Understanding Pure States in Quantum Mechanics: From Atoms to Electrons

[bluecap]

I learned about pure state, mixed state, reduce density matrix, etc. from this now famous paper http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf (thanks to Bill)

I'd like to know something. Atty said somewhere:

"A pure state means that we have prepared many copies of the system of one spin, and each copy of the system is in the same pure state. For example, every copy of the single spin is pointing up"

How do you apply this to an electron in an atom. How do you prepare it such that even if you prepared many copies of the system, it is always same state. Can you do that to the electron that is without any definite state (such as no position eigenstates chosen)? Or must it always have result like the electron always end up in one position eigenstate.. but how do you prepare many copies of the electron with always one particular position eigenstate chosen among infinity? Or can't this be done.. how then do you prepare an electron(s) in atom in pure state?

 

[Simon Bridge]

A pure state can be prepared by operating on an impure state, and filtering.
See: Stern-Gerlach experiment.

 

[bluecap]

A pure state can be prepared by operating on an impure state, and filtering.
See: Stern-Gerlach experiment.

I'm specifically referring to electrons in an atom that includes the entire atom (with nucleus). What is an example where you can prepare it in pure state? What if it is just in superposition of position, how can you make it be in only one position eigenstate out of infinity every time you measure it?

 

[Simon Bridge]

You can't. You can measure the state of the atom, individual electron eigenstates contribute to that.
You can deduce the contributing states... hydrogen is easy for example.

A pure state of position would be an exact measurement of position... the probability of measuring that is zero.

 

[Nugatory]

What if it is just in superposition of position, how can you make it be in only one position eigenstate out of infinity every time you measure it?

You don't need to. A superposition is a pure state.

 

[Simon Bridge]

That would be a better way of putting it... more directly, it is not clear what op is thinking "pure state" means, particularly, what is meant by "position eigenstate".
The quote does not make sense for eg.

 

[bluecap]

That would be a better way of putting it... more directly, it is not clear what op is thinking "pure state" means, particularly, what is meant by "position eigenstate".
The quote does not make sense for eg.

I know superposition is a pure state. But in an atom, it is already in superposition of position. So technically we can refer it as pure state. No problem. But atty said:

"In a pure state, one has many systems all prepared in the same pure state, and there is in principle, a measurement that when performed on each system will give exactly the same result."

For the atom in superposition of position. Cant we say it is already in pure state? But why is there a requirement that a measurement should always give the same result.. what then is that eigenstate that must always be the same result?

 

[Simon Bridge]

What would it mean to prepare a particle, never mind the atom for now, in a single position eigenstate?

 

[bluecap]

What would it mean to prepare a particle, never mind the atom for now, in a single position eigenstate?

That's right. It can't be done so any atom with electrons in superposition of position is a pure state. But then atty said:

"In a pure state, one has many systems all prepared in the same pure state, and there is in principle, a measurement that when performed on each system will give exactly the same result."

What same result is he talking about? Please give an actual example in the electrons in atom in superposition of positions.

 

[Simon Bridge]

You are misunderstanding what you are being told. I am trying to guide you to a better understanding by using questions on simple systems. Please do not jump ahead.
The advantage of providing assistance free of charge is that I get to provide the assistance needed as opposed to that asked for. Therefore I will not give examples with whole atoms until I am confident you will understand me. That is not the assistance you need right now.

If you will not answer questions, I cannot help you.

 

[bluecap]

You are misunderstanding what you are being told. I am trying to guide you to a better understanding by using questions on simple systems. Please do not jump ahead.
The advantage of providing assistance free of charge is that I get to provide the assistance needed as opposed to that asked for. Therefore I will not give examples with whole atoms until I am confident you will understand me. That is not the assistance you need right now.

If you will not answer questions, I cannot help you.

You asked "What would it mean to prepare a particle, never mind the atom for now, in a single position eigenstate?". I answered it can't be done (because you already gave clue that "You can't. You can measure the state of the atom, individual electron eigenstates contribute to that.You can deduce the contributing states... hydrogen is easy for example.." Well. I'm just puzzled because an electron in atom is already in superposition (without doing anything to it). Why do we need "a measurement that when performed on each system will give exactly the same result." Hm... maybe I'm thinking of the electron superposition as actually there occurring even without anyone doing a preparing the state. I wonder if this is the source of my confusion...

 

[Simon Bridge]

I did not ask if it could be done, I asked what it would mean.
You have yet to answer the question.

 

[bluecap]

I did not ask if it could be done, I asked what it would mean.
You have yet to answer the question.

Oh. To prepare a single electron in a position eigenstate means to put it in one position or at least within that allowed by HUP (by some kind of filtering). So I thought this was what it mean to prepare it in pure state such that when measured.. it is always around that one position. But even without doing that. I was asking if an unmeasured electron in atom is already in pure state (or superposition).

 

[bhobba]

In an atom things are a bit subtle.

I agree with everything said before but wish to make 2 pedantic points.

1. Move an atom around and it moves as a whole held together by electrostatic forces. This means the nucleus and electrons are strictly speaking entangled, but for simplicity of analysis in say the hydrogen atom its usually ignored and the electron treated as if in a pure state.

2. We have the phenomena of spontaneous emission which is because the electron couples to the quantum EM field that pervades the universe so is entangled with it - again usually ignored for simplicity. But when reading about spontaneous emission keep it in the back of your mind because its cause is that being entangled with the EM field its not really in a stationary state since we used the pure state simplifying assumption.

Thanks
Bill

 

[Simon Bridge]

Oh. To prepare a single electron in a position eigenstate means to put it in one position or at least within that allowed by HUP (by some kind of filtering). So I thought ... what it mean to prepare it in pure state such that when measured.. it is always around that one position.

Nope. Close. You are describing the properties of a particle confined to a region of space by a potential. ie. a particle confined to a set of apparatus in the lab will always be found someplace within the apparatus.

But even without doing that. I was asking if an unmeasured electron in atom is already in pure state (or superposition).

It is in a superposition of position eigenstates that is also a pure state as opposed to an entangled state.

I think I have a fair understanding of where you are coming from now ... please hang on, the next bit is going to take a little bit to type out.

I suspect a category error, so I'd like to get away from the "whole atom" thing.

 

[Simon Bridge]

$\renewcommand{\ket}[1]{| #1 \rangle}$
Lets get away from specific particles and physical systems just to pin down the situations here - mostly this is semantics.
You do not seem to know what is meant by an eigenstate... or a state.

This needs a recap of basic QM:
http://www.physics.wustl.edu/alford/physics/essentials.pdf
... putting a particle in an eigenstate means to set it up so there is only one a-priori possibility of measurement.
... if that is an eigenstate of position, that will mean setting things up so there is only one possible position ... this is not physically realizable, but we can write it down mathematically, even use it to deduce other things.
The position representation of the wave function is a superposition of position eigenstates, and that is useful.
In general, the eigenstate of a particular measurement will be a superposition of eigenstates of some other measurement.
ie. the position wave-function of an energy eigenstate represents and eigenstate (energy) in terms of a superposition of position eigenstates.

Then you can deal with understanding pure vs mixed states ... which relates to your needs re entanglement.
http://www.lecture-notes.co.uk/susskind/quantum-entanglements/lecture-8/pure-and-mixed-states/
... this is part of a lecture series. If you have trouble with part of the lecture then go back to previous lectures to find the concept you are having trouble with.
tldr: a pure state is one which tells us everything about the system of interest.

I'm not sure what atyy was talking about - @atyy is the best person to ask.
I'm guessing he was attempting to describe the difference between identical repeated measurements and identically prepared systems. But I'm only guessing.

If your contention is that the quoted bit by atyy is not sufficient to describe what is meant by a pure state in all possible circumstances, then you are correct.

A laymans description of QM concepts will, necessarily, incomplete because a lay description would have to be in terms of a subset of Newtonian mechanics that is generally and intuitively understood by laymen using the laymans language. Specifically, the meanings of words will get a bit wishy washy because that is what laymans language is like.

If a complete description was possible in those terms, then we would not need quantum mechanics.
It is possible to get from there to a better understanding. It will mean learning new ideas though, and, when you are done, you will no longer have a laymans description.

 

[bluecap]

$\renewcommand{\ket}[1]{| #1 \rangle}$
Lets get away from specific particles and physical systems just to pin down the situations here - mostly this is semantics.
You do not seem to know what is meant by an eigenstate... or a state.

This needs a recap of basic QM:
http://www.physics.wustl.edu/alford/physics/essentials.pdf
... putting a particle in an eigenstate means to set it up so there is only one a-priori possibility of measurement.
... if that is an eigenstate of position, that will mean setting things up so there is only one possible position ... this is not physically realizable, but we can write it down mathematically, even use it to deduce other things.
The position representation of the wave function is a superposition of position eigenstates, and that is useful.
In general, the eigenstate of a particular measurement will be a superposition of eigenstates of some other measurement.
ie. the position wave-function of an energy eigenstate represents and eigenstate (energy) in terms of a superposition of position eigenstates.

Then you can deal with understanding pure vs mixed states ... which relates to your needs re entanglement.
http://www.lecture-notes.co.uk/susskind/quantum-entanglements/lecture-8/pure-and-mixed-states/
... this is part of a lecture series. If you have trouble with part of the lecture then go back to previous lectures to find the concept you are having trouble with.
tldr: a pure state is one which tells us everything about the system of interest.

I'm not sure what atyy was talking about - @atyy is the best person to ask.
I'm guessing he was attempting to describe the difference between identical repeated measurements and identically prepared systems. But I'm only guessing.

If your contention is that the quoted bit by atyy is not sufficient to describe what is meant by a pure state in all possible circumstances, then you are correct.

A laymans description of QM concepts will, necessarily, incomplete because a lay description would have to be in terms of a subset of Newtonian mechanics that is generally and intuitively understood by laymen using the laymans language. Specifically, the meanings of words will get a bit wishy washy because that is what laymans language is like.

If a complete description was possible in those terms, then we would not need quantum mechanics.
It is possible to get from there to a better understanding. It will mean learning new ideas though, and, when you are done, you will no longer have a laymans description.

Thanks.. but I know what is meant by eigenstate or eigenpositions.
We quantum laymen don't think Newtonianly.. in fact, we imagine things or atoms in ghostly superpositions.. so pure state is our ordinary thought and life.. this is why I wonder what atty was describing when he stated:

"In a pure state, one has many systems all prepared in the same pure state, and there is in principle, a measurement that when performed on each system will give exactly the same result."

I think what atty meant was that since physicists don't assume things as really in pure state (or ghostly superpositions that we laymen come by automatically).. one has to prepare the state and how else can you do that but by making sure there is at least one eigenstate (like spin up) so as to have a concrete way to state it.

Let's wait for atty to clarify his points.

 

[Simon Bridge]

Thanks.. but I know what is meant by eigenstate or eigenpositions.
We quantum laymen don't think Newtonianly.. in fact, we imagine things or atoms in ghostly superpositions.. so pure state is our ordinary thought and life..

... and yet, when asked to say what being in an eigenstate meant, you did not supply the correct answer. The said it involved a distribution close to a particular value and invoked HUP ... all extraneous to the concept. An eigenstate is always associated with eigen(="only one") value.
I do not know what you mean by "ghostly superposition" ... afaik it has no formal definition in physics.
I doubt anyone can explain anything to you unless you can communicate what you mean by this.
Try to make explaining your use of this term a priority.

... this is why I wonder what atty was describing when he stated:

"In a pure state, one has many systems all prepared in the same pure state, and there is in principle, a measurement that when performed on each system will give exactly the same result."

Notice the first sentence contains a possible logical contradiction?
"In a pure state one has many systems..." says that a pure state is a state of many.
"[each one] prepared in a pure state" ... so is each system itself composed of many systems in infinite regression?
But it does loosely describe what is oftenmeant by preparing a bunch of systems in identical states ... in each state there exists a measurement that will give you the same result for all systems. You have observed that this measurement cannot be a measurement of position.
I suspect atyy was speaking about a specific example - but I do not have the original context.

I think what atty meant was that since physicists don't assume things as really in pure state (or ghostly superpositions that we laymen come by automatically).. one has to prepare the state and how else can you do that but by making sure there is at least one eigenstate (like spin up) so as to have a concrete way to state it.

There's a bunch of stuff there.

physicists don't assume things as really in pure state

A pure state (see the susskind lecture) is one where we have complete knowledge (in QM terms mind!) of the system. In Nature we will never have one of those. It is impossible to completely isolate something we are experimenting on.
Instead we try to control the variables so the ones we don't want to affect the experiment have too small-an effect to be measured during the experiment run.
When we want to measure an effect that is really really small where lots of things are going on, then there are big problems to be overcome.

For instance - in principle, a system can be prepared in an excited state and it stays there for ever.
In practice, the excited state decays - often quite fast. This tells us the model is incomplete. ie. we did not have a pure state.

one has to prepare the state

We prepare systems because it is inconvenient to rely on discovering an atom, say, with exactly the qualities we want.
We can do that if we want to - it's just usually really inconvenient. ie if we want a silver atom with magnetic moment axis pointing to the ceiling, we could just get a big bag of silver with random magnetic moments and sort them out one at a time to find the ones with just the right spin ... or we could rotate all the silver atoms with a magnetic field, and separate out the ones that point down.

 

[bluecap]

... and yet, when asked to say what being in an eigenstate meant, you did not supply the correct answer. The said it involved a distribution close to a particular value and invoked HUP ... all extraneous to the concept. An eigenstate is always associated with eigen(="only one") value.
I do not know what you mean by "ghostly superposition" ... afaik it has no formal definition in physics.
I doubt anyone can explain anything to you unless you can communicate what you mean by this.
Try to make explaining your use of this term a priority.

superposition is not classical hence ghost like. for example if you put Trump and Clinton in superposition, Trump can have Clinton lower body. and vice versa. these can't exist in classical world hence they are said to be in ghostly superposition. same as all particles. when they are entangled. they share parts like Trump and Clinton sharing the bodies. so superposition is another reality. Bohmians, Many worlders and even Ensemblers like Hobba want to picture classical world superposition is unclassical and could occur in another domain of reality. Bohm calls it the Implicate Order before our classical physicists ruined and suppressed the idea.

Notice the first sentence contains a possible logical contradiction?
"In a pure state one has many systems..." says that a pure state is a state of many.
"[each one] prepared in a pure state" ... so is each system itself composed of many systems in infinite regression?
But it does loosely describe what is oftenmeant by preparing a bunch of systems in identical states ... in each state there exists a measurement that will give you the same result for all systems. You have observed that this measurement cannot be a measurement of position.
I suspect atyy was speaking about a specific example - but I do not have the original context.

here is atty original context https://www.physicsforums.com/threa...mproper-mixed-states-in-laymens-terms.734987/

There's a bunch of stuff there.

A pure state (see the susskind lecture) is one where we have complete knowledge (in QM terms mind!) of the system. In Nature we will never have one of those. It is impossible to completely isolate something we are experimenting on.

what ? you mean scientists having complete knowledge of computer processor or cpu or any integrated circuit means it is in pure state? but i thought pure state is only ray in Hilbert space of trace 1 and not orthonormal vectors.. pls clarify.

Instead we try to control the variables so the ones we don't want to affect the experiment have too small-an effect to be measured during the experiment run.
When we want to measure an effect that is really really small where lots of things are going on, then there are big problems to be overcome.

For instance - in principle, a system can be prepared in an excited state and it stays there for ever.
In practice, the excited state decays - often quite fast. This tells us the model is incomplete. ie. we did not have a pure state.We prepare systems because it is inconvenient to rely on discovering an atom, say, with exactly the qualities we want.
We can do that if we want to - it's just usually really inconvenient. ie if we want a silver atom with magnetic moment axis pointing to the ceiling, we could just get a big bag of silver with random magnetic moments and sort them out one at a time to find the ones with just the right spin ... or we could rotate all the silver atoms with a magnetic field, and separate out the ones that point down.

thanks