B Do systems continually evolve?

[rede96]

As part of another discussion in the relativity forum about the flow of time one of the points I raised was that afaik if I investigate any system over time I’d find that it was impossible for that system to remain unchanged. Certain properties of the system may not change over time. But if I look all elements of the system, down to the sub atomic level, form a QM point of view is it valid to say that a system must be continually evolving?

 

[hilbert2]

If you have a quantum system in an energy eigenstate, only the complex phase of the wave function changes while the position and momentum probability distributions keep constant. Actually, by simply shifting the potential energy of the system by a constant to make the total energy of that eigenstate zero, you can even make the complex phase to stay unchanged too. But if you have a macroscopic system that consists of a huge number of atoms/molecules, you're not going to be able to put it in an exact energy eigenstate (even though this can be approximated by superfluid helium at very low temperature, or some other Bose condensate).

 

[rede96]

(even though this can be approximated by superfluid helium at very low temperature, or some other Bose condensate).

The thing is I thought all quantum systems exhibit zero point energy, so their energy remains non-zero even when the temperature is absolute zero. For example helium remains fluid at absolute zero because its zero point energy is too great to allow it to crystallise. So in very layman’s terms there must be something moving around for it to be a fluid.

But if you have a macroscopic system that consists of a huge number of atoms/molecules, you're not going to be able to put it in an exact energy eigenstate

This was kind of my point. I was even wondering if there were some laws of physics that would prohibit a macro system from being in an exact energy eigenstate?

 

[hilbert2]

The state where only the zero point energy is left, is one of the eigenstates and is called the ground state. The electrons in such a system may have a nonzero expectation value of squared momentum $<\mathbf{p}^2 >$ but the wave function is not evolving.

It's exceedingly unlikely for a macroscopic system to be in an energy eigenstate just by chance, and it's not possible to measure the system's internal energy with perfect accuracy taking in account every single constituent particle and making it collapse to an eigenstate of the total Hamiltonian.

 

[PeterDonis]

I was even wondering if there were some laws of physics that would prohibit a macro system from being in an exact energy eigenstate?

An exact energy eigenstate of what Hamiltonian? What states are energy eigenstates depends on what the Hamiltonian is. And the Hamiltonian of any real system includes interactions with everything else. In the presence of those interactions, putting the entire system in an energy eigenstate would mean controlling the behavior of everything--in the limit, the entire universe. That's not possible.

Even in a case where the interactions are limited--for example, a system inside a carefully isolated experimental apparatus--it's impossible to precisely control every single interaction: a single stray photon inside the apparatus still constitutes an interaction and can't be controlled. That's why it's not possible in any practical sense to put a macroscopic system in an energy eigenstate: there are just too many interactions that can't be controlled. And even for microscopic systems, there is a limit on how long we can keep them in particular states, because we can't completely isolate them from interactions. The best we can do is to put them into a state that is an eigenstate of an idealized Hamiltonian that just describes the system and does not include any interactions with anything else, and make the interactions small enough that that idealized Hamiltonian is a good enough approximation to the real Hamiltonian for some appreciable length of time.

 

[Khashishi]

Unlike classical mechanics, in quantum mechanics, there's a difference between the quantum state of a system and what you can observe in the system. We can never observe the quantum state directly, so the nature of the quantum state depends on the particular interpretation of quantum mechanics. In particular, there is something called the Heisenberg picture, in which the quantum state does not evolve in time, and the Schrodinger picture, in which the quantum state does evolve in time. Neither is more correct than the other, since they give the same predictions for observations, and it is easy to transform between the two.

In your question, you are probably taking the concept of a classical state (i.e. position and momentum of particles) for granted, whereas quantum mechanics is considerably more weird. For example, particles do not simultaneously have an exact position and momentum.

 

[PeterDonis]

there is something called the Heisenberg picture, in which the quantum state does not evolve in time

That's only because the time evolution is in the operators instead of the state (whereas in the Schrodinger picture the time evolution is in the state instead of the operators). And there is also the interaction picture, which splits the difference, so to speak--part of the time evolution is in the state and part is in the operators.

In other words, if the time evolution is there, it's there; you can't make it go away by changing pictures.

 

[akvadrako]

In other words, if the time evolution is there, it's there; you can't make it go away by changing pictures.

If the state isn't evolving as in the Heisenberg picture, could you say the change in operators represents your evolving perspective? Which is why the OP's investigations will never find the system unchanged.

 

[PeroK]

If the state isn't evolving as in the Heisenberg picture, could you say the change in operators represents your evolving perspective? Which is why the OP's investigations will never find the system unchanged.

The Heisenberg and Schrödinger pictures describe the same physical system. Nothing of physical significance can vary between them - including the extent to which the system is evolving.

 

[akvadrako]

The Heisenberg and Schrödinger pictures describe the same physical system. Nothing of physical significance can vary between them - including the extent to which the system is evolving.

My point was that it's important to remember that the complete physical system isn't just it's state but that combined with how you choose to look at it. Just trying to clarify the distinction between the two pictures and how to understand what the operators represent.

When the OP asks about the system down at the subatomic level, it's not clear if he's just asking about the state of the system, which non-quantumly one might assume is the physical part.

 

[rede96]

My point was that it's important to remember that the complete physical system isn't just it's state but that combined with how you choose to look at it. Just trying to clarify the distinction between the two pictures and how to understand what the operators represent.

When the OP asks about the system down at the subatomic level, it's not clear if he's just asking about the state of the system, which non-quantumly one might assume is the physical part.

My original question was just trying to understand if there were any laws of physics which prevented a system from not evolving over time. Like all parts of the system were somehow frozen in time so to speak.

Classically (afaik) I didn’t think there was anything that prevented this. But I thought from a quantum mechanical point of view it would be impossible for a system to have zero energy or its energy to be in a permanently unchanging state, which I assumed it would have to be in for it never to evolve.

Intuitively I would have thought it was impossible for a system to never evolve in some way. But just wondered what laws of physics cover this.

Since making this post I’ve also started to wonder just what we can say about the way a system evolves without doing a measurement on it.

 

[Khashishi]

Since making this post I’ve also started to wonder just what we can say about the way a system evolves without doing a measurement on it.

Yes, it depends on exactly what you mean by "evolve".

Using quantum mechanics, you can calculate the probability of measuring a certain outcome if you had made a measurement, without actually making a measurement. This probability can change over time. I believe this can reasonably be called "evolving". On the other hand, a system can be in a state (called an energy eigenstate) where the probabilities of outcomes don't change over time. Nevertheless, if you were to make two measurements of this same state at two times, you could get different outcomes simply due to chance. In this case, this can reasonably be called "not evolving" over time, but changing as a result of your measurement. These changes don't happen at any point in time. Quantum collapse breaks the usual rules of causality so a time cannot be assigned.

Zero point energy has nothing to do with it, because something can have energy and not be "evolving" in the sense that probabilities of measurement outcomes aren't changing over time. Something can have momentum and not be going anywhere if it is bounded.

If you define "evolving" by changes in the quantum state, then you run into the ambiguity in post 6, but if you define "evolving" by changing probabilities of measurement outcomes, then there is no ambiguity. You can have systems that are not evolving provided they are insulated from outside interference. You might also be interested in the quantum Zeno effect, in which you use an outside interference to keep a system from evolving when the system would normally evolve.

 

[PeterDonis]

I thought from a quantum mechanical point of view it would be impossible for a system to have zero energy or its energy to be in a permanently unchanging state

As I hope the replies here have made clear to you, this is not impossible. It is just unlikely [edit: for macroscopic systems].

 

[Khashishi]

It's not too unlikely for a small enough system. An atom in its ground state is not evolving (by the probabilities of measurement outcomes definition).

When you get to macroscopic systems, anything with a nonzero temperature is going to be giving off heat.

 

[PeterDonis]

It's not too unlikely for a small enough system

Yes, agreed. I have edited my post to clarify.

 

[rede96]

As I hope the replies here have made clear to you, this is not impossible. It is just unlikely [edit: for macroscopic systems].

Yes thanks. Although I guess there is going to be some threshold as I increase the size of the system where it does become impossible?

 

[PeterDonis]

I guess there is going to be some threshold as I increase the size of the system where it does become impossible?

Not impossible in the absolute sense, but more and more unlikely as the size of the system goes up, to the point where it becomes unlikely enough to be impossible for all practical purposes.

 

[rede96]

Yes, it depends on exactly what you mean by "evolve".

I suppose a better way of phrasing it would be are all systems in a contact state of change?

Taking the case you mentioned where a system is in an energy eigenstate, the outcome of a measurement is still probabilistic. Which suggest that there is something in constant change otherwise the outcome of a measurement would be a certainty.

So the fact that I can make a measurement and then take another measurement at a later time and get a different result would suggest that the system is in a constant state of change.

Does that make sense?

 

[Khashishi]

Taking the case you mentioned where a system is in an energy eigenstate, the outcome of a measurement is still probabilistic. Which suggest that there is something in constant change otherwise the outcome of a measurement would be a certainty.

No, this means you don't understand quantum mechanics. You are assuming the world is deterministic, when quantum mechanics is not (at least in the Copenhagen interpretation). You can measure something in the same state and get different results.

 

[rede96]

No, this means you don't understand quantum mechanics. You are assuming the world is deterministic, when quantum mechanics is not (at least in the Copenhagen interpretation). You can measure something in the same state and get different results.

On the contrary, it’s because the outcome is probabilistic that I’m asking the question. Simple logic says that if I take two measurements of a system separated by some time interval and get two different results something about that system has changed.

So as I understand it, that doesn’t mean if I take two measurements and get the same results nothing has changed. I could get the same result just through chance. So I wasn’t trying to imply the world is deterministic at all.

 

[PeroK]

On the contrary, it’s because the outcome is probabilistic that I’m asking the question. Simple logic says that if I take two measurements of a system separated by some time interval and get two different results something about that system has changed.

That's not the context of two quantum measurements on the same state, which would be taken on identically prepared systems.

A measurement of position, say, would be taken at time $t_1$ on half the systems and give a range of values with an average ( expectation value) of $x_1$.

A measurement of position would be taken at time $t_2$ on the other half and return an average value of $x_2$.

It's the time evolution of this expected value that QM deals with when we say a state is stationary.

In a stationary state, a measurement of energy always returned the same value. A measurement of other observables returns only a constant expected value.

 

[Khashishi]

But you said the system is in a "constant state of change" [emphasis mine]. It changes as a result of you measuring it. That doesn't mean it's constantly changing when you aren't measuring it.

 

[PeroK]

PS a very crude analogy is if you have a whole bunch of coins. It doesn't matter when you toss them, the expected range is 50% heads. You don't know which ones will come up heads. The result of your coin tosses is not dependent on time.

If the coins were changing, then you would get more heads sometimes and more tails sometimes. Perhaps every hour might be a peak of heads etc. This would indicate that the coins are changing over time.

Of course, the difference is that you can toss the same coin repeatedly without affecting its state.

 

[rede96]

PS a very crude analogy is if you have a whole bunch of coins. It doesn't matter when you toss them, the expected range is 50% heads. You don't know which ones will come up heads. The result of your coin tosses is not dependent on time.

If the coins were changing, then you would get more heads sometimes and more tails sometimes. Perhaps every hour might be a peak of heads etc. This would indicate that the coins are changing over time.

Of course, the difference is that you can toss the same coin repeatedly without affecting its state.

Nice analogy, thanks. So using the stack of coins to represent the system, isn't true to say that "heads or tails" is just one property of that system that can be measured and that doesn't mean all properties of the system, such as temperature of the coins, will remain unchanged?

So when I imagine a system to be in a constant state of change I am imagining that it can't be possible for all properties of that system to remain unchanged over time.

 

[Khashishi]

For a macroscopic system, you would be right. But a tiny system like an atom doesn't have a lot of properties. If it is in the ground state, it doesn't have a temperature. It can be fully specified by a few quantum numbers.

 

[rede96]

For a macroscopic system, you would be right. But a tiny system like an atom doesn't have a lot of properties. If it is in the ground state, it doesn't have a temperature. It can be fully specified by a few quantum numbers.

Although I appreciate this line of discussion is probably just academic at best, where is the boundary between macro and micro systems? How small do I need to go before it can be said that all properties of a system can remain unchanged over time?

Even if I think of a Hydrogen atom, the electron and proton have properties like spin. Is it correct to think of the spin state as never changing? Or the momentum and position of the electron never changing?

From what little I understand about QM I know not to think of something in a superposition as having a value until it's measured but as I mentioned before, just the fact that the outcome of a measurement on a particular property of a system is probabilistic must say that property is not in an unchanging state?

 

[Khashishi]

Although I appreciate this line of discussion is probably just academic at best, where is the boundary between macro and micro systems? How small do I need to go before it can be said that all properties of a system can remain unchanged over time?

There's not a hard limit. It depends on how well you can isolate your system. Maybe a few atoms, or more if you can cool your system to nanokelvins.

Even if I think of a Hydrogen atom, the electron and proton have properties like spin.

Let's start with the spin of a single electron in an external magnetic field, for simplicity. If the external magnetic field stays constant, then the spin of the electron will have two energy eigenstates. One is aligned with the magnetic field and one is in the opposite direction. If the spin is in one of these states, it will stay constant in time, provided you don't disturb it. The spin of the electron can also be not aligned with the magnetic field, but then it is not in an energy eigenstate, and it will change in time.

You can choose to measure the spin along any direction you want. If you measure it along the direction of the external magnetic field, then you will force the spin to change to one of the energy eigenvalues, pointed either parallel or antiparallel to the field, and it will stay fixed there until something happens.
Or you can measure it along some other direction, in which case it will be forced away from an energy eigenstate, and it will continually rotate in time until something happens.

An atom is somewhat more complicated because in an energy eigenstate, the spin of the electron is coupled with the orbit of the electron, so it does not have a fixed value.

 

[PeterDonis]

Simple logic says that if I take two measurements of a system separated by some time interval and get two different results something about that system has changed.

And the response to this simple logic, for QM, is that what changed about the system was you measuring it. In classical physics, you can in principle take a system in any state, and make any measurement you like on it, and the system's state will be unaffected by the measurement; so it makes sense to view measurements at different times, that give different results, as reflecting a change in the system that would have happened whether you measured it or not.

But in QM, you cannot take a system in any state and make any measurement you like on it without affecting the state--more precisely, without affecting the state in a way that it would not have been affected if you had not made the measurement. Measurement in QM is inherently an interaction between the system and the measuring device, whose effect on the system cannot be eliminated, even in principle. So when you make successive measurements and they give different results, you can no longer argue that the change was due to the system and would have happened whether you measured it or not; your measurement in itself changes the time evolution of the system, even to the extent of causing a time evolution that would not otherwise have occurred.

 

[rede96]

And the response to this simple logic, for QM, is that what changed about the system was you measuring it.

Thanks for the clear explanation. So what about an atom in decay for example? There doesn’t need to be a direct measurement on the atom, we can just detect the result of the decay when the atom emits a neutrino for example. So in that case the measurement doesn’t effect the system as the atom decays before the measurement takes place.

And so in that special case I could think of the system as changing changing even though I’m not taking a measurement directly?

 

[PeterDonis]

And so in that special case I could think of the system as changing changing even though I’m not taking a measurement directly?

Um, yes. So what? Nobody has been arguing that it is impossible for a system to change if you're not measuring it. That's not what this discussion is about. This discussion is about whether it's possible for a system not to change. Your example isn't relevant to that at all.

 

[rede96]

Um, yes. So what? Nobody has been arguing that it is impossible for a system to change if you're not measuring it.

I wasn’t suggesting anyone was. I was just seeking clarification hence the question mark at the end of the sentence.