# Unitary evolution and environment

1. Dec 20, 2015

### zonde

I have question about unitary evolution. As I understand it is assumed that unitary evolution concerns only quantum system and there are no changes in environment.
First, I would like to ask if my understanding is correct?
Second, from where comes this assumption? Does it comes with Schrodinger equation (postulate of QM)?

2. Dec 20, 2015

### Staff: Mentor

Wigner's Theorem - see attached.

It only applies to isolated systems. Entangled systems eg systems entangled with the environment, are generally not in a pure state (they are in a mixed state) and unitary evolution may not apply. This is the origin of decoherence as the explanation for apparent collapse which of course most definitely is not unitary. The system and environment proceed by unitary evolution but entangled subsystems may not.

Thanks
Bill

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3. Dec 20, 2015

### gre_abandon

The system we dealt with in quantum mechanics is usually a closed system. Sometimes we also study open quantum systems but still the system together with the surroundings constitutes a larger CLOSED system. And Schrodinger equation only applies to a closed system. You are correct on this point :)

We like unitary time evolution because it conserves probability. If the time evolution is not unitary, there might be situation in which a particle gradually disappears as time proceeds.

4. Dec 21, 2015

### zonde

Thanks bhobba and gre_abandon,
So it's both, Wigner's Theorem and Schrodinger equation that applies only to closed systems?

There can be harmonic oscillator and there can be driven&damped oscillator that looks the same way as harmonic oscillator. So we assume that quantum system is like harmonic oscillator (closed system) and we can apply Schrodinger equation.
But if our assumption does not hold and system only looks like the type of harmonic oscillator but actually is the type of driven&damped oscillator (open system). How we can test this assumption? What we get if we apply Schrodinger equation to the second type system?

5. Dec 21, 2015

### Staff: Mentor

Wigner's theorem and Schroedinger's equation applies to systems not entangled with other systems.

I don't understand what you mean by harmonic oscillator analogy. Its to do with entanglement.

Thanks
Bill

6. Dec 21, 2015

### zonde

And if we out of our ignorance apply Schroedinger's equation and Wigner's theorem to the system that is entangled with other system(s)? How we would find that out?

7. Dec 21, 2015

### Staff: Mentor

You cant. When a system is entangled with another system it isnt in a pure state and hence has no wavefunction.

Thanks
Bill

8. Dec 21, 2015

### zonde

So you say that if we will check that the system is in pure state (check that specific filter does not reduce number of particles) we will know it is not entangled?

9. Dec 21, 2015

### StevieTNZ

I may be wrong, but I think what Bill is saying is that if you only take the one system into consideration, you won't find it is in a pure state (when you use a density matrix instead?). However if you take the system + whatever it is entangled with, then the wave-function applying to both systems is in a pure state.

10. Dec 21, 2015

### zonde

Yes, you might be right. I reread my posts and I might have not made it clear enough that my question is about correspondence between QM and reality and not about QM itself.

11. Dec 21, 2015

### Staff: Mentor

I have zero idea what you mean by that.

Consider the following entangled state between system 1 and system 2 - c1 |a>|b> + c2|b>|a> I often speak about in discussing entanglement. Neither system 1 or system 2 are in a pure state so Schroedinger's equation cant apply.

If it's not clear now then I will have to leave it to someone else to explain because that's as good as I can do.

Thanks
Bill

Last edited: Dec 21, 2015
12. Dec 21, 2015

### zonde

It's crystal clear, thank you.
Now that we are on the same page can I go one step further?

Just by logic I can rewrite your statement:
as
"When a system is in pure state it isn't entangled with another system(s)."
Do you agree?

13. Dec 22, 2015

### f95toli

We do this all the time. The way you (usually) find out in an experiment is that your system will initially behave -approximately- the way you would expect it to from solving the SE, but at longer timescales it it will start to deviate until it eventually it behaves like a "classical" system. Hence, if you want to study quantum effects the key is to design your experiment in such a way that your system is as isolated from the environment as possible thereby reducing the amount of decoherence. For many systems we can quite accurately model this because the dominant cause coupling to the environment is easy to identify.

Note that nothing I that I have written here disagrees with what has been said above. However, it is all to easy to get the idea from reading popsci that this is some sort of insurmountable problem when it is -from a practical point of view- no more an issue than in other areas of physics. There is no such thing as a truly closed system, but that does not stop us from (nearly) always making that approximation in e.g. thermodynamics of even Newtonian mechanics.

14. Dec 22, 2015

### gre_abandon

Yes, these theorems both only apply to closed system. Speaking of open quantum system, we usually consider the system we are interested in and the surroundings together as a whole. The time evolution of the whole system, which is unitary, can be written as $$\mathcal{H^{sys}}\otimes \mathcal{H^{env}}$$. Quantum harmonic oscillator is just a model that is extremely useful in physics and it is hard to say that something IS a quantum oscillator. What we can do is just check the theoretical prediction against the experimental results to see if a model works.

15. Dec 22, 2015

### zonde

I totally agree with that. This is actually the basis for my question in post #4. But I am trying to think in what way model could fail if particular assumption would not hold. And the reason is that the model might not be quite consistent so that the failure of model might hide in it's inconsistencies.
Thanks for your explanation. And I think I can now formulate my doubt more clearly. If assumption about system being closed fails it would be more like $H^{sys}$ undergoes unitary evolution but $H^{env}$ undergoes complementary evolution (it might not make any sense however).
So I think I might end the discussion here, at least as far as my questions are concerned.

16. Dec 22, 2015

### zonde

Thanks,
This gives idea how you understand "open system". And I understand that from practical point of view it's not a problem that there are no ideal "closed systems". My doubt was rather that the ideal system (the one that obeys Schrodinger equation) might not be closed.