in a box) - I think that meets every requirement - hopefully not too difficult and gives us lesser mortals something to get our teeth into. We've done 'particle in a box' before on this forum but it turned into a dry exercise. If you are willing then perhaps we can try again and extract some of the essence of the result.
-C2.
Open 'PM' to Mr Homm
Many many thanks for your immense contribution to this forum.
I hope I may be forgiven for paddling out out to sea and waiting to be rescued..
".. how an electron behaves inside a rectangular chunk of metal" (particle in a box) - I think that meets every requirement - hopefully not too difficult and gives us lesser mortals something to get our teeth into. We've done 'particle in a box' before on this forum but it turned into a dry exercise. If you are willing then perhaps we can try again and extract some of the essence of the result.
-C2.
Open 'PM' to Mr Homm
Many many thanks for your immense contribution to this forum.
OK so I'm on my own here ..
Here's a nice particle in a box analysis
http://itl.chem.ufl.edu/4412_aa/partinbox.html
If we get as far as Eq 12 we see (roughly)
Psi(x) = sqrt(1/L) sin (n*pi*x/L)
since I don't care about the absolute amplitude this gives us
Psi(x) = k sin(n*pi*x/L)
Looking at an actual 'cavity' made up of two microwave dishes facing each other we find they 'resonate' when the total path length (that is 2L) is an integer number of wavelengths .. curiously similar to the QM result above
.
We should look at the classical (EM) meaning of 'resonance' .. maybe tomorrow.
-C2.
..Here's a nice particle in a box analysis
http://itl.chem.ufl.edu/4412_aa/partinbox.html
If we get as far as Eq 12 we see (roughly)
Psi(x) = sqrt(1/L) sin (n*pi*x/L)
since I don't care about the absolute amplitude this gives us
Psi(x) = k sin(n*pi*x/L)
Looking at an actual 'cavity' made up of two microwave dishes facing each other we find they 'resonate' when the total path length (that is 2L) is an integer number of wavelengths .. curiously similar to the QM result above
. We should look at the classical (EM) meaning of 'resonance' .. maybe tomorrow.
-C2.
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