As I understand it (e.g. from discussions around the Fermi field theory of the nuclear force), a spin 1/2 particle can emit a spin-1 particle and simultaneously flip its spin (say, spin +1/2 -> photon +1 & spin -1/2); but how does this work with spin-2 particles? Does it need to emit pairs in...
As I understand it, the fundamental unit of interaction in QED is a term with a pair of (spinor) electron factors and a (vector) photon factor, represented in a Feynman diagram as two (anti-)electron lines and one photon line meeting at a vertex.
I get the case where the photon and electron...
Late last year, I started a thread (https://www.physicsforums.com/threads/historical-paper-dump-sites.780669/#post-4907513) to solicit/compile a list of URLs with open access to historical papers; shortly after, I discovered that several of them weren't as open as they had been...This morning, I...
Planck's only lists ten papers; Boltzmann's had none -- though, the link to his article on plato.stanford *does* have a nice-sized "primary sources" section; it doesn't look comprehensive, but it's a good start; maybe I'll plug a few more names into Plato and see if the pattern continues.
The...
Does anyone have any suggestions for finding lists of all papers published by individual physicists?
Usually the Google machine turns up hits pretty quickly, but I've hit a brick wall looking for lists for Max Planck and Ludwig Boltzmann.
netlib.org/bibnet/ is amazing, but it's pretty narrow.
In most introductory QFT treatments, it's stated early on (and without proof) that particles with even integral spin are always attractive, while those with odd integral spin can be repulsive; sometimes this is even cited as evidence that the graviton must be spin 2 (I think Feynman's...
Thanks for the encouragement; I think the problem stems from my incomplete understanding of spin, particularly with regards to multiple particles. I've since found a few references where they show that the two-particle spin operators are I⊗σ and σ⊗I, as opposed to the naive versions I tried...
I suppose that's a last-ditch, break-glass-in-case-of-emergency-type option. But the internet's hours are more convenient.
However, the public libraries I've seen (note: small sample) don't have academic journals. I do remember my old college library had a pretty nice selection of printed...
So...apparently I spoke too soon about how generous Oxford and RSPA were; today, most of the articles I grabbed last week (10/31 and prior) are now pay-walled again. I remembered this happening before, and it looks like all of the articles I have (younger than 70 years at least) from RSPA are...
I'm looking for sites with historical journal articles -- ideally in English, but if there's none available, I can settle for the original. Specifically, I'm looking for stuff by (in no particular order, and certainly not exclusively) e.g. Heisenberg, Pauli, Dirac, Schrodinger, Feynman...
If I'm not mistaken (and I could very well be, as I'm a dilettante dabbler), the corresponding 1st-order diagram would be an incoming electron with momentum p, and an outgoing electron with a momentum (p-k) and photon with momentum k, so the perturbation expansion term would have a destruction...
I'm having some trouble grasping the meaning of the exchange term in the Hamiltonian Heisenberg gives in his classic 1932 paper (the one typically given as the first to describe nucleons via a spin-like degree of freedom; NOTE: I realize this isn't the same as what is today called isospin, but...
The prize goes to dextercioby! I'd downloaded that Darwin paper just this week (If all journals were as open as the Royal Society, the world would be a more wonderful place), but hadn't gotten to it yet (since it was one of a few hundred), but that's definitely a derivation...but, there's a...
@dlgoff, I watched the first half hour of so; I'd forgotten how much of a badass Feynman was...but I didn't catch the part in question.
@vanhees71, I know that was just intended to correvt the record, but it does nicely highlight some of the difficulties that early theorists would have had to...
Hey samalkhaiat,
That's probably the most concise derivation I've seen (and definitely undergrad level, which is also a big plus), so thanks for that...but, I'm actually looking for the historical record -- how it was originally derived without the benefit of a century of quantum mechanics...
After coming across an article by Schwinger where he tries to deduce the spin of a neutron, I started to wonder when the same thing was done for the photon...
Specifically, I've been trying to locate "the" article where the spin of the electromagnetic field was theoretically determined to be 1...
I intended to cover that with the weasle-word "approximate", along with e.g. the deviation of magnetic from geological "North", finite propagation speed of electromagnetic effects, general-relativistic corrections, &c.
This morning I was momentarily struck by the coincidence that magnetic North and "solar East" (the direction of the rising sun) happened to be (approximately) perpendicular, but now it doesn't really seem coincidental; as I understand it, spinning of the Earth causes a current that leads to the...
That may technically be true, but that seems to be a minority viewpoint: virtually everyplace discussing Hamiltonian mechanics w/ hard spheres (including the dynamical billiards Wikipedia page you cited) state plainly that both a "hard sphere" and a barrier (e.g. "An [infinite] potential well")...
Very interesting: thanks for sharing...after suggestions from one or another of my iterations of the question, I had looked at some other similar limit-based solutions with nascent delta functions, but my PDE-disability kept me from getting anything meaningful (similar to your problem in the...
The "point case" was where the force was a delta function instead of a step function...the only point I was trying to make was that it seems the "simpler" finite triangle barrier (with step function force ) seems to have a much more complicated solution when compared to the step function barrier...
So this seems like the same problem as before, except here the "constant" force is over a region instead of at a point...in both cases, you have an equation of the form ##d^2x/dt^2 = -F(x(t)) ##...and in this case, the "regions" are over the dependent variable.
I'll try a few graphs later on...
Actually, the system that brought me here is one where the equations of motion are well known: elastic collisions in one dimension -- either between two particles or between a particle and a potential barrier (possibly infinite).
I'm afraid to give too specific / complete a derivation, as it...
When solving virtually any non-trivial system via Hamilton's equations, it seems that I'm ending up with equations of the form
## ∂H/∂p = dx/dt = p/m ##
## ∂H/∂x = -dp/dt = -F(x) ##
## p(t) = ∫(dp/dt)dt ##
## d^2x/dt^2 = F(x) ##
E.g., everything is given as a function of x, but we need to...
OK -- you won me over ;)
HOWEVER, as I still suck at integrals, I'm having an identical problem integrating a finite (ramp) potential.
## V(x) = Kθ(-x)θ(x + 1)(x + 1) + Kθ(x)θ(1 - x)(1 - x) ##
The first pair of steps restricts the positive slope to (-1,0), while the second restricts the...
If I actually do try to change the variables, that introduces a p into it as follows:
##ΔP_i = ∫±Kδ(D - |x_1 - x_2|)ε(x_1 - x_2)dt##
But, ##dx_i/dt = ∂V/∂p_i##, which by the initial assumption, was simply ##p_i/m##
So, ## dt = dx_i/(p_i/m_i) ##
Thus,
##ΔP_i = ∫±Kδ(D - |x_1 - x_2|)ε(x_1...
Ah hell, you're right. I'll have to go over it all again to be sure...but I guess the good news is that as long as they have opposite signs, it should still come out right.
So that's just a restatement of the definition of the delta function...the integral over any domain containing a zero of...
So I tried working this out from scratch...starting with two particles, then taking the mass of the second to be infinite to get an immobile wall, and confirmed that only if the condition on the height of the barrier is met is energy conserved -- and in that case, the correct Newtonian equations...
Re-derivation
So I tried working this out from scratch...starting with two particles, then taking the mass of the second to be infinite to get an immobile wall, and confirmed that only if the condition on the height of the barrier is met is energy conserved -- and in that case, the correct...
That does sound plausible, and It does match up with the intro paragraph of that last reference -- that walls and other immobile barriers are usually treated as constraints, not as potentials.
However, notwithstanding your persuasive argument about the reversibility of virtual displacements...
You're right. I should have integrated over t surrounding the collision time...if x=0 at t=t0, then the integral would be any interval containing t0 -- eg t0-dt...t0+dt. The end result us still the same -- energy is only conserved for a specific momentum-dependent height (whether that height is...
Sorry, I was mixing topics -- for "interaction" there I meant for two particles, rather than a particle and a wall (hence the delta function for point particles and a two-sided step function for spheres). It was off-topic, but I included it for context,
Yes! That's what I thought, too (and...
I think we might be getting somewhere; that's where a lot of my technical difficulty has come in -- dealing with x/t...particularly since in the elastic case, dp/dt = -2p(t)δ(x(t))...I couldn't figure out how to integrate that without having the explicit form of x(t).
I tried to skip around...
Sure, that's how it is defined; but in the actual equations of motion for which we're trying to find the Hamiltonian, the force (dp/dt) is *exactly* a delta function (since the entire interaction takes place at a single point/instant)...and as far as I can tell, using a function (e.g. Gaussian)...
Sorry, missed this earlier. This looks just like the normal Newtonian derivation, except with the delta function replaced with a gaussian...I don't quite follow what you mean by the delta function not helping -- since it's both exactly what we want (instantaneous localized force) and easier to...
Correct -- well, specifically, the Hamiltonian for it; the equations of motion, and the force, are trivial. I just cannot find a Hamiltonian that gives them.
I started by looking for the hard-sphere interaction for N particles in 3D, but as I ran into frustration after frustration, I started...
Yeah, that's what irks me : there are thousands of derivations for a quantum wave function in an infinite well -- which is by definition unobservable and useful only in very specific, highly technical situations...but there's not a single derivation for a cue ball bouncing against the side of...
Now that's the answer I keep finding everywhere, but I cannot get it to come out right; if V is a *finite* step function (say v0 * step(x)), then dV/dx is v0 * delta(x)...and so dp/dt is -v0delta(x), so the final momentum will be p0 - v0...which is only elastic/energy-conserving if v0 isn't a...
FYI, I tried going at this from a couple of different points of view here (https://www.physicsforums.com/showthread.php?p=4566105#post4566105) and here (https://www.physicsforums.com/showthread.php?t=721040).
Homework Statement
I'm trying to find the Hamiltonian for a system consisting of a single particle moving in 1D elastically colliding with an infinite potential barrier.
By conservation of energy, we know the magnitude of the momentum must be the same before and after the collision; for...
Any help showing this? I get dp/dt=-∂H/∂x, which is depends on V, not the momentum...so how does it reverse it (eg total change = -2p)?
But in the Newtonian case, this is exactly what happens, without any issue: the momentum at x=-dx is p, at x=+dx it's -p, so the force (dp/dt) is -2pδ(x)...
Ah, right -- hadn't thought of that...but that's really the problem then, because in the equations of motion that we know this system produces, momentum *is* conserved (at least if the barrier is taken to have infinite mass -- and in any case, energy is conserved).
Also, it isn't just momentum...
If I use the Hamiltonian given above, the equations of motion don't come out right, because the force is proportional to the height of the well...so as far as I can tell, energy/momentum isn't even conserved.
Edit: finish answering question.
So, I'm trying to figure out if my assumptions are...
Right -- it's a really trivial problem to solve with conservation of energy/momentum or Newton's laws...but I can't get it to come out right in Hamiltonian form.
My understanding is that a classical idealized particle, moving in one dimension, with momentum p and kinetic energy T comes into contact with an infinite step-function potential V, there will be an (instantaneous) elastic collision - the particle's momentum becomes -p, so its energy remains...
So the derivative of the unit step/sign function is the delta function...∂H/∂x = ∂V/∂x = -dp/dt in your case is ∞δ(x(t)), so the impulse (Δp) over any finite amount of time around a zero of x(t) will be -∞...or am I missing something? It seems like to get the right Δp -- which IIRC is -2p(t) in...
My problem with all of these finite, non-delta function interaction potentials is that, while they presumably reproduce the same qualitative behavior, the do not reproduce the exact (trivial) Newtonian equations of motion (e.g.where the momenta are swapped in the center-of-mass system).
To...