Calculating the Quantum Kepler Length of a Particle

[arivero]

One knows that in classical gravity a orbiting point sweeps equal areas at equal times. It can be seen that for macroscopic distances the area swept in a plank time is a lot greater than the minimum quantum of area, which is about (plank length)^2.

Now, I ask, given a particle of mass m, for which radius will a circular gravitational orbit around the particle to have the property of sweeping one plank area in exactly one plank time unit?

Below that radius, it should be posible to use "plank time beats" to divide area into regions smaller that previous. So a fundamental break of physics will happen at "quantum kepler length of the particle m" ;-)

Of course you know which other name this length receives, do you?


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[marcus]

Originally posted by arivero
...\ ;-)

Of course you know which other name this length receives, do you?

compton wavelength
nice

 

[marcus]

Originally posted by marcus
compton wavelength
nice

maybe a little discussion would be appropriate

this is a really nice pensee

(physics haiku?)

in natural units the (reduced) compton a particle with mass M is 1/M

the circular orbit velocity around such a particle is
$$\sqrt{\frac{M}{R}}$$

so the rate area is swept out is

$$R\sqrt{\frac{M}{R}}= \sqrt{MR}$$

setting that equal to one (as Alejandro requires) tells us
that R = 1/M

this is the (reduced) compton

(when dealing with G = hbar = c = 1
units one prefers the reduced wavelength which is the cyclic
one divided by 2pi, as well as the angular format for frequency)

 

[8LPF16]

How does the compton value relate to the Red/Orange line 655nm with no mass?

LPF

 

[marcus]

Alejandro something is catchy or piquant about the implied argument.
I like it.
Alejandro says, in effect, that a circular orbit cannot sweep out
area at a rate that is slower than the Planck rate. He says it must be true for any circular orbit that:

$$\sqrt{MR}> 1$$

because say the sweeping of area is slower than unity, say that it takes 3 time units to sweep out 1 area unit
then one can divide the time interval up and this leads to
being able to describe an area smaller than the area quantum.
(one can use Planck "beats" to divide the indivisible)


so he looks for a paradox

he says fix some mass M and make the orbit radius R so small that

$$\sqrt{MR}< 1$$

now the rate of sweeping of area is less than unity
what prevents this?

and Alejandro discovers that what prevents this situation is that the orbit radius is less than the Compton for that mass.
(the central gravitating body is being localized more restrictively than quantum mechnanics permits it to be localized---every particle is spread out by at least its compton)

 

[marcus]

Originally posted by arivero
One knows that in classical gravity a orbiting point sweeps equal areas at equal times. It can be seen that for macroscopic distances the area swept in a plank time is a lot greater than the minimum quantum of area, which is about (plank length)^2.

Now, I ask, given a particle of mass m, for which radius will a circular gravitational orbit around the particle to have the property of sweeping one plank area in exactly one plank time unit?

Below that radius, it should be posible to use "plank time beats" to divide area into regions smaller that previous. So a fundamental break of physics will happen at "quantum kepler length of the particle m" ;-)

Of course you know which other name this length receives, do you?

If it is correct, Loop Quantum Gravity implies that the Planck quantities (the "natural units" as Planck called them) are real features of the environment
that they are built into spacetime just as the speed of light (which is the Planck unit of speed) is built in.

the theory pinpoints the place at which new physical effects can be expected, namely at Planck energy (2 billion joules)
and it derives quantized spectrums for area and volume where
the minimum measurable amounts of area and volume are order-one multiples of Planck area and Planck volume.

among the new physical effects expected at Planck scale are
modifications of Lorentz symmetry

It now seems that these and other Planck scale effects may be possible to probe in the next few years leading to tests of LQG.
If it checks out this will among other things infuse a greater sense of solidity into Planck units. There will be reason to think that they really are nature's units.

Alejandro's puzzle (why can't area be swept out at slower than Planck unit rate?) is an example of a "Planck Parable", as I would call it.
Or a "Planck units Haiku".

I will try to think of another such story or puzzle, to respond in kind.

If Loop Gravity develops successfully and tests out we will probably see more Planck haiku growing up as a kind of physics folklore.

 

[marcus]

The Wizard's Stove (a Loop Gravity parable)

to understand this you need to go to the NIST website and look
up Planck temperature
or take my word for it that it is 1.4168e32 kelvin

that is the temp T such that kT is Planck energy
and if Planck energy is built into spacetime as an intrinsic
threshold of some new effects
then the temp is also (in the same way) a built in intrinsic feature of nature

On that scale room temperature is about 2e-30
and a nice oven temperature is about 3e-30
(in fact if you work it out 3e-30 of the natural unit temperature
is 425 kelvin, which is 305 Fahrenheit qualifying as a "slow to moderate" oven.

Well in this short short story there is a Wizard who wants his stove to always be at 3e-30. Hope notation is clear, I mean 3x10^-30

have to go for now, continue later

 

[marcus]

the Wizard's Stove continued

Once there was a Wizard who lived in a cave that he'd hollowed out in the core of a comet and the wizard was always cold
He wanted a potbelly stove that always stayed at a steady temperature of 3E-30 natural. This would take the chill out of the cave and be convenient for baking.

What he got to stoke the stove with is interesting. He got a supply of the usual kind of black holes each of which was always at the steady temperature of
3E-30 natural. (that is 425 kelvin or 152 celsius or fahrenheit 305 depending on which conventional human scale you like)

the question is, what was the mass of each of the black holes.
and how big were they?

 

[arivero]

Yes, it is the typical Haiku one finds in elementary quantum mechanics book. Perhaps sometime we will see it in elementary LQG books :-) By the way, I had never seen it, but I do skip a lot of literature on this.

Originally posted by marcus

(physics haiku?)
(when dealing with G = hbar = c = 1
units one prefers the reduced wavelength which is the cyclic
one divided by 2pi, as well as the angular format for frequency)

It becomes even more rythmical if you allow for G, as then the one coming from plank length simplifies against the one coming from Newton universal gravitation law. At the end, Compton length contains, of course, m, h, and c, but no G around. As for author trickery, I confess that the paradox has been choosen a posteriori :-)

I am -paying time- at a cybercafe now, so I hope that by monday the wizard will have all his charcoal supply issues solved!

 

[marcus]

Originally posted by arivero
... I confess that the paradox has been choosen a posteriori :-)

I am -paying time- at a cybercafe now, so I hope that by monday the wizard will have all his charcoal supply issues solved!

A significant confession!

the wizard reflected to himself that the temperature of the usual kind of black hole (that they sell for stoves) is

$$\frac{1}{8\pi M}$$

where M is the mass of the hole

 

[marcus]

at this time a tribe of gypsies was passing through the solar system telling fortunes, selling black holes and love potions, and stealing as gypsies always do.

So the wizard solved the following equation for the mass M, so that he could ask the gypsies for holes of that mass:


$$\frac{1}{8\pi M} = 3*10^{-30}$$

 

[marcus]

$$\frac{1}{8\pi M} = 3*10^{-30}$$

$$8\pi M = \frac{1}{3}*10^{30}$$

$$M = \frac{1}{24\pi}*10^{30}$$

 

[marcus]

this means that each piece of "stove charcoal"
would be the size of a mote of dust and
would have a mass about a twenty-thousandth of that of the earth

and if it would not be too tedious I will work that out
from the previously derived value of M

 

[marcus]

the mass calculated was
M = 1.3E28 Planck

and the Earth's mass is
2.75E32 Planck

that's why I said roughly a twenty-thousandth----1/20,000 Earth mass.

------if you like everything metric------
the Planck mass unit is 22 micrograms as can be seen at NIST site,
so that E9 Planck is 22 kilograms
I get that in kilograms M = 29E19 kg = 2.9E20 kg.
and that is, in fact, roughly a 20,000th of the mass of earth
which is like about 6E24 kg.
----------------------------------------

the radius of an ordinary kind of black hole is 2M
so the ones for the Wizard's stove have a radius of
2.6E28
this is the size of a tiny mote of dust or pollengrain
because E28 is about one sixth of a micron
so we are talking about half a micron radius or micron diameter

 

[marcus]

black holes this size are always nice and warm
in fact fahrenheit 305, OK for baking cornbread
and homefry potatoes, macaroni-and-cheese,
such as a wizard might want for supper.

 

[arivero]

Fine parable. I like this kind of effort to grasp the scale of the units and objects we use; in fact some months ago I did the example of putting the elementary particle masses in amu (atomic mass units) and then to look for the nucleus having the same weight!

As you have completely given the answer to your history, let me to complete mine too. We have got Compton Length. Now, can we get quantum mechanics from it, using similar arguments?

Yes, we can: the Bohr-Sommerfeld quantum condition can be formulated, at least for circles, via a a Newton(*)-Kepler principle: any bound particle sweeps a multiple of Compton Area in a unit of Compton Time. This principle does not need gravity; it works for any central force.

The usual way, instead of this, is to use the obtained wavelength to check for destructive interference, most arguments in this line invoque De Broglie instead of Compton(**), so perhaps our whole thread could need a "fine tuning".

[EDITED:] In fact, note that the argument here above does not depend of $$c$$. Very much as the argumeng starting the thread does not depend of $$G$$.

(*)http://members.tripod.com/~gravitee/booki2.htm
(**) Indeed this was the point raised, subtly, by LPF earlier in the thread.

 

[marcus]

Originally posted by arivero
Fine parable. I like this kind of effort to grasp the scale...

Glad you approve. "kepler length" was witty and subtle (but could be stated in very simple terms) and I wish we could have more idea-puzzles like that.
the "wizard stove" idea actually did not come up to that level but was the best I could think of at the moment
why are subtle presentations of the ideas so difficult to think of?
(because the mind digs itself into ruts it can't get out of)
think of more and maybe we will start a thread.

 

[marcus]

what the Moon thinks she weighs

every lady has some idea of her weight--she may wonder if it is too much or too little, or she may be content with it, but she has some idea

the moon thinks of her weight as a force
measured by scales at her surface*

Now Alejandro you were just talking about circular orbit speed.
The moon knows the circular orbit speed right down at her surface
and she thinks of it in Planck terms (that is as a fraction of the speed of light) because that is natural
she thinks of every physical quantity that way, in those units.

what is the relation between her weight and the fourth power of that speed?

----------------------

* it is hard to imagine the moon standing on bathroom scales placed on her own surface, but she thinks of the force which is her weight as
a million times what one millionth of her would weigh on scales at her surface. or perhaps a billion times a billionth, I'm not sure which.

--------------------
EDIT: hello A. using edit button as suggested to clarify problem.
the key thing is "she thinks of every physical quantity in Planck terms" so the question is, what is the relation between two numbers:
the fourth power of the surface orbit speed and her weight (both expressed in the Planck system of units). no big deal, kind of a silly problem actually, I should solve it probably just as a check that I am stating it right. what I like is the moon having an idea of her own weight.
---------------------
MORE EDIT: well,you didnt take me up on this one, so I will say the answer.
They are the same number.
in Planck units the circ orbit velocity at surface is
$$\sqrt{\frac{GM}{Rc^2}}$$
you see I divided the dimensionful speed by c to get the speed as a fraction of c
And so the fourth power of the speed is
$$(\frac{GM}{Rc^2})^2$$
this is the same as her weight at her own surface
$$(\frac{GM^2}{R^2})$$
divided by the unit of force belonging to Planck units
$$\frac{c^4}{G}$$

 

[arivero]

I believe you have not stated the question rightly; even in plank units, the first equation (orbital velocity) gives the velocity as the square root of $M_earth/R_orbit$ while the second one gives the force as the square of $M_moon/R_moon$... Ok Ok I see.. I get you are intending to use the same Mass and same Radius in both equations, but -to me- it is not rightly stated, probably due to stress (a hard monday, as I see from other threads). You are referring to the "orbital" velocity of a stone thrown across the surface of the moon. And, actually, you only ask for G=1, neither c and h are needed.

In any case, it is fully classical gravity so we are hitting the off-topic wall :-(

[edited 10/Feb] Well, I see, you could try to connect it with the other quantization puzzles we have set up, but it seems too far fetched :-(

 

[arivero]

addenda

I have found this link

http://nedwww.ipac.caltech.edu/level5/Glossary/Essay_plancklt.html

which gives a interesting operational definition of Planck time.

There mass and time of Planck are defined as the quantitues simultanesly compatible with uncertainty principle and with gravitational collapse equation. IE a mass of plank spreaded in a plank volume takes a time of plank to collapse gravitationally into a point.

He says it is adapted from P. Coles' 1999 dictionary.

 

[marcus]

Let's try for more Loop Gravity fables!

Alejandro, thanks for the link
if that collapse-time business is from Cole's dictionary then
I prefer your definition of the "kepler length" of something.
It would make a better dictionary

Or a better "modern physics bestiary"

Let's try for more LQG fables.

What I like, one thing I like, about Loop Gravity is that
LQG and the Planck units are
mutually validating

 

[marcus]

Originally posted by marcus

LQG and the Planck units are
mutually validating

what I mean is in LQG one finds that
space (area and volume) is quantized and it turns out
that the quantization is precisely at Planck scale

so that confirms that the Planck units are a good
idea of units built into nature.

On the other hand Planck units lend some
confirmation to Loop gravity because they keep coming up
in the equations in and around the theory.

 

[arivero]

I disagree. As a mathematical aparatus, LQG does not worry about which fundamental length to choose; the theory does not depend of a concrete value. It does not turn up that quantization uses this scale, it is simply that physical intuition has suggested people to use Planck length. Examples as the one I provided in this thread are not usual in LQG treatises. Of course, I could be missing some reference... no time to read everything :-)

Personally, I am against the natural system of units, because people get used to neglect the h and the c from formulae even when they are describing approximations. For instance the fermionic lagrangian has a h multiplying it, thus it is obvious why there is not classical limit for fermions... but if you forget to put it, some students will have one or two bad days until they do the dimensional adjustment.

 

[arivero]

compton length and Zitterbewegung

Perhaps Zitterbewegung is the most general effect that depens on compton wavelength. There was some comment about it in the past, and I even started a thread recently at the Quantum zone of the forum,

https://www.physicsforums.com/showthread.php?s=&threadid=13365

 

[marcus]

A.R. this is slightly off-topic perhaps, having to do with the
article you mentioned with the quote from Lucretius
and the "relational" pivotal idea

Rovelli recaps those ideas in his book pages 150-155
which are section 5.6 "Relational interpretation of quantum theory"
5.6.1 "The observer observed"
5.6.2 "Facts are interactions"
he illustrates this section with a drawing of Archimedes
or Galileo (who can tell the difference) that it looks like he himself
drew it, or someone in his family who is not an artist. The man is fearless

anyway I am glad he did not let drop the ideas of that 1998 article
you cited, but updated and included them in the book

 

[arivero]

Originally posted by arivero
By the way, I had never seen it, but I do skip a lot of literature on this.

Doing a bit of literature search, a first surprising point is that the recent paper of Markopoulou and Smolin does not use this trick but simply prefers to postulate the existence of a stocastic phenomena.
http://xxx.unizar.es/abs/gr-qc/0311059

The "trialogue", at formula 4.2 from Veneziano, uses gravitational force between strings to underline that G and c could be more fundamental than h.
http://xxx.unizar.es/abs/physics/0110060

Also Novikov/Zeldovich are quoted pointing out that plank length and c should be enough in quantum gravity, because G and h appear always (or at least in the Einstein Hilbert action), as a product $$\sqrt{Gh}$$.

 

[marcus]

Originally posted by arivero
Doing a bit of literature search, a first surprising point is that the recent paper of Markopoulou and Smolin does not use this trick but simply prefers to postulate the existence of a stocastic phenomena.
http://xxx.unizar.es/abs/gr-qc/0311059
...
because G and h appear always (or at least in the Einstein Hilbert action), as a product $$\sqrt{Gh}$$.

Your university at Zaragoza has its own arXiv mirror site!

You refer to
"Quantum Theory from Quantum Gravity"
by Fotini Markopoulou and Lee Smolin

I am not sure what to think about this article. How can the very quantum theory itself arise from spin networks?
is the network or graph of adjacency relations such a mathematically rich structure that it can give rise to quantum theory? I did not know what to make of the abstract:

"We provide a mechanism by which, from a background independent model with no quantum mechanics, quantum theory arises
in the same limit in which spatial properties appear.

Starting with an arbitrary abstract graph as the microscopic model of spacetime..."

any clarification about what they are trying to do?

 

[arivero]

There are about 20 mirrors of arxiv around the world (so, will it survive meteorite clash?). Zaragoza happens to have the spanish one. It is remotelly supported from the original, so no maintenance needed.

It seems that M & S point out that if a stochastic trembling in coordinaates is postulated, then plank constant and schoedinger equation follow. Not surprising to us in this thread. But they do not take risk about the origin of the trembling.

From a fundamental point of view, surely they are just remarking that unification of quantum theory and general relativity is supposed to be a new theory, so both regimes should emerge in the appropiate limits. No news here neither.

 

[Orion1]

Kepler's Length...

Problem:

given a particle of mass m, for which radius will a circular gravitational orbit around the particle to have the property of sweeping one plank area in exactly one plank time unit?




Kepler's Second Law:
$$\frac{dA}{dt} = \frac{L}{2m} = K_k$$

Planck Area: (circle)
$$dA = \pi r_p^2 = \frac{\pi \hbar G}{c^3}$$

Planck Time: (period)
$$dt = t_p = 2 \pi \sqrt{ \frac{ \hbar G}{c^5}$$

Angular Momentum:
$$L = mrv$$

$$v = \sqrt{ \frac{Gm}{r_1}}$$

$$L = mr_1 \sqrt{ \frac{Gm}{r_1}}$$

integral:
$$\frac{\pi r_p^2}{t_p} = \frac{r_1}{2} \sqrt{ \frac{Gm}{r_1}}$$

$$r_1 = \frac{4}{Gm} \left( \frac{ \pi r_p^2}{t_p} \right)^2$$

Solution #1:
Compton Wavelength:
$$r_1 = \frac{\hbar}{mc} = \overline{ \lambda}$$

Planck Mass:
$$m = m_p = \sqrt{ \frac{\hbar c}{G}}$$

Solution #2:
Planck Radius:
$$r_1 = \sqrt{ \frac{ \hbar G}{c^3}} = \overline{ \lambda}$$

$$K_k = 2.422*10^{-27} m^2s^{-1}$$
$$L = \hbar !$$

 

[marcus]

there are actually two masses in the picture
and arivero referred to the central mass
(in which case the mass of the satellite is assumed
negligible, making the angular momentum of the orbit
hard to determine)
but this analysis is very much in a good spirit and
the right direction!
Orion please try it where you have a pair of particles
of equal mass circling each other
maybe they are each m/2 so the combined mass is m
or they could each be m.
I am just waking up so not sure if this is a sensible reply
but it might be.

 

[arivero]

Originally posted by Orion1
Solution #1:
Compton Wavelength:
$$r_1 = \frac{\hbar}{mc} = \overline{ \lambda}$$

Planck Mass:
$$m = m_p = \sqrt{ \frac{\hbar c}{G}}$$

Solution #2:
Planck Radius:
$$r_1 = \sqrt{ \frac{ \hbar G}{c^3}} = \overline{ \lambda}$$

$$K_k = 2.422*10^{-27} m^2s^{-1}$$
$$L = \hbar !$$

The "standard", or at least majoritary, definition for Planck length is "the compton length of a mass of plank". I supposse that you are pointing out that this particular case has a total gravitational action of exactly $$h$$. Is it?

In any case I think it is more important to stress that the Compton Length answer happens for any mass.

Also, it is usually told that $$m_p$$ is the case where Swartzchild radius and Compton radius coincide. Not sure about the meaning of this, here.

 

[Orion1]

Binary Systems...



Compton Radius/Wavelength:
$$r_1 = \frac{\hbar}{mc} = \overline{ \lambda}$$

The Compton Radius is only a solution for a Keplerian-Planck System, for which m<<M.

As m approaches M in magnitude, the system becomes binaric, and all keplerian laws fail and no longer apply.

Kepler's Laws function only as derivatives to the Binary Theorem, therefore are not actually 'laws'.

Solutions to binary systems requires a modification to Newtons Gravitational Law using a center of mass integration.

$$dt_1 = dt_2$$

$$\frac{dA}{dt} = K_k = 2.422*10^{-27} m^2s^{-1}$$

Orion1 Binary System Theorem:
$$\frac{dA}{dt} = \frac{Gm_2^3t_1}{4 \pi (m_1 + m_2)^2 r_1} = K_k$$

$$r_1 = \frac{Gm_2^3t_1}{4 \pi K_k (m_1 + m_2)^2}$$

Limit t₁ = t_p (period)
$$r_1 = \frac{Gm_2^3}{2 K_k (m_1 + m_2)^2} \sqrt{ \frac{\hbar G}{c^5}}$$

Limit m₁ = m₂ = m_p
r₁ = 2.538*10^-35 m
r_p = 1.616*10^-35 m



Also, it is usually told that $$m_p$$ is the case where Swartzchild radius and Compton radius coincide.


Compton Radius:
$$r_c = \frac{\hbar}{mc} = \overline{ \lambda}$$

Schwarzschild Radius:
$$r_s = \frac{2Gm}{c^2}$$

$$r_c = r_s$$

$$\frac{\hbar}{mc} = \frac{2Gm}{c^2}$$

Compton-Schwarzschild Mass:
$$m_{cs} = \sqrt{ \frac{ \hbar c}{2G}$$

Close, but negative.

 

[marcus]

Hi Orion
you said:"Also, it is usually told that [planck mass] is the case where Swartzchild radius and Compton radius coincide."

I have never heard that, what I have read has been that Planck mass is the case where half the Schwarzschild radius and the Compton wavelength coincide.

if you have a link to something online with the alternative definition you could post it, then anyone who is curious about the different defintion could check it out

but it is just a factor of two that we are worrying about

half the Schw. is GM/c^2

and that has to equal the Compton, which in reduced or angular format is hbar c/Mc^2

So one sets the two equal and solves

GM/c^2 = hbar c/Mc^2

I will try to put this in LaTex, to match your clear writing.

$$\frac{GM}{c^2} = \frac{\hbar c}{Mc^2}$$

$$M^2 = \frac{\hbar c}{G}$$

 

[arivero]

factor2

Sorry marcus it was me who raised the sloopy definition; there is a factor 2 (even 4) around all the thread, but I am not worried at this level; we are playing sort of dimensional analysis.

Orion, note that the initial setup does not ask for a whole revolution. Area law forks for a sector of the circle too. Answers are the same, although to look for a complete revolution is an interesting particular case, hmm.

 

[Orion1]

Orion1 Binary Theorem...


the initial setup does not ask for a whole revolution. Area law forks for a sector of the circle too.



$$dt_1 = dt_2$$

$$\frac{dA}{dt} = K_k = 2.422*10^{-27} m^2s^{-1}$$

Orion1 Binary System Theorem:
$$r_1 = \frac{4(m_1 + m_2)^2}{Gm_2^3} \left( \frac{dA}{dt} \right)^2$$

$$r_1 = \frac{}{Gm_2^3} \left( 2(m_1 + m_2) \left( \frac{dA}{dt} \right) \right)^2$$

$$r_1 = \frac{ (2K_k(m_1 + m_2))^2}{Gm_2^3}$$

Limit m₁ = m₂ = m_p
r₁ = 6.464*10^-35 m
r_p = 1.616*10^-35 m
r₁ = 4r_p


Below that radius, it should be posible to use "plank time beats" to divide area into regions smaller that previous. So a fundamental break of physics will happen at "quantum kepler length of the particle m"


$$\frac{dA}{dt} = \frac{}{2} \sqrt{ \frac{\hbar G}{c}} = K_k = 2.422*10^{-27} m^2s^{-1}$$

Planck Mass Fusion:
Limit m₁ = m₂ = m_p
Limit r₁ = r_p
$$\frac{dA}{dt} = \frac{ \sqrt{Gm_2^3r_p}}{2(m_1 + m_2)} = \frac{}{4} \sqrt{ \frac{\hbar G}{c}} = K_1 = 1.211*10^{-27} m^2s^{-1}$$

Planck-Compton Radius: r₁ = r_p
$$r_p = \frac{\hbar}{m_pc} = \overline{ \lambda_p}$$

Gravitation 'quantum shutdown' below the Planck Radius: r₁ <= r_p

r₁ <= r_p
$$\frac{ \sqrt{m_2^3r_1}}{(m_1 + m_2)} = \frac{}{2} \sqrt{ \frac{\hbar }{c}}$$

r₁ <= r_p
$$\frac{m_2^3}{(m_1 + m_2)^2} = \frac{ \hbar}{4cr_1}$$

r₁ <= r_p
$$r_1 = \frac{ \hbar (m_1 + m_2)^2}{4cm_2^3}$$