PDA

View Full Version : Valid quantum relationship? What does it mean?



thorkil2
2008-Sep-22, 01:30 AM
f = c/ λ
E = hf = h c/ λ
λ = h/mv
E = hc/ λ = hcmv/h = cmv

I confess that I have only the most general knowledge of quantum theory, so this may be nonsense--apples and oranges, but there is a rationale. I would like to know whether there is any meaning, or if I've stuck two things together that don't belong.

Photon energy is E = hf
Since c = λf, then E = h c/ λ

deBroglie's formula equates wavelength with momentum, or λ = h/mv. I realize we may be talking about two different things, but then, they may not be so different. Both relate a specific wave nature and both share h as a factor. So I stuck them together to see what popped out.

λ = h/mv
E = hc/ λ = hcmv/h = cmv


What I get (as you can see) is something that looks remarkably like
E =〖mc〗^2 , but only if v = c. I'm not sure how to interpret that, or whether it has any meaning at all. Curiosity is far stronger than fear of asking foolish questions.

Digix
2008-Sep-22, 02:00 AM
what is that variable v in your equation?
from that
E = hc/ λ = hcmv/h = cmv I suppose it must be some speed to keep correct dimensions.

since you are calculating wavelength of photon, so in that case V=C and there is nothing interesting.
you just made some circular equations.

you can however use similar way for calculating wavelengths of electrons or bricks, but it will not be so simple in these cases.

thorkil2
2008-Sep-22, 02:07 AM
The v derives from λ = h/mv, which is the deBroglie equation for matter waves, wherein wavelength is inversely proportional to momentum. I was examining this with an interest in how EM and matter waves relate, or whether they do, so it wasn't just to calculate the energy of a photon, but I suppose in the end that may be all I've done.

Digix
2008-Sep-22, 02:19 AM
yes, you just got E=mc^2

Ken G
2008-Sep-22, 02:40 AM
E = hc/ λ = hcmv/h = cmv
There's your problem. The first equation for E assumes you have a m=0 particle moving at the speed of light, and the second one assumes the particle does have a rest mass and is not moving at the speed of light. So the final expression doesn't apply to light, nor to nonrelativistic particles. However, it can apply to light if you replace the deBroglie wavelength with h/p instead of h/mv.

thorkil2
2008-Sep-22, 05:38 AM
Thanks. That clarifies it and puts it back to what my intuition was telling me in the first place. I knew that I was dealing with a built-in conflict between massless EM and a rest frame mass, but was uncertain what to make of the mutually shared h and λ factors, or whether there was any comparability between matter and EM waves. It seemed there ought to be, but the conflicts didn't make this a very likely candidate for deriving the connection. Curiosity satisfied; back to my element (though I'm less sure about even that of late).

Ken G
2008-Sep-22, 06:03 AM
I think there is a connection between matter and EM waves, but you might have to delve into the Dirac equation and relativistic quantum mechanics to find it. It's not trivial, it was tacked onto quantum mechanics later on. E=mc2, on the other hand, is not inherently quantum mechanical, as you can pretty much tell since there's no h in it. I think what you are basically asking is, what happens when we use quantum mechanical expressions that have h in them, but the h cancels-- does that erect a bridge from the quantum domain to the classical domain? I think in some sense it does, and must be related to the concept of correspondence, which basically says that since our only way of probing the quantum domain is with classical apparatus, all we can ever know about the quantum domain is how it projects onto classical concepts. I think that principle is poorly understood, which leads to a lot of confusion about what Bohr's "Copenhagen interpretation" is really all about.

thorkil2
2008-Sep-22, 06:50 AM
I think it was the cancelling h's that intrigued me the most about the problem. I stumbled onto this question while I was hunting for insight into another. I was doing a review of (very) basic quantum theory to examine another conundrum: whether time has to be quantized. It seems to me (and I arrive at this somewhat unwillingly) that it must be quantized because of motion. If you look for the change boundary between motion and (relative) non-motion by subdividing temporal interval, there must come a boundary point on one side of which is non-motion, and on the other side, motion; where a thing is either moving or not moving (all relative to a designated relative rest frame). Sorry, digression, but these things sometimes lead me off in odd directions, and I guess I'm rambling now. Thanks very much for your insights.

Ken G
2008-Sep-22, 01:36 PM
Yeah, I don't know if time needs to be quantized, but I do think that Zeno's motion paradoxes are also widely misunderstood. Most people think Zeno's paradoxes are mathematical, and can be solved with the invention of limits as used in calculus, but I think they are ontological paradoxes (paradoxes about what is, rather than about how we can calculate things, so mathematics is a higher-order issue). That's more along the lines of how you are thinking about it, in terms of that "boundary".

I think it was for similar reasons that Zeno conceived the arrow paradox, which basically says that if a moving arrow is instantaneously in the place where it is, how is it different from a nonmoving arrow? And if it is not in the place where it is, where is it? Those questions, as ontological challenges to reality, uncannily mimic the uncertainty principle, which places motion and placeness into complementary opposition. But Zeno has come up in other threads so I probably shouldn't try to turn your thread into another one of those!

Digix
2008-Sep-22, 03:09 PM
actually nothing is quantized, except energy transfers.

QM just says that if we have point A and point B and we like to transfer some amount of energy, the longer time you use for transfer the more precise it will be.

there is no quantization on energy when transfer times are infinite

thorkil2
2008-Sep-22, 04:10 PM
But Zeno has come up in other threads so I probably shouldn't try to turn your thread into another one of those!


My fault, actually; I changed the subject. Shouldn't continue posting when fatigue and later hour gang up on me. I had not thought of this in terms of Zeno, but I woke up this morning to a different approach (from my own) that I will need time to think through clearly.

thorkil2
2008-Sep-22, 04:22 PM
actually nothing is quantized, except energy transfers.

QM just says that if we have point A and point B and we like to transfer some amount of energy, the longer time you use for transfer the more precise it will be.

there is no quantization on energy when transfer times are infinite

Interesting thought. I'm way out of my league in any discussion of QM, but it seems like a somewhat pat answer to a question (quantization of time) that continues to be addressed in serious discussions of cosmology. On the other hand, while I try to stay current, perhaps my sources are outdated.

Ken G
2008-Sep-22, 04:30 PM
there is no quantization on energy when transfer times are infiniteThat's not right, there is no uncertainty in the energy when the transfer times are infinite, but whether or not the energy is quantized is a different issue. The quantization of action (h) is still the core of quantum mechanics, and whether or not that translates into quantizing the energy depends on the action. For example, an atom can only emit quantized energies, especially if it has an unlimited time to do the emitting, because of the quantization of the action of the electron in the atom over that unlimited time.

Digix
2008-Sep-22, 04:41 PM
atoms cant use infinite time for emission, just like any other resonator usually cant give or take energy in infinite time, because they are internally dynamic. and there is only half of period available to transfer some amount of energy.
so I think this does not object for the general rule. which is derived from
E = h/t
everything else is just specialized cases.
also we can do lots of perverted thought experiments were we do things that object to some of specialized QM rules

Ken G
2008-Sep-23, 12:59 AM
atoms cant use infinite time for emission, just like any other resonator usually cant give or take energy in infinite time, because they are internally dynamic. and there is only half of period available to transfer some amount of energy.That's irrelevant because we can look for the limiting physics as that time tends to larger and larger amounts. The quantization does not go away-- it gets sharper and all the clearer.



so I think this does not object for the general rule. which is derived from
E = h/t
everything else is just specialized cases.It is the action that is quantized, so t does not have the meaning you think it does when the system is periodic (like an atom). In that case, the t and E in your equation refer to uncertainties, not quantized values (to get the latter, t is to be interpreted as the period, not the duration, of the behavior).

Tensor
2008-Sep-23, 02:59 AM
If you look for the change boundary between motion and (relative) non-motion by subdividing temporal interval, there must come a boundary point on one side of which is non-motion, and on the other side, motion; where a thing is either moving or not moving (all relative to a designated relative rest frame).

I struggled mightly with this in High School physics. My teacher threw in something about "the wheel isn't moving, but it's just starting to move" and for some reason, at that time, I couldn't wrap my mind around that concept. Which set me back for a while as I tried to get this straight in my mind. Sounds similar to your boundry above



Sorry, digression, but these things sometimes lead me off in odd directions, and I guess I'm rambling now. Thanks very much for your insights.

Hey, we all start rambling at one time or another. It's always interesting to go back and look at those two or three months later. :)

thorkil2
2008-Sep-23, 08:02 AM
I struggled mightly with this in High School physics. My teacher threw in something about "the wheel isn't moving, but it's just starting to move" and for some reason, at that time, I couldn't wrap my mind around that concept. Which set me back for a while as I tried to get this straight in my mind. Sounds similar to your boundry above.

The whole idea of "examining" the boundary is almost meaninlessly hypothetical anyway because we run into uncertainty long before we have a chance of encountering that boundary--if it exists at all (and in light of uncertainty--I'm not sure the word "exists" has any meaning in this context). So I think I'm stuck with my quandary. I've never thought of Zeno's so-called paradoxes has having any merit either in Physics or philosophy because they fail to recognize that motion is every bit as much an ontological state as rest--in fact, motion is probably a more meaningful concept than rest. It's entirely possible to look at the Universe and say that rest is an odds-out artificial condition, since everything is in motion, so any rest state must be specified arbitrarily from an ever-moving relative field of choices (though convenience is usually substituted for arbitrariness--if they aren't the same thing). :)

thorkil2
2008-Sep-23, 08:07 AM
Egad, I can't believe I've carried the original post so far afield. Does that make me guilty of hijacking my own post, and should I now ban myself, or will a slap in the forehead do?

Tensor
2008-Sep-23, 11:31 AM
Egad, I can't believe I've carried the original post so far afield. Does that make me guilty of hijacking my own post, and should I now ban myself, or will a slap in the forehead do?


Just borrow this :doh:

Digix
2008-Sep-23, 02:09 PM
That's irrelevant because we can look for the limiting physics as that time tends to larger and larger amounts. The quantization does not go away-- it gets sharper and all the clearer.
I was not talking about that, since this limiting physic is on top of that.
You will not be able to perform transfer of less than E = h/t energy anywhere in any situation where t is maximum time allowed for that transfer E is minimum energy
If you know exception tell it.



It is the action that is quantized, so t does not have the meaning you think it does when the system is periodic (like an atom). In that case, the t and E in your equation refer to uncertainties, not quantized values (to get the latter, t is to be interpreted as the period, not the duration, of the behavior).
t and E in my equation is taken from photon energy formula E = hf. I just replaced f with t since process does not need to be repeatable.
then we just assume that atom is in constant action and everything is fine.
uncertainty is same as quantized values, since you cant take partial quantas you cant measure them.
if you look at some quantized system in the other way there will be problems when we decide to split it.
for example let say we have LC resonator, we can charge C in the way that it will contain 1.3 of quanta for LC frequency, but then we connect them and resonator can only emit 1 quanta but then we disconnect L from C and measure them and we see that system still have 0.3 quanta energy left.
of course you cant do same with atoms

Ken G
2008-Sep-23, 02:22 PM
It's entirely possible to look at the Universe and say that rest is an odds-out artificial condition, since everything is in motion, so any rest state must be specified arbitrarily from an ever-moving relative field of choices Yes, that's a good point. "At rest" is just a concept we invented, and then got all bothered that we couldn't tell how it was "actually possible". But there's another way to frame Zeno's paradox that is essentially the same, without mentioning anything being "at rest": If two objects have a relative motion, where is that information "stored" or "carried"? Realism requires that the objects must have an actual attribute we call relative motion, but how can two objects share what is a single attribute? It requires that the distance between them is a real thing as well, even if there is nothingness between them, and it suffers from the problem that different observers won't even agree what that distance is, or how rapidly it is changing! I think realism died with Einstein, if not with Zeno himself, though ironically it was Einstein (with deBroglie and Bohm) who labored to restore realism to quantum mechanics.

Ken G
2008-Sep-23, 02:32 PM
You will not be able to perform transfer of less than E = h/t energy anywhere in any situation where t is maximum time allowed for that transfer E is minimum energyThat isn't what you said-- had you said that, I would not have needed to correct it. What you said is that as t goes infinite, the quantization on E goes away, which is false in general, and I gave an example (an atom).


t and E in my equation is taken from photon energy formula E = hf. I just replaced f with t since process does not need to be repeatable.My point was that there's a big difference between t, the way you described it, and 1/f. The former, you said, was an elapsed time, while the latter is the period of repetition of a periodic system. When people confuse those, they get all confused about the uncertainty principle, which is what I'm trying to clarify.

uncertainty is same as quantized values, since you cant take partial quantas you cant measure them.This is just the kind of confusion I'm trying to help with.
for example let say we have LC resonator, we can charge C in the way that it will contain 1.3 of quanta for LC frequency, but then we connect them and resonator can only emit 1 quanta but then we disconnect L from C and measure them and we see that system still have 0.3 quanta energy left.Perhaps you really mean there can be a 30% chance that it still contains one quantum of energy, but I don't see the connection with the claim that as time gets larger, the quantization goes away.

Digix
2008-Sep-23, 03:56 PM
That isn't what you said-- had you said that, I would not have needed to correct it. What you said is that as t goes infinite, the quantization on E goes away, which is false in general, and I gave an example (an atom).
I am talking about energy transfers not about atoms or other dynamic storage systems. atom is very complex system some kind of nonlinear resonator where frequency depends on energy.

if we have 2 or any amount of energy storage elements, and some connections between them. If we connect them so that they form temporary resonator, with frequency f or period t. Each energy transfer between them will be quantized.
I do not attempt to use E = h/t in other situations



My point was that there's a big difference between t, the way you described it, and 1/f. The former, you said, was an elapsed time, while the latter is the period of repetition of a periodic system.
why does it need repetition? t is action time


When people confuse those, they get all confused about the uncertainty principle, which is what I'm trying to clarify.This is just the kind of confusion I'm trying to help with.Perhaps you really mean there can be a 30% chance that it still contains one quantum of energy, but I don't see the connection with the claim that as time gets larger, the quantization goes away.
since we have infinite time to measure halted resonator, I don't see how do you apply uncertainty.
after all there is no resonator anymore just plain static storage elements. and I don't see any reason for energy quantization in stationary capacitor.

thorkil2
2008-Sep-24, 07:24 PM
Yes, that's a good point. "At rest" is just a concept we invented, and then got all bothered that we couldn't tell how it was "actually possible". But there's another way to frame Zeno's paradox that is essentially the same, without mentioning anything being "at rest": If two objects have a relative motion, where is that information "stored" or "carried"? Realism requires that the objects must have an actual attribute we call relative motion, but how can two objects share what is a single attribute? It requires that the distance between them is a real thing as well, even if there is nothingness between them, and it suffers from the problem that different observers won't even agree what that distance is, or how rapidly it is changing! I think realism died with Einstein, if not with Zeno himself, though ironically it was Einstein (with deBroglie and Bohm) who labored to restore realism to quantum mechanics.

I should also have noted while making motion "ontological" that the concept of motion requires at least two frames, and a "correct" concept requires three. If you create a hypothetical empty space, then place a single object in it (we dispense with niceties like molecular motion for this experiment), there is no way to define motion (But be careful: your consciousness of the object adds a second frame). Two objects in the space can move with relation to each other--let's say orbitally, but observations from either surface will be identical yet different in that they define what's moving and what's not moving oppositely. The observation point on either surface (in this simple model) is a default rest frame, because there is no way for the observer to measure anything but the motion of the other body. The two objects can be seen moving around a common center only by adding a third external frame of reference--from the center itself, or the observer's consciousness of the bodies--and so it grows. So maybe the information storage is (how to put this...) a function of the existence minimally of that third frame of reference--not two, but three. That's clear as mud, but I can't think of a better way of stating it at the moment. Have to give it some more thought

Ken G
2008-Sep-25, 03:43 AM
Then consider Mach's principle, and whether that is an ontological requirement too. Do we even know how a universe with only two objects in it behaves, even if we allow ourselves to be a third object of no importance, a "fly on the wall"? We've never done an experiment in a universe like that.

thorkil2
2008-Sep-25, 06:30 AM
Then consider Mach's principle, and whether that is an ontological requirement too. Do we even know how a universe with only two objects in it behaves, even if we allow ourselves to be a third object of no importance, a "fly on the wall"? We've never done an experiment in a universe like that.

Nor will we, but that does not prevent us from considering it hypothetically. I wasn't creating a Universe, just a very simple set of relationships and conditions under which they might be examined for whatever insight they might yield.

loglo
2008-Sep-28, 08:06 AM
Nor will we, but that does not prevent us from considering it hypothetically. I wasn't creating a Universe, just a very simple set of relationships and conditions under which they might be examined for whatever insight they might yield.

Delurking because Ken's remark about Mach's principle reminded me that a lot of work has been done by Edward Anderson on Relational Particle models which explore these simple kinds of "universes" . see for eg Relational particle models as toy models for quantum gravity and quantum cosmology (http://arxiv.org/pdf/gr-qc/0509054)

There is a really interesting paper of his giving the 8th-12th routes to Relativity in there somewhere. If you are into that kind of thing. :)