- #1
Laci
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- TL;DR Summary
- No macroscopic objects with spread center of mass were ever seen, although QM prefers such states.
One of the strange features of Quantum Mechanics is that for his formulation one needs the classical physics that actually should emerge as its macroscopic limit. All experiences with quantum objects have to be analyzed through classical "glasses".
Naturally, then the question arises: where is the threshold between the microscopic and the macroscopic worlds?
In the first half of the twentieth century the big successes of quantum theory were obtained mainly in the field of ("elementary") microscopic particles: electrons, atoms, nuclei. Thereafter, the most spectacular evolution took place in understanding the properties of condensed matter on the basis of many-body quantum mechanics. This step revolutionized our technology. Now is undergoing another exceptional development in producing ever smaller pieces of solid matter. We are approaching the microscopic world from above. Therefore the answer to above stated question may be very close.
In this context I would like to underline some simple but ignored aspects. Solid state theory is conceived as the quantum mechanical description of the stable state of an enormous number of interacting electrons and ions. All we treat is however only the relative motion, while the center of mass motion is considered irrelevant and left to the classical description. Moreover, we consider mostly infinite systems or closed ones in a periodical box.
A somewhat appeasing argument was given in almost all old handbooks about the stability of a wave package of the center of mass for macroscopic masses. Indeed if at an initial time t=0 one has a Gaussian one dimensional wave package
$$\psi(x,0)=\frac{1}{(2\pi )^{\frac{1}{4}}\sqrt{d}}e^{-\frac{x^2}{4d^2}}$$
then after a lapse of time t it decays according to the Schrödinger equation as
$$\psi(x,t)=\sqrt{\frac{\pi}{d^2+\frac{\imath\hbar t}{2m}}}e^{-\frac{ x^2}{4 (d^2 +\frac{\imath \hbar t}{2m})}} $$
This means that the average quadratic width grows as
$$\langle x^2\rangle_{t} =d^2\left[1+\frac{1}{2}\left(\frac{\hbar t}{md^2}\right)^2\right] \enspace . $$
One may see, that for ##m \to \infty## the width does not changes at all. However, the decisive parameter is not the mass m alone but the product ##md^2##. For any finite mass m at ##d \to 0 ## the wave package instantly decays.
Leaving aside such ideal limits, let us consider a numerical example for sake of illustration. A body of m=1g and an initial linear imprecision of its center of mass of 1 Angström after 200 million years will be smeared out as wide as to ##2\times 10^4## Angström. This is not yet frightening, but worth to think about.
Even more provoking is the question: Why all the macroscopic objects we know have a very precise center of mass position? Ultimately, why there are no plane waves of macroscopic objects, although energetically more favorable. In our example of the Gaussian package the average kinetic energy is
$$\frac{\hbar^2}{2m}\langle k^2\rangle =\frac{\hbar^2}{8md^2} \enspace .$$
It looks like a "super-selection" rule for macroscopic objects. "God allowed them only in this state?"
On our current way toward micro-miniaturization we should get an answer to this question. Where is the threshold between micro and macro? One cannot exclude the possibility, that the macroscopic physics is not just the simple limit of the quantum theory. (In a mathematical sense it is surely not the ##\hbar \to 0 ## vanishing limit.)
I am aware that just touching this point is a heresy. In our human-centered world-image motivated by the exceptional scientific progresses of the last centuries we cheer to construct a unified picture of the world without loopholes and crevices. One looks even for "God's equations" forgetting that mathematics has to be connected to experiment by some interpretation and this is the most difficult one. However, all theories of physics of the past were just fragments of knowledge with loose connections between them. Of course, thermodynamics has to do with statistical mechanics, but this is a rather subtle one. Whether one may derive every-day quantum theory from the modern quantum field theory of elementary particles remains a question of belief. The unification of gravitation and quantum theory looks merely a dream. I am afraid, that we are mislead by our need for harmony and exaggerated belief in our brains.
Naturally, then the question arises: where is the threshold between the microscopic and the macroscopic worlds?
In the first half of the twentieth century the big successes of quantum theory were obtained mainly in the field of ("elementary") microscopic particles: electrons, atoms, nuclei. Thereafter, the most spectacular evolution took place in understanding the properties of condensed matter on the basis of many-body quantum mechanics. This step revolutionized our technology. Now is undergoing another exceptional development in producing ever smaller pieces of solid matter. We are approaching the microscopic world from above. Therefore the answer to above stated question may be very close.
In this context I would like to underline some simple but ignored aspects. Solid state theory is conceived as the quantum mechanical description of the stable state of an enormous number of interacting electrons and ions. All we treat is however only the relative motion, while the center of mass motion is considered irrelevant and left to the classical description. Moreover, we consider mostly infinite systems or closed ones in a periodical box.
A somewhat appeasing argument was given in almost all old handbooks about the stability of a wave package of the center of mass for macroscopic masses. Indeed if at an initial time t=0 one has a Gaussian one dimensional wave package
$$\psi(x,0)=\frac{1}{(2\pi )^{\frac{1}{4}}\sqrt{d}}e^{-\frac{x^2}{4d^2}}$$
then after a lapse of time t it decays according to the Schrödinger equation as
$$\psi(x,t)=\sqrt{\frac{\pi}{d^2+\frac{\imath\hbar t}{2m}}}e^{-\frac{ x^2}{4 (d^2 +\frac{\imath \hbar t}{2m})}} $$
This means that the average quadratic width grows as
$$\langle x^2\rangle_{t} =d^2\left[1+\frac{1}{2}\left(\frac{\hbar t}{md^2}\right)^2\right] \enspace . $$
One may see, that for ##m \to \infty## the width does not changes at all. However, the decisive parameter is not the mass m alone but the product ##md^2##. For any finite mass m at ##d \to 0 ## the wave package instantly decays.
Leaving aside such ideal limits, let us consider a numerical example for sake of illustration. A body of m=1g and an initial linear imprecision of its center of mass of 1 Angström after 200 million years will be smeared out as wide as to ##2\times 10^4## Angström. This is not yet frightening, but worth to think about.
Even more provoking is the question: Why all the macroscopic objects we know have a very precise center of mass position? Ultimately, why there are no plane waves of macroscopic objects, although energetically more favorable. In our example of the Gaussian package the average kinetic energy is
$$\frac{\hbar^2}{2m}\langle k^2\rangle =\frac{\hbar^2}{8md^2} \enspace .$$
It looks like a "super-selection" rule for macroscopic objects. "God allowed them only in this state?"
On our current way toward micro-miniaturization we should get an answer to this question. Where is the threshold between micro and macro? One cannot exclude the possibility, that the macroscopic physics is not just the simple limit of the quantum theory. (In a mathematical sense it is surely not the ##\hbar \to 0 ## vanishing limit.)
I am aware that just touching this point is a heresy. In our human-centered world-image motivated by the exceptional scientific progresses of the last centuries we cheer to construct a unified picture of the world without loopholes and crevices. One looks even for "God's equations" forgetting that mathematics has to be connected to experiment by some interpretation and this is the most difficult one. However, all theories of physics of the past were just fragments of knowledge with loose connections between them. Of course, thermodynamics has to do with statistical mechanics, but this is a rather subtle one. Whether one may derive every-day quantum theory from the modern quantum field theory of elementary particles remains a question of belief. The unification of gravitation and quantum theory looks merely a dream. I am afraid, that we are mislead by our need for harmony and exaggerated belief in our brains.
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