Description of isolated macroscopic systems in quantum mechanics

[mephistomunchen]

TL;DR Summary

Is the evolution of an isolated system always periodic in QM?

If we prepare a macroscopic system (something like Shrodinger's cat) in a known quantum-mechanical state and we let it evolve for a very long time completely isolated, for what I understand the position of all it's particles will become more and more spread in space.

But if the evolution of the system is unitary, shouldn't it end up returning to the initial state after a very long time?
If not, then how will the indeterminacy of the system's state will evolve? Will it reach a state that is a superposition of all possible states with even probabilities?

 

[Paul Colby]

This theorem might be of interest.

 

[PeroK]

Summary:: Is the evolution of an isolated system always periodic in QM?

If we prepare a macroscopic system (something like Shrodinger's cat) in a known quantum-mechanical state and we let it evolve for a very long time completely isolated, for what I understand the position of all it's particles will become more and more spread in space.

But if the evolution of the system is unitary, shouldn't it end up returning to the initial state after a very long time?
If not, then how will the indeterminacy of the system's state will evolve? Will it reach a state that is a superposition of all possible states with even probabilities?

A cat, for example, is not a collection of free particles. In order to model a cat using only QM first principles is practically impossible. Instead, you must use QM to produce a working theory of organic chemistry and so on. The cat then evolves according to that set of higher-level principles.

 

[mephistomunchen]

A cat, for example, is not a collection of free particles. In order to model a cat using only QM first principles is practically impossible. Instead, you must use QM to produce a working theory of organic chemistry and so on. The cat then evolves according to that set of higher-level principle

This theorem might be of interest.

This theorem might be of interest.

Thank you for the reference!

So, if the evolution of the state vector is deterministic and after a long enough time returns arbitrarily near to the original state, it means that the evolution has to be periodic. Am I right?

 

[mephistomunchen]

A cat, for example, is not a collection of free particles. In order to model a cat using only QM first principles is practically impossible. Instead, you must use QM to produce a working theory of organic chemistry and so on. The cat then evolves according to that set of higher-level principles.

I understand. Surely the cat is not the right example of a system to be modeled quantum-mechanically. But should a QM system necessarily be modeled a collection of free particles?
In classical mechanics you can choose the degrees of freedom of your system arbitrarily to describe a complex system (for example the engine of a car), and if you have a complete set of degree of freedom (variables), you can build an Hamiltonian that gives you the equation of motion.
I would guess that in QM it should be possible to do a similar thing. If you take a molecule (instead of a cat), for example, and you have a low enough temperature to limit it's degrees of freedom, you could build an hamiltonian that contains only the set of variables (and their derivatives) that actually can change at that temperature. And that would be a complete description of the system.
So, this would be some sort of "molecular machine" that moves according to QM instead of classical mechanics.
Is this right? Or has a QM system necessarily to be described as collection of free particles?

 

[PeroK]

I understand. Surely the cat is not the right example of a system to be modeled quantum-mechanically. But should a QM system necessarily be modeled a collection of free particles?
In classical mechanics you can choose the degrees of freedom of your system arbitrarily to describe a complex system (for example the engine of a car), and if you have a complete set of degree of freedom (variables), you can build an Hamiltonian that gives you the equation of motion.
I would guess that in QM it should be possible to do a similar thing. If you take a molecule (instead of a cat), for example, and you have a low enough temperature to limit it's degrees of freedom, you could build an hamiltonian that contains only the set of variables (and their derivatives) that actually can change at that temperature. And that would be a complete description of the system.
So, this would be some sort of "molecular machine" that moves according to QM instead of classical mechanics.
Is this right? Or has a QM system necessarily to be described as collection of free particles?

I think you need to be more specific about what you're asking. A molecule isn't a macroscopic system - and, in any case, may be in an eigenstate of the Hamiltonian and stable indefinitely.

In QM the position of a free particle becomes more spread out over time, which seemed to be the basis of your original analysis. That doesn't mean that the constituent particles in a molecule naturally drift apart.

 

[Paul Colby]

So, if the evolution of the state vector is deterministic and after a long enough time returns arbitrarily near to the original state, it means that the evolution has to be periodic. Am I right?

Depends on the system. Simple systems with few states will cycle on time scales that are meaningful. Complex systems, like cats, have enough states the time needed to cycle will exceed the expected lifetime of the universe. In these cases the answer is no.

 

[mephistomunchen]

Depends on the system. Simple systems with few states will cycle on time scales that are meaningful. Complex systems, like cats, have enough states the time needed to cycle will exceed the expected lifetime of the universe. In these cases the answer is no.

OK, I understand. So, it's really a question of complexity of the system. Thanks for the explanation.

 

[mephistomunchen]

I think you need to be more specific about what you're asking. A molecule isn't a macroscopic system - and, in any case, may be in an eigenstate of the Hamiltonian and stable indefinitely.

In QM the position of a free particle becomes more spread out over time, which seemed to be the basis of your original analysis. That doesn't mean that the constituent particles in a molecule naturally drift apart.

Well, I was trying to understand how is it possible to reconcile the fact that the system's status becomes more and more spread out over the time with the fact that the unitary evolution is really periodic. So, it should return to the non-spread status periodically. This seems to me somehow contradictory.

If the system is a set of free particles, then it's easy to understand that the positions become spread over all available space and will never return to the initial position in a reasonable amount of time.
But what if I have a system that has a lot of degrees of freedom (let's say 10 real variables that describe some angular positions, to fix the idea - plus related momentums), but it's not made of free particles?

The system that I have in mind, for example, is this: take a molecule with a complex structure (for example a protein), that at a certain temperature can move only with 10 degrees of freedom - that are angles between various triplets of atoms: so there will be some rigid parts of the molecule that don't move and some that can rotate relatively to each-other. The position of the molecule is not important, we can consider it fixed in space.

To make the experiment, let's say that we "take a picture" of the position of the status with a fast flash of light able to determine our 10 parameters with some small indeterminacy, leaving the 10 momentums with much more indeterminacy.
Then, the isolated system will start to evolve in time, and our 10 position parameters will start to spread, since there is a lot of indeterminacy in the 10 momentums.

But then what will happen to evolution of the 10 position parameters? will they remain spread forever, or will they return to the original status of determined position and spread momentums?

If the evolution in time is periodic, probably there will be a discrete (even if very big) set of eigenvalues of energy, and the status of the system can even be described as a superposition of probability amplitudes of the energy eigenstates.
So, in the end the thing that I would like to understand is: is it possible to describe any finite system in this way, however big, treating it basically as a superposition oscillators with relative amplitudes, or is there something fundamentally different in systems that are "big enough"?

Well, sorry for the very long explanation :-)

 

[PeroK]

Well, I was trying to understand how is it possible to reconcile the fact that the system's status becomes more and more spread out over the time with the fact that the unitary evolution is really periodic.

Why do you think that? The underlined phrase is just something you've invented, as far as I can tell.

 

[mephistomunchen]

Why do you think that? The underlined phrase is just something you've invented, as far as I can tell.

I mean: in the initial state the positions are sharply determined and the velocities are spread: that's how we prepared the system. Then, since the velocities are spread, this will cause the positions to become spread too. Isn't this true?

 

[PeroK]

I mean: in the initial state the positions are sharply determined and the velocities are spread: that's how we prepared the system. Then, since the velocities are spread, this will cause the positions to become spread too. Isn't this true?

It sounds like you are confusing the quantum state of a free particle with the quantum state of a bound system - you may not know where you'll find a molecule, say, but it'll still be the same molecule (bound state) wherever it turns up.

 

[mephistomunchen]

It sounds like you are confusing the quantum state of a free particle with the quantum state of a bound system - you may not know where you'll find a molecule, say, but it'll still be the same molecule (bound state) wherever it turns up.

Sorry, but I don't understand. Why the bound system should be different?
P.S. After all, any system can be considered to be bounded if you enclose it in a big-enough box, right?

 

[PeroK]

Sorry, but I don't understand. Why the bound system should be different?

There is nothing in QM that says that a molecule spreads out over time. The position you may find an isolated molecule will spread out, but the molecule is not (as I stated back in my first reply) a collection of unrelated free particles. It's a bound system, which behaves as a unit.

I think your model of a molecule is of a number of free particles that all go their separate ways according to the uncertainty principle. That model is clearly wrong and at odds with basic chemistry: atoms, molecules and larger strutures form and remain stable.

A basic understanding of bound quantum systems, starting with the hydrogen atom will tell you that. A hydrogen atom is not a free proton and a free electron that are free to dissociate from one another.

 

[mephistomunchen]

There is nothing in QM that says that a molecule spreads out over time. The position you may find an isolated molecule will spread out, but the molecule is not (as I stated back in my first reply) a collection of unrelated free particles. It's a bound system, which behaves as a unit.

I think your model of a molecule is of a number of free particles that all go their separate ways according to the uncertainty principle. That model is clearly wrong and at odds with basic chemistry: atoms, molecules and larger strutures form and remain stable.

A basic understanding of bound quantum systems, starting with the hydrogen atom will tell you that. A hydrogen atom is not a free proton and a free electron that are free to dissociate from one another.

Well, for the hydrogen atom the parameters can be three real numbers: distance of the electron from the proton and two angles that determine the direction: the usual polar reference system centred on the proton. For a complex molecule made of parts that can only rotate relative to each-other, the parameters would be angles that describe the relative positions of the various atoms.
However, thanks for your answer.

 

[PeroK]

Well, for the hydrogen atom the parameters can be three real numbers: distance of the electron from the proton and two angles that determine the direction: the usual polar reference system centred on the proton.

That's classical mechanics, not quantum mechanics. In QM, the state of the hydrogen atom is defined by four quantum numbers: Energy ($n$), total orbital angular momentum of the electron ($l$), the magnetic quantum number ($m$), and the spin of the electron ($s$). That's the basic model at least.

It's not defined by the position and classical orbit of the electron.

 

[PeterDonis]

if the evolution of the system is unitary, shouldn't it end up returning to the initial state after a very long time?

Why would it? Unitary evolution is not the same as being periodic.

 

[mephistomunchen]

That's classical mechanics, not quantum mechanics. In QM, the state of the hydrogen atom is defined by four quantum numbers: Energy ($n$), total orbital angular momentum of the electron ($l$), the magnetic quantum number ($m$), and the spin of the electron ($s$). That's the basic model at least.

It's not defined by the position and classical orbit of the electron.

In the Shrodinger equation for a particle in a spherical potential field in spherical coordinates, the state vector is a function of four variables: time, R, and two angles (sorry, I still didn't learn how to write formulae in this site).
For a complex molecule, the state vector would be a function of time and a bunch of angles, and the potential field would be a very complex function of these variables. The expressions for conjugated angular momentums can be derived from classical expressions following the standard quantization rules.

 

[PeterDonis]

This theorem might be of interest.

As the article notes, the application of this theorem to quantum systems is very limited. Basically, it would only apply to very simple systems that never interact with anything else, ever.

 

[PeterDonis]

if the evolution of the state vector is deterministic and after a long enough time returns arbitrarily near to the original state, it means that the evolution has to be periodic.

Strictly speaking, it only has to be "almost periodic"--it doesn't return to the exact same state.

However, as I noted in post #19 just now, the application of this theorem to quantum systems is very limited. It certainly doesn't apply to cats.

 

[PeterDonis]

as a QM system necessarily to be described as collection of free particles?

Certainly not. In fact, no quantum system of any interest is described as a collection of free particles. All of the interesting things in QM involve interactions.

You are correct that it is often possible to pick out a small number of degrees of freedom of a quantum system and construct a useful model that just uses those and ignores all the others. A simple example is the usual model of the hydrogen atom, which ignores the motion of the atom's center of mass and only treats the "internal" state of the electron in the Coulomb potential of the proton. For small quantum systems, such models will often predict characteristically "quantum" behavior (such as the discrete energy levels of the hydrogen atom). However, for a system as large as a cat, any model that picks out a usefully small number of degrees of freedom and ignores the others will end up being a classical model, not a quantum model.

 

[PeterDonis]

Why the bound system should be different?

Because it includes interactions. Interactions can drastically change the behavior of a quantum system as compared to a collection of free particles.

any system can be considered to be bounded if you enclose it in a big-enough box, right?

No. "Bounded" in space is not the same as "bound system". "Bound system" means a system which has significant interactions between its parts that change the way their wave functions evolve; unitary evolution of a bound system looks very different from unitary evolution of a system of free (non-interacting) particles.

 

[mephistomunchen]

Because it includes interactions. Interactions can drastically change the behavior of a quantum system as compared to a collection of free particles.No. "Bounded" in space is not the same as "bound system". "Bound system" means a system which has significant interactions between its parts that change the way their wave functions evolve; unitary evolution of a bound system looks very different from unitary evolution of a system of free (non-interacting) particles.

Yes, I understand.
However, I was referring to the fact that the effect of "spreading" of the values of a parameter that has a very sharp squared amplitude function is true even for bound systems: the squared amplitude of the conjugate momentum for that parameter would be a smooth function

 

[PeterDonis]

the effect of "spreading" of the values of a parameter that has a very sharp squared amplitude function is true even for bound systems

What parameter do you think has a "very sharp squared amplitude function" for a bound system? What if there isn't any such parameter for a bound system?

 

[PeroK]

Yes, I understand.
However, I was referring to the fact that the effect of "spreading" of the values of a parameter that has a very sharp squared amplitude function is true even for bound systems: the squared amplitude of the conjugate momentum for that parameter would be a smooth function

In the Shrodinger equation for a particle in a spherical potential field in spherical coordinates, the state vector is a function of four variables: time, R, and two angles (sorry, I still didn't learn how to write formulae in this site).
For a complex molecule, the state vector would be a function of time and a bunch of angles, and the potential field would be a very complex function of these variables. The expressions for conjugated angular momentums can be derived from classical expressions following the standard quantization rules.

I think you should consider the possibility that you have misunderstood or misinterpreted what you have learned about QM.

If you insist that you have understood the material fully (and that it is we who are wrong), then I don't believe the contradictions you are finding will disappear. In fact, I think you'll only find more examples where QM appears to contradict itself - as those contrardictions are a result of your misunderstanding of QM and not with QM itself.

 

[mephistomunchen]

What parameter do you think has a "very sharp squared amplitude function" for a bound system? What if there isn't any such parameter for a bound system?

I assumed it's possible to take a "picture" of the molecule at a certain time by interacting with short wavelength photons, so that the positions of it's atoms (and then the angles between them) are determined with good accuracy

 

[mephistomunchen]

I think you should consider the possibility that you have misunderstood or misinterpreted what you have learned about QM.

If you insist that you have understood the material fully (and that it is we who are wrong), then I don't believe the contradictions you are finding will disappear. In fact, I think you'll only find more examples where QM appears to contradict itself - as those contrardictions are a result of your misunderstanding of QM and not with QM itself.

Surely I have misunderstood something, I just don't know what :-). Anyway, thanks for all explanations!

 

[PeroK]

I assumed it's possible to take a "picture" of the molecule at a certain time by interacting with short wavelength photons, so that the positions of it's atoms (and then the angles between them) are determined with good accuracy

You can't have a picture of a molecule the way you can have a picture of a chessboard, with clearly defined images of every piece in its starting square. Measuring the position of particles in a molecule is about what you can infer from a few scattered photons.

For example, it would be very useful to know the ground state of ammonia. But, you can't just take a picture showing where everything is. Information about a molecule is inferred from theoretical models and things like emission and absorption spectra.

In any case, if you do pinpoint a particle in a molecule, then the system must be thrown into a superposition of all energy states - including unbound states. Effcetively you have a very significant external interaction that may have destroyed the molecule.

Instead, what QM is really telling us is that we can only infer a limited amount of information about a microscopic system - especially when it comes to the precise position of sub-atomic particles. The development of the model of the hydrogen atom was achieved not by taking accurate pictures of the atom and seeing where the electron is, but by studying the emission spectra under various conditions.

This is why it's important not to see microscopic systems as classical-type systems with a bit of uncertainty thrown in, but as fundamentally quantum, with essentially unknowable classical parameters, such as position of each particle. Instead, you have to describe them in terms of what can be measured - hence for hydrogen we have the four quantum numbers described above, rather than a classical set of coordinates for the electron.

 

[PeterDonis]

I assumed it's possible to take a "picture" of the molecule at a certain time by interacting with short wavelength photons

Doing so will destroy the molecule; photons of short enough wavelength to pin down the position of individual atoms inside the molecule are energetic enough to split the molecule apart. And at the level of atoms, photons of short enough wavelength to pin down the position of individual electrons inside an atom are energetic enough to ionize the atom. In both of these cases, you won't get any useful information from the photons.

 

[A. Neumaier]

Summary:: Is the evolution of an isolated system always periodic in QM?

If we prepare a macroscopic system (something like Shrodinger's cat) in a known quantum-mechanical state and we let it evolve for a very long time completely isolated,

Your assumption is meaningless. A macroscopic system is never even approximately isolated. You'd need to switch off the whole surrounding (particles, fields, etc.).

 

[mephistomunchen]

Your assumption is meaningless. A macroscopic system is never even approximately isolated. You'd need to switch off the whole surrounding (particles, fields, etc.).

I know that is not practically achievable, but I don't think it's meaningless. Sometimes idealized experiments are very useful to understand the essential points of a physical theory.

Shrodinger's cat experiment, after all, is not considered to be meaningless, right?

As I see it, there's something missing in QM related to the explanation of what happens in the measurement process.

I now the theory (collapse of the wave function when the system is measured and unitary evolution of isolated system), but I think there should be a way to describe the measurement process as a quantum interaction between the system and the observer.

I was trying to find such a description that makes sense, but all that I was able to find are explanations that this "doesn't make sense"...

 

[atyy]

It seems that a pure state may evolve into another pure state whose properties appear thermal (pure state quantum statistical mechanics.).

https://arxiv.org/abs/1503.07538
Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems
C. Gogolin, J. Eisert

 

[atyy]

But if the evolution of the system is unitary, shouldn't it end up returning to the initial state after a very long time?

In post #2, @Paul Colby gives a reference to a quantum Poincare recurrence theorem (which I wasn't aware of). In any case, if it is like the classical case, Poincare recurrence doesn't imply periodicity, since the time evolution into the state and out of the state may be different each time. Also, the Poincare recurrence time may be very long, greater than the lifetime of the universe, so that it is not relevant for physics.

 

[mephistomunchen]

In post #2, @Paul Colby gives a reference to a quantum Poincare recurrence theorem (which I wasn't aware of). In any case, if it is like the classical case, Poincare recurrence doesn't imply periodicity, since the time evolution into the state and out of the state may be different each time. Also, the Poincare recurrence time may be very long, greater than the lifetime of the universe, so that it is not relevant for physics.

Yes, I see.
My intuition would be that any system will have some spectrum of eigenvalues of the Hamiltonian.
If the spectrum is discrete, it means that the time evolution has to be the superposition of a finite set of periodic transformations. That doesn't mean that the system is periodic anyway, of course, because the ratio between the periods may be not rational. But anyway, it's a superposition of repeating behaviours (in some sense nothing really happens that will not be undone).

If instead the spectrum of eigenvalues is continuous, the behaviour of the system can be always changing, and even irreversible processes could be possible if the quantum system is seen form a macroscopic level.

Is this correct?

 

[mephistomunchen]

It seems that a pure state may evolve into another pure state whose properties appear thermal (pure state quantum statistical mechanics.).

https://arxiv.org/abs/1503.07538
Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems
C. Gogolin, J. Eisert

This seems very interesting. Only I am not sure if I'll be able to understand it
Thank you very much!