Misc. DIY: Simulate Atoms and Molecules

[andrewr]

A little more on spinning bloch spheres http://www.animatedphysics.com/videos/spinningblochsphere.htm" (objects inside an empty shell do not feel force from the shell) you get a derived pauli exclusion principle that allows 2 electrons into the proton shell. It they wander outside the proton shell, they get pulled back in by the normal coulomb force but once inside they are no longer attracted to the center. A single electron, once inside this type of sphere, will do not gain any more kinetic energy or need to radiate energy as it is no longer seeing any force. Also don't forget that the bloch sphere can represent anti-particles. If you have a BS spinning in one direction, its axis points either up or down. The antiparticle will be the same BS spinning in the opposite direction but its axis is pointed the same way as the particle.

I don't follow this idea. A bloch sphere is a state vector tool as far as I know. It has a fixed radius because a global phase in QM has no meaning. I think you spoke about the nucleus having multiple spin states earlier, and in the literature on Bloch spheres (eg: that I have found) there is mention that the benefits don't extend higher spin orders -- although, I find that puzzling as I would expect each proton to be represented by a single bloch sphere and a second equation would handle the interaction/entanglement issues.

Protons also have orbits with each other -- I think a brilliant woman was involved in that deduction, Maria Goeppert-Mayer, and the mathematics are quite similar to that of electron orbitals. The so called, "strong" force trumping the EM field, but otherwise playing a similar role. I don't see why, then, if a Bloch sphere models a spin 1/2 particle why several of them would not correctly model the nucleus -- although for my purposes, I don't intend to simulate these orbitals as the concepts are too difficult for a first try.
The bloch sphere, as far as I know, isn't intended to model the angular momentum of orbits -- so I am not sure why 7/2 momentum, etc. have anything to do with this. I expect the electron when orbiting, has lower energy when the intrinsic spin of the electron cancels with the orbital motion and thus would automatically lead to a reduction in the overall magnetic moment. Since the Bohr magneton is a measure of the "orbital" effect, I find it no surprise that the electron has a moment nearly double. The exact value of the ratio is something of a clue as to the interaction between electron spin and orbital angular momentum.

The shell theorem is based on the concept of uniformly distributed charge; The nucleus is several point charges in orbits. I know that as an "approximation" one can consider the charge of an electron spread out in the shape of the probability -- but I have seen mention made that this approximation predicts wrongly in many cases. Slater determinants basically assume that interactions are based on the "average" electric field... that is one problem with them; but then so is determining "where" the charge needs to emanate from as a replacement.

My view is that an electron's field might indeed be spread out because of HUP -- but not over an entire atom for the distances are too large. Thus a true shell is not possible to make. I suppose it could be possible for two electrons of similar wavelength to intertwine, for in that case a single electron might produce a shell around the other electron -- but, the energy required to get them that close would seem to be *nuclear!* and I have no idea how they could be gotten apart again either...

I may not understand this correctly, but is the extra wide scatter at x=0 something odd that happened in the 1922 experiment or is it seen all the time?

I have the same question. In the original analysis I read before studying the actual experiment -- the author indicated that the spike was to be ignored as a facet of the *original* SG magnet shape. However, my analysis is based solely by noticing that the gradient is symmetric about the photo's center x axis. (x,+y = x,-y in photo coordinates) Atoms from both sides of the slit would be attracted toward center -- and any atoms colliding would stop precessing and re-orient their trajectory toward the center because of the gradient. Thus, higher density of atoms along the center line would be unavoidable in any experiment which has a symmetry -- which is all that I have seen... and thus, I would expect a spike. Perhaps only the exact shape of the spike is unique to SG, and later experiments have a different distribution; The author I read did not explain in detail. This is one of the reasons I am looking for data from reproductions of the experiment, like MIT's, ... I'll let you know if/when I find any -- but it is important that I reconstruct an actual experiment, and not a dumbed down analysis.
I have in mind that precession changes the angle of the magnetic loop with time, so that even classically the tilt will always sample the horizontal as well as the vertical gradient on all sides of the magnetic field and that modeling it as a static loop of wire at a fixed angle is surely going to give the wrong result.

In my own analysis, I simplified the QM spin in order to extract only the Z motion in the experiment (Z experiment = x in the photo), but I know that the equation I wrote down was unacceptable in reality -- I just didn't want to do the mathematics for the complete gradient so I ignored part of it. OTOH I was just hoping to get an opinion if the principle of my method is anywhere near "standard". I will find it in the literature, eventually, -- but I had hoped to shorten the time by asking questions and sharing information.

I went back to look at Rice.edu's paper, and they removed it -- I guess they didn't like the traffic; So I can't read that analysis again either -- but I noticed one thing at the Stanford site (besides the really grainy pictures) They rotated the oven slit / OFF magnet photo 180 degrees from the way it was on the postcard (LH side photo) -- so perhaps someone there noticed the same thing I did concerning the density not matching the other photo.

 

[alxm]

Since the Bohr magneton is a measure of the "orbital" effect

The Bohr magneton is the measure of intrinsic, spin-related magnetic moment, not orbital magnetic moment.

I know that as an "approximation" one can consider the charge of an electron spread out in the shape of the probability -- but I have seen mention made that this approximation predicts wrongly in many cases.

No, you just minterpreted what was written then.

Slater determinants basically assume that interactions are based on the "average" electric field... that is one problem with them

It makes no such assumptions whatsoever, and as I already told you in this thread, there are exact methods which use Slater determinants. Again you're saying that existing methods don't work when you clearly don't know anything about them.

My view is that an electron's field might indeed be spread out because of HUP -- but not over an entire atom for the distances are too large.

False. Do the math. In fact, try the Feynman lectures on physics where he derives the Bohr radius using the HUP.

 

[alxm]

The posters here seem extremely well versed on the subject that I am interested in asking some questions about.

Not the ones in this thread. Which is why it was moved to the crackpot section for "Independent Research".

I am trying to garner a practical understanding of electron spin. I am thinking that the atomic model really should include photons.

So study QED and its applications in this.

Is the electron pairing explained by a reciprocating photon/s that pull/s the electron into an orbital path at the same time as changing (photon) direction?

No, electron pairing is due to the Pauli Principle, which is due to the spin statistics theorem, and ultimately Special Relativity.

If this is the case, what is the nature of the interaction with the nucleus, and is there photonic polarisation and inversion occurring? Is a photon a record of a quantum event manifested as a moving spatial distortion?

This sounds like Star trek-type technobabble.
The nature of an electrons interaction with the nucleus is electrostatic attraction, and to a limited extent, a magnetic interaction.

 

[andrewr]

The Bohr magneton is the measure of intrinsic, spin-related magnetic moment, not orbital magnetic moment.

Yes, you're right. I stated that wrong. I stated it correctly earlier.

No, you just minterpreted what was written then.

And you have come into the thread *just* to be negative again -- and aren't giving useful information to solve the problems I am interested in.

It makes no such assumptions whatsoever, and as I already told you in this thread, there are exact methods which use Slater determinants. Again you're saying that existing methods don't work when you clearly don't know anything about them.

Oh, "nothing" is a serious exaggeration -- and "don't work" is a black and white critique -- perhaps you want other's to believe I said that? wishful thinking?

Again -- I already told you, I don't believe you; do you feel better that I can repeat what I said too?
Oooh ... I also said in reply to you, that if you had no more to say, there wouldn't be any more of this stuff in my thread -- but here you are. You didn't just leave it at a correction on the Bohr Magneton...No.
You had to attack my person.

False. Do the math. In fact, try the Feynman lectures on physics where he derives the Bohr radius using the HUP.

Being able to numerically solve for a "classical" Bohr radius which doesn't have any meaning since the electron doesn't "orbit" -- proves nothing. Why do you bring that up? Feynman's point is good for a bedtime story ... its comforting for those who like circular orbits and can't see them any more; but it is also irrelevant.

I Believe Schrodinger originally tried to say that ψψ* and EM intensity were directly related -- and they aren't. HUP says "I don't know and perhaps even I can't know" not "it is PERFECTLY uniform.".
Besides, even if what you say about HF/SD was "perfectly" true in some strange sense -- I really don't care. I am not interested in doing them.

 

[andrewr]

edguy,
Thanks for all the help. I think I'm going to move on to a different site; Whether or not alexm is right is irrelevant -- the lack of tactfulness is not going to end.

radguy -- best wishes.

--Andrewr.

 

[alxm]

And you have come into the thread *just* to be negative again -- and aren't giving useful information to solve the problems I am interested in.

I gave you useful information - learn basic quantum mechanics, since you clearly have serious gaps in your knowledge of it. Then learn quantum chemistry and how existing methods work, which you clearly don't know either. Then you can think about improving them.

We're talking about a subject which has occupied some of the best minds in physics for the last 80 years, something which entire careers of many Nobel laureates has been dedicated to. If you think you can up with something new and useful without knowing QM properly, then you're delusional.

Again -- I already told you, I don't believe you; do you feel better that I can repeat what I said too?

This has been proven mathematically (Löwdin, 1955). It's not a matter of opinion, you don't need to believe me. You need to study more before you make judgment on things you obviously don't know about.

Oooh ... I also said in reply to you, that if you had no more to say, there wouldn't be any more of this stuff in my thread -- but here you are.

That was before you started to 'teach' other people stuff that was clearly wrong.

You didn't just leave it at a correction on the Bohr Magneton...No. You had to attack my person.

I didn't start making personal attacks, you did. Read the thread.

Being able to numerically solve for a "classical" Bohr radius which doesn't have any meaning since the electron doesn't "orbit" -- proves nothing. Why do you bring that up? Feynman's point is good for a bedtime story ... its comforting for those who like circular orbits and can't see them any more; but it is also irrelevant.

The Uncertainty Principle is quantum mechanics. He used a quantum mechanical rationale and analytically arrived at a quantum mechanical result. The Bohr radius has a physical meaning in quantum mechanics, and every introductory textbook derives it. It does not mean that the electron 'orbits' in the classical sense, or that they're circular. You're just showing once again that you don't understand basic QM. Or basic chemistry for that matter.

Besides, even if what you say about HF/SD was "perfectly" true in some strange sense -- I really don't care. I am not interested in doing them.

You haven't shown any understanding of what a Slater determinant is and why it's used, nor proposed any method of doing what-it's-used-for. You seem to think you can mix a classically definied position with quantum mechanics and/or that the concept of an electrons 'position' has any meaning on an atomic scale, you think that you can use a single-particle Hamiltonian to describe an atomic or molecular system, that you can insert Bloch sphere results into an atomic/molecular Hamiltonian (!), that the HUP only pertains to experiments, that methods mathematically proven to be exact are not, that Slater determinants are something they're not, that DFT methods are something they are not, that variational methods have 'well-known problems' in the face of the fact that all the most accurate results have been obtained that way since 1929.

In short, everything you've posted is a rambling incoherent mess of poorly understood and blatantly misunderstood concepts. And you meet any attempt at correcting these mistakes with arrogant derision, dismissal and personal attacks. Then you complain how people aren't helping you or answering your questions? That's flat-out delusional, crackpot behavior.

 

[andrewr]

edguy,

I don't want to leave you totally hanging -- (I hate dead end threads when searching for stuff...) so I am using the ignore feature and going to post what I learn, but not bother to read condescending black and white remarks. I really don't care if I am heckled behind my back by someone that I can't understand.


I correct mistakes when noticed; my apologies in advance for all my shortcomings. . Reader beware... and I think I appreciate why 1000+ people look and only a very few dare to participate. Mercilessness... eh?

I had one question answered so far (elsewhere), and painlessly.

On the SG experiment, it is correct that the extra high density of atoms near the center-line happens on modern as well as old experiments. My classical thinking about collisions of atoms is not the only possibility. Unfortunately, even though you wanted to rule out spin-flip as a mechanism -- it happens.
Spin flip which in NMRI can be caused by RF frequencies near the precession rate (a misnomer perhaps, but no worse than "spin") of the atoms in the Magnetic field. When molecules are still separate, but close and in slightly different magnetic gradients -- they can also be coupled since each one's EM (Or QM?) field can interact.
When two resonant oscillators are coupled but with very slightly different frequencies in electronic circuits each resonator becomes far more broadband in its filtering characteristics; that effect, I think, is similar to the kind of thing that appears to be destabilizing the precession of atoms in the SG experiment and allowing spin-flip to cause atoms to re-orient.

I was introduced to a paper hosted by Cornell University as something to look at for a full QM view of what is going on. arXiv.org. The specifics are modeled (I don't know how accurately) by Bose-Einstein condensation. The author there calls it "magnification" and computes some distributions (QM) for an example magnetic field distribution. See: " Quantum mechanical description of Stern-Gerlach experiments" published 2004-09-29. Potel, G; Barranco, F; Cruz-Barrios,S ; and Gmez Camacho, J.
It's available through deep-dyve for a buck as of today -- or if you are willing to dig, arXiv.org for free.

Also, my analysis on the spin state is an approach often taken -- but there is a caveat I need to study, and it wasn't explained very clearly yet. So, I am encouraged to at least attempt it and see where it fails.


--Andrew.

 

[andrewr]

Hello all!

I came across some papers on Newton which I found quite interesting. In my physics books I have "Newtons rings" and in each book there is a formula based on waves which explains them.
What is terribly interesting about this -- is that Newton didn't believe in the wave theory of light. In fact, he ridiculed the few scientists who did -- and because he had popularity, prestige and arrogance -- no one questioned his experiments to disprove wave theory of light. What I find more interesting is that scientists who published accurate data were cowed to silence by carelessly taken data which Newton produced. His science was on par with the strong handedness church of Gallileo's time. The net result is that science was retarded in the discovery of the wave nature of light on the order of century because of pride...

On the angular momentum questions -- in which I was derailed by a typo -- the quantum condition is that each state of angular momentum differs by exactly Plank's reduced constant. In itself, this doesn't seem surprising -- except for the fact that Plank's constant has the units Joule*Seconds and not just Joules.
Angular momentum is classically related to ... momentum = m*v, which also means to fixed energy by (mv)**2/2m. eg: If momentum and mass are known, energy is exactly known.

The Bohr magneton represents a fixed magnetic strength -- and when Bohr began his investigations, he did not know about "spin". So, like Newton's rings ... there is something historical involved and the name is anachronistic. Consider my mistaken statement in that light.

When I refreshed my memory on how to compute angular momentum expectation values, I used the classical hydrogen with fixed nucleus example to work with. Solutions to Schrodinger's equation depend only on radius alone -- the rotational modes of oscillation varying not because of the inward potential but because of the condition that if one rotates an atom 360 degrees -- the wave function of that new position needs to be exactly the same as if it were not rotated. This means that angular momentum solutions are a product of the radial distribution and a function... So the symmetry makes circularly oriented oscillations to be integerial in number of nodes regardless of the nuclear charge for stationary states.

Again, in my physics books the Bohr orbit is shown as having exactly N nodes around a ring -- and this appears to be quite common.

I am getting what appear to be different answers concerning how to deal with the probability ψ**2 and χ associated with angular momentum. For angular orbital momentum, the most common approach is to compute the quantum momentum and radius from nuclear center to produce the familiar L=r x p.
Angular momentum is the cross product of radius and linear momentum at time t classically. The magnitude is radius times momentum.

In principle, that's fine -- it means that when measured, a probability of rotational motion can be found. But what is curious is that spin is not treated analogously. That is: generally one starts with the spin states and makes a new variable which is a product: state=ψ*χ
However for angular (orbital) momentum, one simply adds the term: l*(l+1)/kr**2 to the radial potential generated by the nucleus --- regardless of its magnitude.

I am curious about the difference in approach for, hypothetically, if there is a quantum equivalent to classical spin -- I would expect one to compute something akin to momentum's curl at all locations in space to find out how much tendency there is for an electron to "spin" on its axis. For that's what one does classically -- for example: The spin momentum of the Earth is the effective mass and velocity distribution taken relative to the Earth's center -- rather than the sun. I haven't found anyone do this QM wise yet, and I think it would yield some useful information about the relationship of classical to QM spin.

Such information, along with the "ensemble" interpretation, might be useful in determining how to interpret the meaning of spin h/4*π. (spin one half). etc.

 

[andrewr]

Hi Everyone!

Unfortunately the other forum I joined closed, and life took over for 6 months and I couldn't work on the project --- so I am back, still having the same basic question; but perhaps a bit wiser.

I thought I would start out by still keeping alexm on my ignore list, and posting a modification to the Schrodinger equation that has proved useful to me. Normally, ψ, is the variable that the equation is worked out in; eg: the time invariant version, but that form wastes a lot of computation time. I have found it practical to use a substitution where ψ=ke**φ; and rewrite the Schrodinger time invariant equation as:<br /> -\frac{\hbar^{2}}{2m}\left\{ \varphi_{r}^{2}+\frac{2\varphi_{r}}{r}+\varphi_{rr}-l(l+1)r^{-2}\right\} =E-V<br />

This version is the radial equation, which can easily be converted to Cartesian coordinates by substituting a vector in for r=[x,y,z] proves to simplify the numerical version of the program I use to try and solve bonding of atoms. In the case where the electron has no angular momentum l=0, for example, this equation will correctly reduce to 13.6Ev for E and the central potential for the nucleus.
eg: The solution is just:

<br /> \varphi=-\frac{e^{2}}{4\pi\epsilon_{0}}\frac{m}{\hbar^{2}}r<br />

Yielding:
<br /> -\frac{m}{2\hbar^{2}}\left(\frac{e^{2}}{4\pi\epsilon_{0}}\right)^{2}+\frac{\frac{e^{2}}{4\pi\epsilon_{0}}}{r}=E-V(r) <br />

Which is much simpler to solve for using numerical methods than what I tried earlier in the thread.
The only difference between 1D (Cartesian) and 3D radial forms is the single divide 2phi by r term that shows up; so that term is the result of adding second derivatives in x,y,z in Cartesian coordinates and is distinctly the term associated with angular momentum. I still haven't figured out if there is an analogous way to include spin of an electron using this type of equation, but at least I know how to do all of the orbital angular momentum correctly now using simpler numerical integration.

 

[andrewr]

There is a thread on physics forums which attacks the problem in the general direction I am interested in pursuing, and perhaps some of you know a bit more about this than I do, and could give some hints.

See this thread: https://www.physicsforums.com/showthread.php?t=227376"

And in particular, jayryablon linked this article:
http://jayryablon.files.wordpress.com/2008/04/ohanian-what-is-spin.pdf

Which shows that there is a way to compute spin using r x p type mathematics, rather than becoming stumped on the fact that spin of fermions is fractional, and since it is 1/2 for electrons, that the solution (in a basis of spin bra-kets) is 720 degrees until a rotation operator brings one back to the original orientation -- rather than 360 degrees as would be the case for physical rotation in 3D space.

So, to tie this back to my comments earlier in the thread -- there appears to be a way that the spin of an electron is related to the orbital angular momentum of the electron; not that they are identical, but that there is a relationship.

I have come across articles, similar to the one on "vortons" earlier mentioned by another poster, that attempt to use a particular configuration of wave propagation and general relativity to explain how the internal structure of an electron might be capable of producing the external field described by Ohanian; I don't have the link here with me, but the conclusion was nearly the same: they indicate in these articles that an electron might be a photon twisted back on itself by a warping of space (eg: the need for GR). But neither of these approaches lends itself to the numerical modelling of atoms with the Schrodinger equation directly. So, I have to extract any useful knowledge from them and convert it into the framework of the Schrodinger equation; eg: a modification of the previous posts's equation -- to take spin of the electron into account.

I am choosing to re-organize the Schrodinger equation in terms of energy, rather than carrying around psi in each term of the equation because it makes it easier to work with and to understand the ramifications of the individual terms (at least to me). For example, since planks reduced constant times a phase rate gives momentum, it is clear that the squaring of such a quantity divided by 2m gives energy according to p**2/2m. The first term in the equation is just that, a squaring of momentum divided by 2m, so I instantly know how it arrives at a computation of energy.

Since Ohanian's remarks are about a circulating momentum field, there is some hope that I will be able to spot/inspect equations translated from his paper into the form I just listed last post (or a modification of it) -- such that the meaning of r x p type momentum and spin might be translated from the Maxwell equation / Dirac rooted analysis that Ohanian gives, into something derived from the Schrodinger equation. How, exactly, to accomplish this -- I am not certain, yet, but that's part of the fun of trying.

In my previous equation, I am simply listing one part of a solution of Schrodinger's equation by separation of variables into a factor that depends only on radial R(r) and angular Y(θ,\phi) where the meaning of phi in my equation is distinct (by font) from that of the azimuthal angle mentioned here. Only the R(r) portion is handled by the previous equation, including an adjustment for the angular momentum that would occur if L was non-zero, but not including the energy represented by variation in phase that is purely tangent to the radius (eg: on a spherical surface).

It is often argued (and well received) that the very quantization of orbital angular momentum is caused by the single valuedness requirement of the variation in phase as one makes a complete circuit around the point r=0 of the radial equation. Again, this is just Bohr's (but not Sommerfelds) criteria in the circular orbits; the de-broglie wavelength must have an integerial number of wavelengths in order for it to be stable. (exist).

Ohanian, then, is arguing something subtly different; The bulk of a wave can't produce the angular momentum of "spin" but the difference in waves at the edge of a "packet" can and does produce the equivalent of "spin"; Since the equations of state of a nucleus are eigenvectors, they aren't necessarily a packet in the same sense as Ohanian, or at least a single steady state solution might not be.

Motion requires a superposition of stationary states; and so I suspect that in order to detect spin, perhaps more than one state is required since Ohanian mentions a "wave packet".

Any thoughts?

 

[andrewr]

DIY:SAM: Spining electrons and analogy with optical circular polarization

The use of light to make force microscopes, or "optical tweezers", has caused a significant amount of papers including the subject of spin and angular momentum at least in light. Since there is a connection between the electron and the photons it emits, even to the point that many believe an electron is a photon bent back on itself -- somehow/GR -- I am looking in this direction for how the mathematics is typically handled.

There is a researcher, Timo A. Nieminen, who actively uses both the spin and angular momentum of photons to manipulate microscopic particles; in one of his papers, he comments on the ways which present day literature approaches the calculation in contradictory appearing ways: see p 3. and p. 4.
http://arxiv.org/abs/0812.2039"

"... The reverse of this procedure, obtaining the spin and orbital terms starting from r x (E x H)/c, involving the same surface terms, had already been shown by Humblet (1943). For and physically realisable beam, these surface terms vanish, and despite the controversy, both expressions for the angular momentum density yield the same total angular momentum and torque exerted on an object (Zambrini and Barnett, 2005; Nieminen et al. 2007c). ..."

I take his comment to mean that the typical controversy in the community/literature has a simple resolution, and cites papers along with one of his own which go into tensors (which is beyond my skills). But the articles are available on arxiv should you wish to know more. The other papers, in addition to Ohanian, indicate the same general theme:

Angular momentum and spin are two separable types of angular momentum. The "spin" does not depend on where the "origin" or "axis" is placed in order to compute; but "orbital" angular momentum does. Maximum spin is always found in a pure circularly polarized wave (photon);

I am following Timo's work, because as an experimenter, he is verifying the things he states in his actual research -- and using it to predict empirical results correctly. Ultimately, what his work suggests to me is that spin can be extracted from the helical nature of a traveling wave. In electrodynamics, polarization can be thought of as a superposition of two plane waves; For example if a wave is traveling in Z+, then the E fields in both the X and Y directions (two plane waves), and the phase relationship among them is what determines "polarization" (spin).

Theoretically, then, I ought to be able to represent spin by factoring Schrodinger's equation into parts representing TEM (Transverse excitation mode) type waves which it lacks. The phase between these two copies of the Schrodinger equation -- would in some way (distributed or local I don't know) be able to represent spin;

Following up on this idea, I came across two other papers which pique my intuition.
The first is also from Timo;http://arxiv.org/abs/physics/0408080"
The important thing to notice here is that focusing the beam causes the TEM00 wave to be at an angle, which causes some of the wave to be longitudinal rather than transverse; and this focusing decreases the spin, but increases the angular momentum of the wave; causing a net conservation.
The mathematics for an electron ought to be very similar.

I may be wrong, but it appears to me that a single photon would naturally carry spin -- and therefore it is more proper for light to be thought of a circularly polarized naturally, and all other polarizations being from a superposition of circularly polarized light waves. In Electrodynamics, we tended to start from plane polarized light (TEM waves)-- assuming it to be the simplest and construct circularly polarized light from two plane waves. But, my gut intuition is that since electrons must always have spin, and light comes from them -- I would assume that a single un-entangled photon would be also have to have spin -- and therefore be circularly polarized. I am not sure how this could be found out/tested. I tried looking up synchrotron radiation and bremsstrahlung, but there was insufficient information to come to a conclusion. In bremsstrahlung single photon/electron emissions occur at the maximum energy of the deceleration, all others can emit multiple photons and wouldn't answer the question.

Also, seeing as the edges of the light packet are very important in computing spin -- I did a search on Larmor precession, and its relationship to packets. I am looking for inconsistencies in experiment (empirical verification) that are caused by packet edges such as Ohanian mentioned. I came across this article, which follows the same pattern of the "packet" being different from the plane wave.
http://arxiv.org/abs/1007.5179"

The mathematics of this last paper are within standard, and introductory Quantum Mechanics, so that it isn't difficult to follow the argument; but it is very close to the issues that I have been trying to solve about simulating "spin" in atomic orbitals and is worth looking at because it underscores again, that most treatments are over simplifications.

There are two main points I'd like to present at this point, which are causing me indecision about exactly what is required to model "spin":

1) Most arguments for spin not being classically (r x p) compatible, come from the geometry of spherical harmonics; eg: There must be an integerial number of nodes/wavelengths vibrating around a circle in order for the function to be single valued, (The so called Bohr orbital condition). That is, the typical argument starts from electron spin being 1/2 -- and then moves to spherical harmonics being intergers -- to finally conclude that the two are incompatible. This comes from the PDE solution boundary conditions required to solve spherical harmonics.

2) There is an algebraic theory of spin; it is a copy of the logic of fundamental commutation relations used in angular momentum, where [Sx,Sy]=ih_Sz, [Sy,Sz]=ih_Sx, [Sz,Sx]=ih_Sy. Building up from these mathematical relations, the spin is found algebraically to be a multiple of 1/2, even though the logic is the same as that from angular momentum. The algebraic theory of the angular momentum (orbital)seems to me, then not to be able to exclude a momentum of 1/2 as a valid orbital angular momentum. (The sum of a spin and an orbital angular momentum which cancel would be 1 - 1/2 = 1/2 ... for example for a net angular momentum of 1/2.)

I have also noticed that there is extra information present in the radial equation of a no-angular momentum hydrogen atom (last post) which suggests that the electron diffuses around in space in the 1S orbital of hydrogen; and that a possible way to attack the problem of "where" is the electron might be by using the idea of diffusion vs. drift velocity as is used in semiconductor physics. (Which I am more familiar with...) I'll post on this later, and see about setting up some very basic simulations to test out my idea with.

 

[andrewr]

Short update:

I have been running various simulations of the basic introductory problems to QM using different mathematical models and comparing results. The formula I gave for the radial Schrodinger equation has a flaw in it (that no one else noticed?). The phase variable is based on the momentum of the particle at all locations in space; and to convert that value to energy is accomplished in by the following three terms which is slightly flawed.

<br /> \varphi_{r}^{2}+\frac{2\varphi_{r}}{r}+\varphi_{rr}<br />

There was nothing but a change of variables and algebraic manipulation involved in my coming up with the above expression; but the solution is only valid, for example, in the lowest two energies of the simple harmonic oscillator -- and in some orbitals (all S) of the Hydrogen atom.

I don't know whether to call the problem a "gradient catastrophe" or a "discontinuity"; but in any event, I wanted to caution others about it. I have not re-worked the radial equation in it's entirety, but I know that the problem shows up mainly in the second derivative; By reworking the substitution from probability space, taking φ=log( |ψ|**2) instead of φ=log( ψ ) appears to eliminate the problem.

I still haven't been able to discover a difference between spin and angular momentum, mathematically, that amounts to anything more than saying angular orbital momentum is computed around a preferred origin, while spin is computed rxp style around all points in space. In a sense, spin is any angular momentum NOT oriented around the preferred origin (typically the nucleus). It is not a coincidence that the effect of spin is magnetically the same as that of orbital motion; both produce 1 Bohr magneton regardless of a supposed difference in "mechanical" angular momentum between spin and orbital angular momentum.

I have been discovering, however, that spin does appear to play a role in the relativistic correction of the wave function; and I am tentatively going to assign the meaning of "probability" of finding a particle at a particular location to always include the ability of that particle to affect another particle traveling perpendicularly to the first. Logically, (by analogy), if a standing wave exists in a coax cable there are node points where one could place a perpendicular tap (T joint) in the coax cable that will receive no power whatsoever from the standing wave -- although analysis of the Poynting vector would suggest that power is flowing through that node point. Any coax, though, attached as a "y" joint and not a T will be able to receive some power from that point. I hope to post some graphs soon.

Cheers!