Understanding Atomic Systems: Nucleus and Electrons in a Constant State

[Rainbows_]

Only an atom with a nucleus and electrons combined produce a definite state. Neither the nucleus nor electrons have definite state. How do you understand this? We know there is electronic transition from ground state to excited state. How could the electrons do that independent of the nucleus since they are supposed to share one quantum state?

 

[Lord Jestocost]

It’s colloquial speech to talk about the eigenstates of electrons in atoms. These are always eigenstates of the entity „nucleus-electrons“ which could in principle be calculated by finding the solutions of the Schrödinger equation for the whole entity.

 

[Rainbows_]

It’s colloquial speech to talk about the eigenstates of electrons in atoms. These are always eigenstates of the entity „nucleus-electrons“ which could in principle be calculated by finding the solutions of the Schrödinger equation for the whole entity.

For composite objects like nucleus/electrons with neither having any definite state.. must we picture them as like mixing coffee and milk where they become something else (an emergence) or is it like the Earth and moon in time exposure that lacks position but still there.. only our ignorance not able to track the positions of them?

 

[Vanadium 50]

That's very poetic. But I have no idea what you mean by that.

 

[Lord Jestocost]

For composite objects like nucleus/electrons with neither having any definite state.. must we picture them as like mixing coffee and milk where they become something else...

Never conceive of quantum physical "entities". You cannot describe quantum phenomena in terms of "classical" conceptions and notions. You will always come to a dead end! Take it as it is.

 

[Rainbows_]

How does the concept of statistical ensembles describe composites? are they not just statistics of marbles?

 

[Lord Jestocost]

How does the concept of statistical ensembles describe composites? are they not just statistics of marbles?

According to the "Principles of Quantum Statistical Mechanics". Have a look, e.g., at "An Introduction to Statistical Thermodynamics" by Terrell L. Hill.

 

[Rainbows_]

That's very poetic. But I have no idea what you mean by that.

composite system = entangled system
can they be described by ensembles of some kind?

 

[Vanadium 50]

I still have no idea what you mean - coffee and milk?

Why not write plain, clear, complete sentences?

 

[Rainbows_]

I still have no idea what you mean - coffee and milk?

Why not write plain, clear, complete sentences?

How are entangled system mixed.. do their parts become completely mixed like in coffee and milk or like in Earth and moon only very fast?

 

[Nugatory]

composite system = entangled system
can they be described by ensembles of some kind?

That's not helping much with understanding what you're asking:
- An entangled system is not the same thing as a composite system.
- Ensembles don't describe systems, they're a mathematical concept used to assign a precise meaning to the probabilistic predictions of quantum mechanics.

The general idea (and I am simplifying a lot for this B-level thread) here is: The quantum system, whether composite or not, has some state function. We do mathematical manipulations on this state function and we get something along the lines of "30% chance that a measurement will give us result X, 70% chance that a measurement will give us result Y". That sounds nice, but what exactly does it mean? What sort of experiment could tell us whether that statement is accurate? If we're just looking at one measurement, there's no good answer. Instead, we imagine that we have prepared a very large number of identical systems in the same way so that they're all described by the same state; and then we separately measure each one of them; and the quantum mechanical statement is actually "30% of the large number of identically prepared systems gave us result X when measured and 70% gave us result Y when measured".

 

[Nugatory]

How are entangled system mixed.. do their parts become completely mixed like in coffee and milk or like in Earth and moon only very fast?

A quantum system doesn't have "parts"; that's the point of your first post in this thread.

We say that a quantum system is entangled if its state function has a particular mathematical property. Again, simplifying a lot, if the state function can be written as a superposition of "Measuring property $A$ will yield result $A_1$ every time and measuring property $B$ will yield result $B_1$ every time" and "Measuring property $A$ will yield result $A_2$ every time and measuring property $B$ will yield result $B_2$ every time" then we say that the system is entangled on the observables $A$ and $B$. Informally (but be warned that "informal" explanations of quantum mechanics must not be taken too seriously!) measuring either $A$ or $B$ collapses the superposition into a state in which we know what the other one would be if we were to measure it.

Most of the interesting examples of entanglements involve systems in which the two observables are measured at different locations: "The detector here reads "X" and the detector over there, 100 meters away, reads "Y".

 

[ZapperZ]

For composite objects like nucleus/electrons with neither having any definite state.. must we picture them as like mixing coffee and milk where they become something else (an emergence) or is it like the Earth and moon in time exposure that lacks position but still there.. only our ignorance not able to track the positions of them?

But this goes back to the Schrödinger Cat scenario. If the issue is simply our ignorance, then the result would not matter if we know or not know which slit the quantum particle goes through. Yet, it DOES make a difference between knowing and not knowing, because it affects the interference pattern.

Zz.

 

[PeterDonis]

it DOES make a difference between knowing and not knowing

The word "knowing" here might cause confusion. What makes a difference is whether or not there are devices at the slits that interact with the quantum particle. The presence or absence of that interaction changes how the wave function evolves, and therefore affects whether or not there is an interference pattern. But no "knowledge" of what is happening by any human is necessary.

 

[Rainbows_]

The nucleus is entangled to the electrons.. so the electrons shouldn't have any wave function and has no definite state. But we know electrons form the outermost shells of the atom.. so they still have separate identities.. or chemistry won't be possible.. how is this paradox explained?

 

[PeterDonis]

The nucleus is entangled to the electrons

The electrons are not just one thing. They have multiple degrees of freedom. The nucleus is not entangled to all of those degrees of freedom. At most, the position of the nucleus constrains the position of the electrons, because of the Coulomb potential between them. But that leaves plenty of freedom for the electrons to do chemistry.

 

[Rainbows_]

The electrons are not just one thing. They have multiple degrees of freedom. The nucleus is not entangled to all of those degrees of freedom. At most, the position of the nucleus constrains the position of the electrons, because of the Coulomb potential between them. But that leaves plenty of freedom for the electrons to do chemistry.

Thanks for the assistance. What dictate what are the degrees of freedom of the electrons where the nucleus would be entangled with (perhaps an example)?

 

[Nugatory]

The nucleus is entangled to the electrons.. so the electrons shouldn't have any wave function and has no definite state. But we know electrons form the outermost shells of the atom.. so they still have separate identities.. or chemistry won't be possible.. how is this paradox explained?

The electrons have no wave function of their own, but just about all of their interesting properties are observables of the combined nucleus/electrons quantum system, which makes it natural to talk about the electrons in isolation, as @Lord Jestocost described in the first reply in this thread.

Another consideration is that we can try solving the problem as if the nucleus were a fixed positive charge that doesn't change at all in the interaction. That's not quite right - we know that the nucleus is part of the quantum system and as the wave function of that system evolves so does the expected result of measurements of various properties of the nucleus. However, it is a very good approximation, quite good enough to explain just about all of chemistry, and it greatly simplifies the calculations. So you'll often see people saying that they've solved the electron/nucleus problem to explain the behavior of the electrons, when they've really gotten the right answer by solving the easier but equivalent problem. You're expected to figure out which it is from the context.

 

[PeterDonis]

What dictate what are the degrees of freedom of the electrons where the nucleus would be entangled with

The interaction between the nucleus and the electrons.

 

[bhobba]

The nucleus is entangled to the electrons.. so the electrons shouldn't have any wave function and has no definite state. But we know electrons form the outermost shells of the atom.. so they still have separate identities.. or chemistry won't be possible.. how is this paradox explained?

You noticed that hey? Yes the electron is, strictly speaking, entangled with the nucleus, but it is via the EM field so that really is the important thing to use in modelling the situation.

But I can tell you that interaction with the field most certainly does lead to interesting things:
http://www.physics.usu.edu/torre/3700_Spring_2015/What_is_a_photon.pdf

The above is beyond the B level but do give it a scan - you will still likely get something from it.

Thanks
Bill

 

[Lord Jestocost]

The nucleus is entangled to the electrons.. so the electrons shouldn't have any wave function and has no definite state. But we know electrons form the outermost shells of the atom.. so they still have separate identities.. or chemistry won't be possible.. how is this paradox explained?

There is no paradox. When two quantum mechanical (qm) entities "Atom_1" and "Atom_2" interact to form a new entity "Molecule_12", the eigenstates of this new entity can again in principle be calculated by finding the solutions of the appropriate Schrödinger equation for the whole entity "Molecule_12", considering all participating qm entities "neutrons", "protons" and "electrons" and all interactions between them. That would indeed be a huge task, so you havo to rely on approximations as indicated by @Nugatory. When you find that the ground state energy of the "Molecule_12" is lower than the sum of the ground state energies of "Atom_1" and "Atom_2", you know that a stable molecule can be formed.

 

[vanhees71]

Check out this very nice paper (published in the American Journal of Physics) concerning the hydrogen atom:

https://arxiv.org/pdf/quant-ph/9709052