Why no plane waves of macroscopic bodies? The micro-macro threshold...

In summary, the answer to the question where is the threshold between micro and macro is that it may not be a simple limit of the quantum theory.
  • #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.
 
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  • #2
Laci said:
A somewhat appeasing argument was given in almost all old handbooks
Can you give some specific references?

Also, you mention "old" books, but a lot of work has been done in this area since QM was first formulated. Have you looked at any modern textbooks?
 
  • #3
Well, I am 84 and my first QM books were the russian ones. It is not an easy task to retrive them. Usually there is just a sentence about it in the frame of Ehrenfest's theorem like in the book of Schiff. However, You are right, I should look into some modern textbooks. Perhaps You are aware of some new argument? On the other hand, I was active also in the area of quantum dots, wires, wells and saw no thoughts at all about this aspect. Therefore I have some doubt about "work in this direction" i.e. experiments. On the other hand, theoretical treatments should be based on something else as QM and its Copenhagen interpretration and so far there is no accepted alternative. For the while I see it merely as a philosophical question.
 
  • #4
Laci said:
Perhaps You are aware of some new argument?
Not so much any new argument as a much more detailed understanding of what is actually going on in macroscopic objects, or more generally in systems with very large numbers of degrees of freedom that cannot be individually tracked. For example, decoherence theory is quite a few decades old now and contains a huge body of work in this area.

There is also, as you note, a growing body of work exploring the regime in between what is traditionally considered "microscopic" and what is traditionally considered "macroscopic". Double slit experiments showing interference patterns have been done with buckyballs (carbon-60), and evidence of Bose-Einstein condensates has been found in experiments involving, IIRC, about ##10^{12}## atoms.
 
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  • #5
Laci said:
...
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.
Albert Einstein made significant progress in physics examining the 'loophole or crevice' of the precession of the orbit of Mercury among other phenomena.
Laci said:
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. ...
Then 'be of good cheer'. Anecdotally, August Kekulé struggled to understand the arrangement of carbon atoms in benzene before dreaming of carbon atoms 'holding hands and dancing' in a ring. Perhaps useful unification theories will emerge from similar dreams.

An advantage of scientific inquiry over theological and philosophical speculation, forgive my naiveté, understanding science requires mathematics without the burden of belief systems.

Laci said:
Naturally, then the question arises: where is the threshold between the microscopic and the macroscopic worlds?

EM students frequently asked me: where is the dividing line between infrared and visible light, or visible and ultraviolet light?

After repeating canonical measurements in millimeters and Angstrom units, my actual answer described the electromagnetic spectrum as a continuum; the notion of visible light being an anthropocentric construct of our limited senses. Perhaps this analogy has some application to your broader question.
 
  • #6
I must confess, that I do not have any knowledge of the decoherence theories.

The buckyball experiment is indeed very interesting showing that its c.m. behaves according to QM. I would then expect no individual bucky-ball should be precisely localised. What says electron microscopy about?
Nevertheless remains the problem about non-existence of smeared out true macroscopic objects!
 
  • #7
Klystron said:
Albert Einstein made significant progress in physics examining the 'loophole or crevice' of the precession of the orbit of Mercury among other phenomena.

Then 'be of good cheer'. Anecdotally, August Kekulé struggled to understand the arrangement of carbon atoms in benzene before dreaming of carbon atoms 'holding hands and dancing' in a ring. Perhaps useful unification theories will emerge from similar dreams.

An advantage of scientific inquiry over theological and philosophical speculation, forgive my naiveté, understanding science requires mathematics without the burden of belief systems.
EM students frequently asked me: where is the dividing line between infrared and visible light, or visible and ultraviolet light?

After repeating canonical measurements in millimeters and Angstrom units, my actual answer described the electromagnetic spectrum as a continuum; the notion of visible light being an anthropocentric construct of our limited senses. Perhaps this analogy has some application to your broader question.
Einstein's case is typical. The Lorentz invariance of the Maxwell equations was alread well-known and Einstein added only their physical interpretation.
 
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  • #8
PeterDonis said:
Not so much any new argument as a much more detailed understanding of what is actually going on in macroscopic objects, or more generally in systems with very large numbers of degrees of freedom that cannot be individually tracked. For example, decoherence theory is quite a few decades old now and contains a huge body of work in this area.

There is also, as you note, a growing body of work exploring the regime in between what is traditionally considered "microscopic" and what is traditionally considered "macroscopic". Double slit experiments showing interference patterns have been done with buckyballs (carbon-60), and evidence of Bose-Einstein condensates has been found in experiments involving, IIRC, about ##10^{12}## atoms.
PeterDonis said:
Can you give some specific references?

Also, you mention "old" books, but a lot of work has been done in this area since QM was first formulated. Have you looked at any modern textbooks?
 
  • #9
PeterDonis referred to: "decoherence theory"
I am afraid I misinterpreted this line of resarch. In now suppose, you spoke about the behaviour of subsystems of large mechanical systems. In my club it was quoted as theory of open systems. In the classical case it has to explain irreversibility, while in the quantum mechanics it has to explain irreversibility, as well as the loose of coherence. This is indeed an aspect we know from our everyday macroscopic experience and had to be understood in the frame of mechnics (classical or quantum mechanical). The literature is enormous, but I would like to quote two instructive exact results within solvable models. In the quantum mechanical respectively classical frame :
E.B. Davies, Quantum theory of open systems. Academic Press, London (1967) pp.1-171
L.A. Bányai, Dissipation and irreversibility in a solvable classical open system. Electron in a dc field interacting with LO/LA phonons, Eur. Phys. J.B. 92:6, pp.1-8 (2019)

Indeed this helps to understand an aspect of our everyday macroscopic experiment, but not the one I am refering to here. I reformulate it as: "Why the center of mass of macroscopic bodies is always well defined, contrary to the possibilities offered by Quantum Mechanics?"
 
  • #10
Laci said:
Why the center of mass of macroscopic bodies is always well defined
Why do you think it is "well-defined"? What experimental facts tell you this? Just saying "well, macroscopic bodies follow classical trajectories" is not sufficient, because we don't actually know that. All of our actual measurements have error bars of finite width. Are those error bars smaller than the error bars quantum uncertainty predicts?
 
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  • #11
PeterDonis said:
Why do you think it is "well-defined"? What experimental facts tell you this? Just saying "well, macroscopic bodies follow classical trajectories" is not sufficient, because we don't actually know that. All of our actual measurements have error bars of finite width. Are those error bars smaller than the error bars quantum uncertainty predicts?
Not at all, but energetically are more favourable largely smeared center of masses, even plane waves. Who forbids them? By the way, if there would be no such a "rule" we could not do anything with the objects around us or with each other. Think about a glass of watter as a cloud spread in the room due to the smeared state of its c.m. Could You grasp it? QM allows and prefers this situation.
 
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  • #12
Laci said:
energetically are more favourable largely smeared center of masses, even plane waves
Why?
 
  • #13
Because have lower average energy and nothing forbids them.
 
  • #14
Now I must stop the chat, since here in central Europe it is night 2:23 am. Good night!
 
  • #15
Laci said:
Because have lower average energy
Please show your work.
 
  • #16
The average kinetic energy of the Gausssian package is inversely proportional to the quadrat of its width (see my note). Therefore it is lower for wider packets.
Did I understood properly Your reference to my "work" ? Or You meant something else.
 
  • #17
There is no "size limit" above which QT were all of a sudden invalid and classical physics would take over. With enough accuracy in preparation and observation you can observe "quantum behavior" of large macroscopic objects. E.g., the precision of the LIGO gravitational-wave detectors is such that the quantum fluctuations of the mirrors (objects with mass in the order of 10s of kg!).
 
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  • #18
This is very interesting, however still does not explain my main question about the center of mass states of macroscopic objects (why there are no smeared out states?) , nor does it clarify the ambigous relation between quantum and classical mechanics. One cannot formulate QM without using classical physics to analyze measurments. Or we should admit that also the c.m. of macroscopic bodies obey fully QM, but "god" created all of them in states of very high localization.
 
  • #19
Laci said:
The average kinetic energy of the Gausssian package
Why are you limiting yourself to Gaussians? What makes you think a Gaussian is the right kind of wave function for a macroscopic object?

Or even a microscopic object, for that matter? Consider a hydrogen atom. It is a stable quantum system with a stable "width". Is it described by a Gaussian? Does QM predict that the atom gets wider with time?
 
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  • #20
Laci said:
The average kinetic energy
Kinetic energy is not the only energy involved. A macroscopic object is a bound state of a huge number of individual quantum systems (something like ##10^{25}## or more of them). What happens to the potential energy of that binding if the "width" you are using in your equations increases?
 
  • #21
PeterDonis said:
Kinetic energy is not the only energy involved. A macroscopic object is a bound state of a huge number of individual quantum systems (something like ##10^{25}## or more of them). What happens to the potential energy of that binding if the "width" you are using in your equations increases?
As You know, the motion of the c.m. is separable and has no influence on the energy of the relative motion. It adds to it. The same is true about an atom. You will never find an atom in rest, because its c.m. energy would be infinite. The example with the gaussian is just an illustration. Other wave packets would decay even faster.
One has not only wave packets of an atom, but even plane waves, as in difraction experiments and these are the only stationary states for the c.m. motion!
Atomic, molecular or condensed matter physics studies mainly the relative motion of its components. Of course, in the presence of an external field (potential) the sytem is not translational invariant and the two kind of motions are coupled. Otherwise not.
 
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  • #22
Laci said:
Not at all, but energetically are more favourable largely smeared center of masses, even plane waves. Who forbids them? By the way, if there would be no such a "rule" we could not do anything with the objects around us or with each other. Think about a glass of watter as a cloud spread in the room due to the smeared state of its c.m. Could You grasp it? QM allows and prefers this situation.
Macroscopic objects are smeared out. But not relative to their size. If you apply the Heisenberg uncertainty principle, you find that numerically the amount of smearing out is undetectably small. Your supposition that uncertainty is proportional to size is false; and, this assumption is not supported by QM at all.
 
  • #23
If any object would be in rest its average c.m. kinetic energy would be infinite. This is the consequence of the Heisenberg uncertainty relation. It has nothing to do with its mass or with his internal structure (relative motion). The only stationary states of the c.m. motion of any object in the absence of an external potential are plane waves.
Indeed the smearing out of the c.m. of macroscopic objects is "undetectable small" . This is the heart of the problem! Why is this anormal situation? All energetically preferred states are those with smeared out c.m. and the lowest stationary state would be a plane wave of momentum 0. This we see by all atomic objects, but not by the macroscopic ones. Have You an explanation for this anomaly?
If You agree, that in a translational invariant hamiltonian the c.m. motion is separable and I hope You are aware of it, then there is nothing more to say.
 
  • #24
Laci said:
Indeed the smearing out of the c.m. of macroscopic objects is "undetectable small" . This is the heart of the problem! Why is this anormal situation?
That's what QM predicts. Learm some QM. Do the maths. Stop attacking a straw man that consists of nothing but your misconceptions about QM.
 
  • #25
I am sorry that You repeatedly misunderstand my statements.
Elementary QM says, that in a translationary invariant system the motion of the c.m. is separable. This is true also in classical mechanics. Therefore, the total energy of the system consists of the sum of the two energies (c.m and relative motion) . The only stationary states of the free motion of the c.m. are plane waves, not wave packets. Among the non-stationary states (wave packets) the average (conserved) energy is lower for more smeared out ones. This follows from the Heisenberg uncertiainity and the example with the Gaussian packet (having the ideal uncertainity) just illustrate this.
If You recognize this elementary knowldege, we my continue to discuss the matter. Otherwise it is meaningless.
 
  • #26
Laci said:
Otherwise it is meaningless.
I agree, your posts are meaningless. Do the maths, then come back.
 
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  • #27
good by
 
  • #28
Laci said:
The only stationary states of the free motion of the c.m. are plane waves, not wave packets. Among the non-stationary states (wave packets) the average (conserved) energy is lower for more smeared out ones.
Neither of these statements is exactly true.
 
  • #29
Laci said:
good by
Before running off, you should consider taking a look at this recent article: https://arxiv.org/abs/2209.07318.

Newton's equations from quantum mechanics for macroscopic bodies in the vacuum​

Kenichi Konishi

The author captures, I think, the essence of your question with this statement:

"The following discussions...shed light to the poorly-defined but familiar "puzzle" why microscopic particles are "usually" in momentum eigenstates, whereas macroscopic ones are "always" in a position eigenstate."​

then goes on to explain why one can safely ignore the uncertainty principle for the motion of the center-of-mass of macroscopic bodies:

"What is most remarkable is the fact that a macroscopic body, e.g., made of ##N=10^{23}## or ##N=10^{51}## atoms and molecules, satisfies the same Heisenberg's uncertainty relations for its CM position and momentum, as that for a single constituent atom. Heisenberg's uncertainties - the effects of quantum fluctuations - do not pile up.

But this means that for an experimental determination of the position and momentum of the CM of a macroscopic body, Heisenberg's uncertainty constraints may be regarded [as] unimportant. To have some concrete ideas, one may take for instance, somewhat arbitrarily, the experimental uncertainties ##\Delta X,\Delta P## of the order of$$\Delta P\sim10^{-2}(\mathrm{~g})\times\frac{10^{-4}\mathrm{~cm}}{\mathrm{sec}}\sim10^{-6}\frac{\mathrm{g}\mathrm{cm}}{\mathrm{sec}};\quad\Delta X\sim10^{-4}\mathrm{~cm}\tag{2.15}$$Their product is$$\Delta P\cdot\Delta X\sim10^{-10}\text{erg sec}\tag{2.16}$$many orders of magnitude larger than the quantum mechanical lower bound, ##\hbar\sim1.05\times 10^{-27}## erg sec. Thus even allowing for much better precision by several orders of magnitudes than the rule-of-thumb (2.15), both for the position and for the momentum, one can still regard the Heisenberg uncertainty relations [as] insignificant.​
To sum up, one can effectively take$$\Delta P\approx0,\quad\Delta X\approx0\tag{2.17}$$for macroscopic bodies. In other words, the position and momentum of (the CM of) a macroscopic body can be measured and determined simultaneously with an ”arbitrary” precision, at macroscopic scales. These data serve as the initial condition for Newton's equation of motion."​

And regarding the spreading of the wave-function of macroscopic free particles:

"Here the essential factor is the mass. While for a free electron with mass ##\sim10^{-27}\mathrm{~g}##, the time needed for doubling its wave packet size (starting from a ##1\mu\mathrm{m}##) is ##10^{-8}## sec, the corresponding time for a macroscopic body of ##1\mathrm{~g}##, with the initial ##\mathrm{CM}## wave packet of the same size ##1\mu\mathrm{m}##, exceeds the age of the universe. This means that once its position and momentum are experimentally measured with "arbitrary" precision, their uncertainties do not grow in time: a macroscopic body follows a classical (well-defined) path: a trajectory."​
 
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  • #30
renormalize said:
Before running off, you should consider taking a look at this recent article: https://arxiv.org/abs/2209.07318.

Newton's equations from quantum mechanics for macroscopic bodies in the vacuum​

Kenichi Konishi

The author captures, I think, the essence of your question with this statement:

"The following discussions...shed light to the poorly-defined but familiar "puzzle" why microscopic particles are "usually" in momentum eigenstates, whereas macroscopic ones are "always" in a position eigenstate."​

then goes on to explain why one can safely ignore the uncertainty principle for the motion of the center-of-mass of macroscopic bodies:

"What is most remarkable is the fact that a macroscopic body, e.g., made of ##N=10^{23}## or ##N=10^{51}## atoms and molecules, satisfies the same Heisenberg's uncertainty relations for its CM position and momentum, as that for a single constituent atom. Heisenberg's uncertainties - the effects of quantum fluctuations - do not pile up.

But this means that for an experimental determination of the position and momentum of the CM of a macroscopic body, Heisenberg's uncertainty constraints may be regarded [as] unimportant. To have some concrete ideas, one may take for instance, somewhat arbitrarily, the experimental uncertainties ##\Delta X,\Delta P## of the order of$$\Delta P\sim10^{-2}(\mathrm{~g})\times\frac{10^{-4}\mathrm{~cm}}{\mathrm{sec}}\sim10^{-6}\frac{\mathrm{g}\mathrm{cm}}{\mathrm{sec}};\quad\Delta X\sim10^{-4}\mathrm{~cm}\tag{2.15}$$Their product is$$\Delta P\cdot\Delta X\sim10^{-10}\text{erg sec}\tag{2.16}$$many orders of magnitude larger than the quantum mechanical lower bound, ##\hbar\sim1.05\times 10^{-27}## erg sec. Thus even allowing for much better precision by several orders of magnitudes than the rule-of-thumb (2.15), both for the position and for the momentum, one can still regard the Heisenberg uncertainty relations [as] insignificant.​
To sum up, one can effectively take$$\Delta P\approx0,\quad\Delta X\approx0\tag{2.17}$$for macroscopic bodies. In other words, the position and momentum of (the CM of) a macroscopic body can be measured and determined simultaneously with an ”arbitrary” precision, at macroscopic scales. These data serve as the initial condition for Newton's equation of motion."​

And regarding the spreading of the wave-function of macroscopic free particles:

"Here the essential factor is the mass. While for a free electron with mass ##\sim10^{-27}\mathrm{~g}##, the time needed for doubling its wave packet size (starting from a ##1\mu\mathrm{m}##) is ##10^{-8}## sec, the corresponding time for a macroscopic body of ##1\mathrm{~g}##, with the initial ##\mathrm{CM}## wave packet of the same size ##1\mu\mathrm{m}##, exceeds the age of the universe. This means that once its position and momentum are experimentally measured with "arbitrary" precision, their uncertainties do not grow in time: a macroscopic body follows a classical (well-defined) path: a trajectory."​
You repeat the standard argument related to Ehrenfest theorem, that for very large masses a narrow wave packet remains stable against astronomical times. I did not conradicted this at all. On the contrary I explicitely mentioned it. If it is very narrow it remains for all means narrow. My question is: Why do not exist strongly smeared out wave packets of macroscopical centers of mass, since energetically they should be preferred? Quantum mechanics do not exclude at all such states and still they were never observed. It means such states cannot be created!? Why?
 
  • #31
Mentor Note -- Changed thread prefix "A" Advanced (graduate school level) to "I" Intermediate (undergraduate level)
 
  • #32
I fully agree with this decision. Hopefully, young students still know the basics of Quantum Mechanics.
 
  • #33
Laci said:
the motion of the c.m. is separable and has no influence on the energy of the relative motion
The Hamiltonian is separable. But that does not mean the state is separable; it doesn't mean you can just ignore the "internal" part of the Hamiltonian when looking at the stationary states of the system. A hydrogen atom's stationary states are not plane waves, even though its Hamiltonian is separable; the separability of the Hamiltonian is most commonly made use of to set the energy of the center of mass motion to zero, i.e., to justify working in the center of mass frame. But even if you don't work in the center of mass frame, the atom's stationary states are not plane waves.
 
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  • #34
I am sorry, You are wrong. You should be more careful with mathematical formulations. " A hydrogen atom's states are not (or are not) plane waves." is not a meaningful statement.

If a Hamiltonian is separable, then its eigenstates are products of the eigenfunctions of the separated degrees of freedom. This is trivial and You must have learned it already on the example of the motion in a fixed Coulomb potential. Here, due to the spherical symmetry of the Hamiltonian one may separate the radial and the angular motions and the eigenfunctions are ##R_{nl}(r)Y_{lm}(\theta, \phi)## , while the energy in this peculiar case has an ##n^2## degeneracy . If the potential is not Coulombian , the degeneracy in ##l## may be lifted.

In the case of a translational invariant Hamiltonian, like the one of Coulomb interacting charged particles in the absence of an external potential, one may separate the c.m. motion from the relative motion and the stationary states are products of the plane waves for the c.m. motion and the stationary states of the relative motion. The relative motion may be bounded i.e. a true normed eigenstate. The energy of the stationary states are the sum of these stationary eigenenergies. (Plane waves are however not eigenstates, since they are not normed.)

In the case of the hydrogen atom (bound state of electron and proton) the stationary states are the product of the above discussed Eigenfunctions depending on the relative coordinates (##r, \theta, \phi ##) of the electron to the proton (however with the reduced mass ##\frac{1}{\mu}=\frac{1}{m_e}+\frac{1}{m_p}## ) multiplied with the plane wave of the c.m. cooordinate satisfying the Schrödinger equation with the total mass ##M=m_e+m_p##. The total energy is the sum of the kinetic energy of the c.m. plus the eigenenergies of the relative motion. All this You may find in any textbook on QM or You may perform the trivial calculus.
Of course , You may construct normed states (wave packets) of the c.m. motion out of plane waves, but they are not stationary. Any of these atomic wave packets have still a conserved average energy. This energy decreases with the initial width of the packet.
Therefore ona may find hydrogen atoms in any of these states with stationary plane waves or broadening wave packets of the c.m. The motion of the c.m. does not affect the internal (relative) motion of the electron and proton.
After You swallowed the above explanations we may continue the discussion.
 
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  • #35
Laci said:
I fully agree with this decision. Hopefully, young students still know the basics of Quantum Mechanics.

The prefix is about your level of knowledge, so that everyone can adjust their responses to that.
 
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