Phase space and phase angles in a Quantum picture.

[Hazzattack]

Hi guys,

I was hoping someone may be able to clear something up for me. I have been reading a paper on Quantum decoherence and was curious about what particular point (it could easily just be me misunderstanding)

It is commonly noted that the superposition of waves represents all of the possible outcomes.

Therefore, if we represented this in phase space, it would consist of all the possible states of a system, each one corresponding to a distinct point in the said phase space.

What i don't fully understand is what is meant by the phase angles between these points? I'm trying to think of this from a more visual side. - aren't all these points orthogonal? yet the author speaks about phase angles between these as if they are dependent on one another

Hope the question makes sense.

p.s. there might already be a visual representation of this online somewhere, if you happen to know of its existence please let me know

Thanks in advance.

 

[bhobba]

It is commonly noted that the superposition of waves represents all of the possible outcomes.

That's an issue right there. Superposition is not that at all - it simply implies the so called pure states form a vector space. Its got nothing to do with waves etc etc. Phase angles etc come into it because its a complex vector space and is responsible for interference effects. What decoherence does at an intuitive level is that phase information gets jumbled up by interacting with other stuff and on the average you end up with a phase of a big fat zero. Mathematically it comes from the process of tracing over the environment.

To understand the modern view you need to consult a good modern textbook. I recommend the first 3 chapters of Ballentine - Quantum Mechanics - A Modern Development:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20

But regarding phase space, yes QM can be formulated along those lines, and in fact is the most powerful and sophisticated mathematical formulation to understand exactly what quantisization is. You will find this approach in tomes like Varadarajan - Geometry of Quantum Theory:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

Be warned however - it is mathematically sophisticated - meaning it's HARD. Do not attempt it unless you have a good background in math and understand QM at the level of Ballentine. Still its the most sophisticated and powerful version we have in understanding QM and Classical Mechanics at the mathematical level, and how they are related.

Thanks
Bill

 

[naima]

Chapter 15 of Ballentine's textbook is about QM in phase space and Wigner functions. It is brilliant;

 

[Hazzattack]

Thanks for the recommendations. I think I may have been slightly vague in that statement, which maybe confused what I actually meant. I was merely suggesting that the superposition principle is a fundamental theorem that enables any solutions to be written as a linear combination of those solutions. Implying you have systems with many possibilities (i.e schrodinger's cat) until wave function collapse takes place. I also should have clarified that I was considering all this from a decoherence perspective.

There might be another question you can help me with.

In the case of the Stern-Gerlach experiment we witness 2 possible spin orientations. However, once again, if spin up is a solution and spin down is also a solution then any linear combinations of these should also be a solution. If that is the case shouldn't there be many possible orientations of spin? - my reasoning for why we may only ever 'see' two is because decoherence suppresses other channels which allows "a bit of spin up and a bit of spin down".

I suppose what I'm asking is given infinite amount of time, would it be possible to physically realize other spin orientations?

Thanks again!

 

[Nugatory]

In the case of the Stern-Gerlach experiment we witness 2 possible spin orientations. However, once again, if spin up is a solution and spin down is also a solution then any linear combinations of these should also be a solution. If that is the case shouldn't there be many possible orientations of spin? - my reasoning for why we may only ever 'see' two is because decoherence suppresses other channels which allows "a bit of spin up and a bit of spin down".

We can see other orientations just by rotating the Stern-Gerlach apparatus to other angles. However, no matter what angle we choose, we always get exactly two beams coming out of it.

You should think of a superposition of up and down not as "a bit of spin up and a bit of spin down", but rather as "a bit of a chance of getting spin up and a bit of a chance of getting spin down, if you measure on the up-down axis".

 

[Hazzattack]

So why exactly do we get only two beams out? - is this purely a physical thing, in the realms of theory is it possible to get more than two beams coming out (while keeping the equipment fixed).

In Youngs slit experiment an interference pattern can be seen, in a way this can be thought of as a physical manifestation of the superposition principle. Allowing not only the classical gaussian distribution we would see behind each slit but also super position states (states which would be forbidden classically). Decoherence gives an understanding as to why with macroscopic objects we don't ever see these superposition states, the channels are suppressed and decohere rapidly. However, this doesn't mean that those channels never existed, only that the probability of physically seeing it is unbelievably small - to the point where we would have time frames approaching infinity before we saw anything 'weird' happen.

This then made me think about the Stern-Gerlach experiment. Theoretically shouldn't it be possible to get more than just these two spin states (regardless of if we only see two physically).

 

[bhobba]

Theoretically shouldn't it be possible to get more than just these two spin states (regardless of if we only see two physically).

Its not part of non-relativistic QM, but relativistic QM (aka QFT or in the case of electrons QED) explains why electrons have spin +/- 1/2 - ie only 2 values. Since electrons are charged this manifests in electrons interacting with magnetic fields in one of 2 ways - called, relative to the apparatus, spin up and spin down. Again explaining that is the job of QED. But its an experimental fact that's the way it is - you only get two values.

Its a basic fact of QM that a system with 2 possible states can be expanded in any orthogonal basis - the exact basis being determined by the orientation of the apparatus. As I explained its simply a manifestation that it forms a complex vector space.

If you want to apply decoherence to it interaction with the magnetic field transforms that superposition into an improper mixed state. How it does that QED undoubtedly explains. Generally decoherence suppresses interference terms to such a low value it's undetectable - at least with current technology - but who knows what future progress will bring.

Thanks
Bill

 

[atyy]

So why exactly do we get only two beams out? - is this purely a physical thing, in the realms of theory is it possible to get more than two beams coming out (while keeping the equipment fixed).

In Youngs slit experiment an interference pattern can be seen, in a way this can be thought of as a physical manifestation of the superposition principle. Allowing not only the classical gaussian distribution we would see behind each slit but also super position states (states which would be forbidden classically). Decoherence gives an understanding as to why with macroscopic objects we don't ever see these superposition states, the channels are suppressed and decohere rapidly. However, this doesn't mean that those channels never existed, only that the probability of physically seeing it is unbelievably small - to the point where we would have time frames approaching infinity before we saw anything 'weird' happen.

This then made me think about the Stern-Gerlach experiment. Theoretically shouldn't it be possible to get more than just these two spin states (regardless of if we only see two physically).

Unitary evolution of the state vector alone doesn't tell you what outcomes are seen when you make a measurement. Therefore quantum mechanics introduces additional postulates to describe measurements - these are the Born Rule and the Projection postulate - which are presented as Axiom 3 in http://www.theory.caltech.edu/~preskill/ph219/chap2_13.pdf .

Decoherence does not remove the need for these additional postulates. In textbook quantum mechanics, we must divide the universe into classical and quantum parts. Decoherence helps explain why we get (essentially) the same results regardless of whether we consider the system, or the system and apparatus to be in the quantum part of the universe. (There is a possibility that decoherence removes the need for the additional postulates if one uses the many-worlds interpretation.)