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View Full Version : Restrictions on uncertainties in QM
As we know, SRT puts some limits on possible speed of propagation of any interactions in Nature. It definitely restricts any changes of the physical features of things no matter do we any measurements of them or not. Therefore, speaking on uncertainties of physical characteristics that can occur at any measurement in the frame of QM, we have to have some definite limits on those uncertainties limited by the requirements of SRT.
For instance, the uncertainty Δx of coordinate of particle can not exceed the value cΔt, where Δt is the physical time this particle has for the development of any uncertainty. Correspondingly, the uncertainty of momentum, ΔP, which naturally equals to Δ(M[v]v) = Δ[Mov/(1-v˛/c˛)˝] = MoΔv/(1-v˛/c˛)˝ + Mov˛Δv/[c˛(1-v˛/c˛)^3/2] = [Mo/(1-v˛/c˛)^3/2]Δv = M[v]Δv/(1-v˛/c˛), can not exceed value M[v]c/(1-v˛/c˛). Because, according to SRT, Δv ≤ c. Therefore, we have to have the following restriction for uncertainties:
ΔPΔx ≤ M[v]c˛Δt/(1-v˛/c˛)...........................(1)
Together with Heisenberg's inequality it gives:
M[v]c˛Δt/(1-v˛/c˛)/ ≥ ΔPΔx ≥ h ....................(2)
There arrives a problem: What wave functions are satisfying to these inequalities?
Data, if you agree with this set up of problem, I will post my thoughts about it, you should know before we will start our investigation, OK?
Actually, I think you were right in the other thread now. Sorry :)
Δ[Mov/(1-v˛/c˛)˝] = MoΔv - Mov˛Δv/[c˛(1-v˛/c˛)^3/2] = [Mo/(1-v˛/c˛)˝]Δv
should be
Δ[Mov/(1-v˛/c˛)˝] = MoΔv/(1-v˛/c˛)˝ + Mov˛Δv/[c˛(1-v˛/c˛)^(3/2)] = [Mo/(1-v˛/c˛)^(3/2)]Δv = M[v]Δv/(1-v˛/c˛) = &˛M[v]Δv ≤ &˛M[v]c
so
&˛M[v]c˛Δt ≥ ΔPΔx ≥ h
or &˛EΔt ≥ ΔPΔx ≥ h
where & = 1/(1-v˛/c˛)˝, as far as I can see. Other than that it looks pretty reasonable. I don't see anything wrong. I'll take a look tomorrow at how to construct functions satisfying that.
Data, I found the same error just a seconds before you posted and fixed it up.
So, we can continue now.
Of course, we can do the same with second Heisenberg’s inequality:
ΔEΔt ≥ h
Because
ΔE = Δ(M[v]c˛) = Δ{Mo c˛/(1- v˛/c˛)^1/2} = Mov Δv/(1- v˛/c˛)^3/2 = PΔv/(1- v˛/c˛)
So, we have:
ΔE ≤ Pc / (1- v˛/c˛)……………………(3)
And the uncertainty in energy should obey the following inequalities:
Pc / (1- v˛/c˛) Δt ≥ ΔE Δt ≥ h……………(4)
And here arrives one middle-side problem, we have to analyze:
If we denote
Je = Pc / (1- v˛/c˛) and Jp = M[v]c˛/(1-v˛/c˛)
Then we see that always Je ≤ Jp.
PS: You just follow my mistake and left sign"-" in line with two terms; it should be "+"; so, correct it.
Data,
I'll take a look tomorrow at how to construct functions satisfying that.
Before you will start to do that, please, read my message that I will prepare for you here.
Good night!
I will. Thanks for pointing out that mistake! I did it by hand correctly, but I don't know how to use symbols here yet so I copied your line and forgot to change it.
Data,
I guess, we have now two big businesses that should be done:
1. We need an accurate definition of notion “uncertainty” (may be as average deviation from average?) as in classic Theory of probability so in None-relativistic QM (NQM); we need simplest, clear and accurate derivation of the Heisenberg’s principle (as it is done in good textbooks, at least); and we need (!) analysis of possibilities of finding the wave functions that are satisfying to our upper limits on uncertainties.
2. There will be a lot of arguing with our approach that will be based upon such a logical scheme:
All your upper limits came from SRT
But NQM is a non-relativistic limit of SRT-theory (QFT)
As we know, such a limit means c rushing to infinity and the rest energy is subtracted from total energy.
If we will take this limit in your expressions for upper limits of uncertainties, we will get in both cases infinities.
That makes your “discovery” trivial: anything is less then infinity!
Such “logic” is absolutely wrong way of thinking, but we have to be prepared to prove that.
So, I guess, we should for now separate our efforts and develop both these issues. My proposal is: you should do issue #1, I should do issue #2. We should post each more or less significant accomplishment immediately, but main project for each of us should stay as I said above: you –issue #1, I – issue #2. Do you agree with that?
I'm still here, I'll post something tomorrow, hopefully~
Data,
since you are busy, I did some research and Ithink it will be a good basis for our further work.
Let me reveal a simple proof of the Heisenberg’s Principle of uncertainty.
First of all, let as define the notion uncertainty of the physical property A.
As the QM teaches us, each observable physical property A has its linear operator, Â. Because this property is observable this operator is Hermitian, i.e. Â٭ = Â what means that the Hermitian conjugated operator is identical with original operator. (Let us recall that Hermitian conjugation means transposition and complex conjugation). The sense and role of such operators is reflected by the following rule: if Ψa is a wave function of the “pure” state of property A (i.e. the wave function of quantum state in which object has exactly the value A for the considered property), then
 Ψa = A Ψa ………………………..(1)
Therefore,
Ψa٭ Â٭ = A Ψa٭ .................
................(2)
If we have so called “mixed” state in regard of property A (i.e. not a “pure” state of property A) in which value of property A can be any with a correspondent probability), then the average value of property A in this mixed state is:
Aaverage = <Ψ٭ Â Ψ > …………..…(3)
Where < > means all integrations and all summations over all arguments of the wave function Ψ of that mixed state.
Often people use the Dirac’s notations and write down (3) as following:
Aaverage = <Ψ| Â |Ψ > …………….…(3)’
The uncertainty ΔA of property A in quantum state described by the wave function Ψ we call the quantity that being squared gives:
(ΔA)˛ = <(Â – <Â>)˛> ………………….(4)
Because we have
<( – <Â>)˛> = <² – 2<Â>  + (<Â>)˛> = <²> – 2<Â> <Â> + (<Â>)˛ =
= <²> – (<Â>)˛,
one can conclude that
(ΔA)˛ = <²> – (<Â>)˛………………….(5)
i. e. uncertainty of property A is exactly the average squared deviation of the expected value of A from its average value.
Attention! If the wave function Ψ describes a pure state of operator Â, i.e. ÂΨ = AΨ, the uncertainty ΔA identically is zero!
After this short reminder of basic notions of QM, let me start a simple proof of the Heisenberg’s principle.
Let us the considered object is in the quantum state with wave function Ψ. Then for the product of uncertainties of two observable properties, G and H, we will have by definition:
(ΔC)˛ (ΔH)˛ = (<Ĉ˛> – (<Ĉ>)˛)*(<Ĥ˛> – (<Ĥ>)˛) =
= <(Ĉ – <Ĉ>)˛>*<(Ĥ – <Ĥ>)˛> ……………………….(6)
Attention! If the wave function Ψ describes a pure state of operator Â, i.e. ÂΨ = AΨ, then we have identically <²> – (<Â>)˛ = 0 and <( – <Â>)˛> = 0!
Here we will use so called the Schwartz’s inequality:
<Ŵ˛>*<Ŷ˛> ≥ |<Ŵ*Ŷ>|˛……………………………(7)
valid for any pare of the Hermitian operators Ŵ and Ŷ.
Attention! If the wave function Ψ describes a pure state of operator Ŵ, i.e. ŴΨ = WΨ, then we have identically <Ŵ˛>*<Ŷ˛> = |<Ŵ*Ŷ>|˛!
The Schwartz’s inequality is a direct generalization of the usual inequality ab ≥ |ab|, which is valid for any pare of vectors of the nD-Euclid space, a and b, for the Hilbert space of the eigenfunctions of the Hermitian operators.
Applying (7) to (6) one will get:
(ΔC)˛ (ΔA)˛ = <(Ĉ – <Ĉ>)˛>*<(Â – <Â>)˛> ≥ |<(Ĉ – <Ĉ>)*(Â – <Â>)>|˛…(8)
A simple algebra shows that
<(Ĉ – <Ĉ>)*(Â – <Â>)> = <[Ĉ, Â]> + < Â Ĉ > – <Ĉ>*<Â> ……….(9)
where [Ĉ, Ĥ] is a commutation of Ĉ and Ĥ, i. e.:
[Ĉ, Ĥ] = ĈĤ – ĤĈ
Therefore, we have:
(ΔC)˛ (ΔA)˛ ≥ |<( 0.5[Ĉ, Â] + 0.5{Â, Ĉ} – <Ĉ>*<Â>)>|˛……….(10)
But [Ĉ, Â] is an operator iŴ and {Â, Ĉ} is an operator Ű, where both operators Ŵ and Ű are some Hermitian operators, i.e. <Ŵ> and <Ű> are real numbers. Therefore, we have:
(ΔC)˛ (ΔA)˛ ≥ | 0.5 i <Ŵ> + 0.5< Ű> – <Ĉ>*<Â>|˛| =
= (0.5<Ű> – <Ĉ>*<Â>)˛ + 0.25< Ŵ >˛……….…………………….(11)
Attention! If the wave function Ψ describes a pure state of operator Â, i.e. ÂΨ = AΨ, then we have identically <( 0.5[Ĉ, Â] + 0.5{Â,Ĉ} – <Ĉ>*<Â>)> = 0 and in (11) the sign “=” “works”.
Because
(0.5<{Â, Ĉ}> – <Ĉ>*<Â>)˛ ≥ 0 ……………………………………(12)
one gets from (11):
(ΔC) (ΔA) ≥ 0.5 |<[Ĉ, Â] >|………………………..……………..…(13)
Let Ĉ and  are a pare of the Hermitial operators that represent two conjugated dynamical variables, C and A. Then, by definition, Ŵ = h. According this result we finally get:
(ΔC) (ΔA) ≥ h/2, if Ψ is not describing a pure state of Ĉ or  ..….(14)’
and
(ΔC) (ΔA) = 0, if Ψ is describing a pure state of Ĉ or  …………….(14)’’
for any pare of two conjugated dynamical variables, C and A.
This derivation of the Heisenberg’s principle you can find in many good books. And now I will show something that I never sow in any textbook.
If we look carefully on equation (6), we will see that relation
(ΔC)˛ (ΔA)˛ = (<Ĉ˛> – (<Ĉ>)˛)*(<²> – (<Â>)˛)
leads to the trivial inequality
(ΔC)˛ (ΔA)˛ ≤ (<Ĉ˛>*(<²> – (<Â>)˛)) = <Ĉ˛>*(ΔA)˛………………(15)
Attention! If the wave function Ψ describes a pure state of operator Â, i.e. ÂΨ = AΨ, the sign “=” “works”.
or
(ΔC) |ΔA| ≤ |√<Ĉ˛>| * |ΔA|……………….(15)’
or
(ΔC) ≤ |√<Ĉ˛>| ………………………………….…(15)’’
Correspondingly one could get
(ΔA) ≤ |√<²>| ………………………………….…(15)’’’
what is a trivial consequence of the definition of uncertainty itself (see (5)!).
Therefore, no matter what happens with the considered object we can state that
the uncertainties of any pare of its conjugated dynamical variables have to be restricted by the inequalities:
|√<Ĉ˛>|*|√<²>| ≥ |√<Ĉ˛>|*(ΔA) ≥ (ΔC)*(ΔA) ≥ 0.5 h ……..(16)’
and
ΔY ≤ |√<Ŷ˛>| ………………………………….…(16)’’
where Ŷ is a Hermitian operator of any observable.
As we sow at derivations of those trivial (by its mathematical contents!) inequalities, there is no question like that: “What wave functions are providing validity of these inequalities?” Because immediate answer is the following: “Any right solution of the corresponding wave-function-equations should do that”.
Let us notice that equality can occur in (16) only if <Ŷ> = 0.
The interesting physical consequences appear at the comparing of those inequalities with the Principle of causality and with requirements of SRT, as we did at initial setting of this thread.
geistkiesel 02-24-05, 01:30 AM "As we know, SRT puts some limits on possible speed of propagation of any interactions in Nature. It definitely restricts any changes of the physical features of things no matter do we any measurements of them or not. . . . "
Yuriy,
The validity of the statement is seriously questioned when considering the results of experiments and other research classifiable as "EPR Experiments".* The twin photon-photon interaction times of photons moving at relative velocity of 2c were scrutinized. Zero total angular momentum of the twin photon system was confirmed from measurment. After an indefinite time of flight the measured polarization angles of the twin photons measured +1,-1.
There are two immediate problems: Gross Violations of SRT speed of light postulates and, That QM Theory predicts the results of EPR experiments.
Other experimental results confirm QM predictions of EPR.
Assuming some professional familiarity with EPR considerations, the lack of EPR reference in the post suggests that SRT considers EPR violations of restrictions placed on "any interaction in Nature" as insignificant.
Yuriy,
Will you please confirm or deny that EPR violations of restrictions placed on "any interaction in Nature" is of no significance to SRT?
*Einstein, Podolsky and Rosen Experiments
Geistkiesel
As you can see, Data, the naďve simplicity of (16) is very delusive.
1. The first of all, we should pay attention that (16) ‘ actually is right only for quantum states, which are not the pure states of any of figured variables: neither C, nor A. Otherwise, (16)’ does not “work” (see derivation of (14)!). We can not object: may be that is a deficiency in chosen way of derivation, and there is more precise one that will show that
(ΔC)*(ΔA) ≥ 0.5 h
for any states, including the pure states of variable C and/or A. And then we should conclude that
in the limit of a pure state of some dynamical variable the uncertainty of its dynamical conjugated variable should be equal to infinity,
what is exactly the consequence of the Heisenberg’s Principle that the Quantum Mechanics states. Let us agreed with such approach.
2. Let us now apply (16)’’ to the coordinate of particle in pure state of a momentum. As we know, ΔX – the uncertainty of a coordinate – in this state should be infinitely large. But, the direct application of SRT says that this uncertainty can not exceed value of c*Δt, where Δt is a period of time we spent to measure (to make sure that we indeed deal with a pure state of momentum) momentum of this particle (the procedure that guaranties that particle will be in the pure state of momentum). So, we have to write down the following uncertainty:
ΔX ≤ c*Δt ………………………………….…(17)
which reflects a simple requirement of SRT: no body can move with speed more then speed of light in vacuum.
Of course, this inequality contradicts to the fundamental assertion of the whole Quantum Mechanics: as close the state on particle is to the pure state of some dynamical variable, as the uncertainty of its dynamical conjugated variable should be higher; in the limit of a pure state of some dynamical variable the uncertainty of its dynamical conjugated variable should be equal to infinity.
What the Hell! We are certain that measurement of the momentum can not take the infinite time. Therefore, according (17), ΔX should be finite! Occurring logical contradiction shows that there is some lack of misunderstanding of the fundamentals of QM.
I thought about that a long time, and as I guess, I know the resolution. But before I will tell you it, we need to analyze the problem “with fresh eyes”. So, what you think? What resolutions you can offer? Any critics, any suggestions, any questions….
Data,
because you still are busy I have to work alone. There are some more of mine reflections…
1. The problem we came can be formulated as the following one.
SRT guarantied that no body, no matter is it under observation or is not, can move with speed more than speed of light in vacuum, c.
In other hand, the classic QM insist that the uncertainty of the position of any particle should be infinite as a result of any measurement of any physical characteristic of this particle, A, if the Hermitian operator of this characteristic, Â, is not commutative with coordinates of this particle, i. e. [x, Â] ≠ 0. Particularly, after measurement of the momentum of any particle the uncertainty of its coordinate should be equal to infinity. Not “limited by some very big but finite number”, but infinity.
These two assertions obviously are contradicting to each other conceptually, philosophically … and mathematically.
2. Some people can object of that conclusion saying that “Classic QM is a limit of SRT at c rushing to infinity. If we do this limit in the restriction of uncertainty arriving of SRT, we will get not
ΔX ≤ c*Δt ………………………………….…(17)
but
ΔX ≤ ∞ …………………………………….…(18)
And that is exactly what QM says.”
Actually this is a trick, nothing else. First of all we should correct such critics that limit of classic Physics arrives from SRT Physics not at taken a limit c → ∞, but limit (v/c) → 0. (BTW, taken any limits of the dimensional variables is mathematical nonsense: before taken of any limit we have to normalize the all of dimensional variables and only then we can apply the limits to those dimensionless ratios.) At limit (v/c) → 0 (17) stays the same (unchanged) and does not convert to (18)!
But (18) indeed is very important for whole customized interpretation of the non-relativistic (i.e. classic QM) because any real calculation of ΔX in the pure state of a momentum gives infinite value for it!
3. To avoid the obvious paradox of inconsistency of (17) and (18), we should return to the very basics of SRT and QM.
Let us consider two observers of the same particle: one in the reference frame where this particle moves with speed v << c, and another one in the reference frame where the same particle has a speed v comparable with c. So, the first observer considers the particle as non-relativistic one, and the second observer considers it as essentially relativistic one. Therefore, the second observer can not describe this particle by the laws of the non-relativistic QM and has to use the Quantum Fields Theory (QFT) in full measure.
According to QFT even the free propagation of this particle is a very complex process … because of its interaction with… vacuum! For example, this propagation contains as a possible one the evolution that is shown in this picture (http://www.sciforums.com/attachment.php?attachmentid=3912&stc=1). As you see, no one involved body moves faster than light in vacuum, no one “hidden” parameter is introduced, and nevertheless after time to we can find our particle on the distance Xo that is much long than product
X* = cto! And that evolutions are real, not a fantasy! And namely these kind evolutions are providing the fact that observer in that RF sees
ΔX ≤ ∞ …………………………………….…(18)
and not
ΔX ≤ c*Δt ………………………………….…(17)
But what is “to see particle as non-relativistic”? It means see its story as the first observer does it! So, we should ask ourselves: “What sees the first observer, if the second one sees what we just described and as the picture shows?”
Due to the main Principle of Relativity, the first observer has to see the same chain of events happened with the particle. But each interaction with the vacuum fluctuation that is shown in picture is an event! Therefore, the first observer sees the same evolution of particle: appearance of particle at appropriate time to’ (do not forget the time dilation!) in appropriate point X*’ (do not forget the length contraction!), which is much longer then a corresponding distance Xo’!
The main conclusion that we have to do is the following:
The consideration of non-relativistic QM of particles does not exclude the physical processes of interaction of these particles with the physical vacuum.
And that is the resolution of our paradox of inconsistency between (17) and (18).
(Now everyone should recognize how important the discussion on vacuum, that lethe and me had in this Forum and unfortunately could not finish, is!)
The derivation of (17) had assumed that we deal with the same quantum state of particle all the time. But each interaction of this particle with vacuum fluctuation “pushes” it off the previous quantum state and “puts” it in some new quantum state!
Are you still with me? Should I continue?
Quick question: How does the limit v = c fit the uncertainity principle. If something exists at 'c' then it would seem that both state and position are knowable but perhaps not measureable.
geistkiesel 02-27-05, 02:14 PM Data,
because you still are busy I have to work alone. There are some more of mine reflections…
1. The problem we came can be formulated as the following one.
SRT guarantied that no body, no matter is it under observation or is not, can move with speed more than speed of light in vacuum, c.
In other hand, the classic QM insist that the uncertainty of the position of any particle should be infinite as a result of any measurement of any physical characteristic of this particle, A, if the Hermitian operator of this characteristic, Â, is not commutative with coordinates of this particle, i. e. [x, Â] ≠ 0. Particularly, after measurement of the momentum of any particle the uncertainty of its coordinate should be equal to infinity. Not “limited by some very big but finite number”, but infinity.
These two assertions obviously are contradicting to each other conceptually, philosophically … and mathematically.
2. Some people can object of that conclusion saying that “Classic QM is a limit of SRT at c rushing to infinity. If we do this limit in the restriction of uncertainty arriving of SRT, we will get not
ΔX ≤ c*Δt ………………………………….…(17)
but
ΔX ≤ ∞ …………………………………….…(18)
And that is exactly what QM says.”
Actually this is a trick, nothing else. First of all we should correct such critics that limit of classic Physics arrives from SRT Physics not at taken a limit c → ∞, but limit (v/c) → ∞. (BTW, taken any limits of the dimensional variables is mathematical nonsense: before taken of any limit we have to normalize the all of dimensional variables and only then we can apply the limits to those dimensionless ratios.)
But (18) indeed is very important for whole interpretation of the non-relativistic (i.e. classic QM) because any real calculation of ΔX in the pure state of a momentum gives infinite value for it.
3. To avoid the obvious paradox of inconsistency of (17) and (18), we should return to the very basics of SRT and QM.
Let us consider two observers of the same particle: one in the reference frame where this particle moves with speed v << c, and another one in the reference frame where the same particle has a speed v comparable with c. So, the first observer considers the particle as non-relativistic one, and the second observer considers it as essentially relativistic one. Therefore, the second observer can not describe this particle by the laws of the non-relativistic QM and has to use the Quantum Fields Theory (QFT) in full measure.
According to QFT even the free propagation of this particle is a very complex process … because of its interaction with… vacuum! For example, this propagation contains as a possible one the evolution that is shown in this picture (http://www.sciforums.com/attachment.php?attachmentid=3912&stc=1). As you see, no one involved body moves faster than light in vacuum, no one “hidden” parameter is introduced, and nevertheless after time to we can find our particle on the distance Xo that is much long than product
X* = cto! And that evolutions are real, not a fantasy! And namely these kind evolutions are providing the fact that observer in that RF sees
ΔX ≤ ∞ …………………………………….…(18)
and not
ΔX ≤ c*Δt ………………………………….…(17)
But what is “to see particle as non-relativistic”? It means see its story as the first observer does it! So, we should ask ourselves: “What sees the first observer, if the second one sees what we just described and as the picture shows?”
Due to the main Principle of Relativity, the first observer has to see the same chain of events happened with the particle. But each interaction with the vacuum fluctuation that is shown in picture is an event! Therefore, the first observer sees the same evolution of particle: appearance of particle at appropriate time to’ (do not forget the time dilation!) in appropriate point X*’ (do not forget the length contraction!), which is much longer then a corresponding distance Xo’!
The main conclusion that we have to do is the following:
The consideration of non-relativistic QM of particles does not exclude the physical processes of interaction of these particles with the physical vacuum.
And that is the resolution of our paradox of inconsistency between (17) and (18).
(Now everyone should recognize how important the discussion on vacuum, that lethe and me had in this Forum and unfortunately could not finish, is!)
The derivation of (17) had assumed that we deal with the same quantum state of particle all the time. But each interaction of this particle with vacuum fluctuation “pushes” it off the previous quantum state and “puts” it in some new quantum state!
Are you still with me? Should I continue?
Without exception
including the firstparticle from the test gas source the virgin, unpolarized spin-1 particle, for example, the spin states of particle transitions through Stern-Gerlach inhomogeneous magnetic field sand gradient volumes are identifiable to 100% certainty with any combination of two channels completely obstructed, or completely wide open, regardless of the number of or combiation of sequential S and T SG segments aligned along the direction of particle motion. This applies to all alien-to-domestic polarizations, (S -> T,) or domestic-to-domestic (S -> S' ) polarizations.
that contrary to popular opinion spin-1 particles polarized along the lab frame (or SG segment ) z-axis have no x-component of observed spin states [/b ]
[b]That all of of the observed state components of the spin-1 particle are nonlocal
That the observed state has no observed components and that the observed states is merely the result of complex process of nonlocal force centers
100% prediction accuracies of particle spin states
nonlocal force centers are identified and physically located wthin any degree of spatial resolution or accuracies demanded
reformation of the spin state, during depolarization of test particles gves strong experimenal results suggsting the that earth;s magnetic field system is basically a purely nonlocal affect, including restoring forces to the compass polarized orientaion of he compass needle
Some may find it difficult to believe that god does not now, nor did she in the past, ever
play dice with the universe
The folllowing transitions are crucually and criticdally nonlocal in the most fundamental consideration and require application of forces at a distance through local/nonlocal force centers,
S -> S';
T -> T';
S -> T' -> S;
S -> T' -> T' + B -> T;
S -> S' -> S' + B -> S;
where B is the physical channel obstruction operator or (B) .
Do you want to deal with these issues, then take your logical next step, go right ahead.
You are the first on my list. You are places there from your comments months ago that you were proficient to a PhD level with SG transition prcesses. So, if you prefer this matter can be accomplished rationally , or would you rather I just stampde ahead, it is your option?
Geistkiesel.
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