The Wave Function of Our World: How Does It Emerge?

[stevendaryl]

I'm not denying that - but you were wondering why macroscopic objects didn't make super-positions, say between registering 'on' and 'off'. They would pass the test. I refer to Dirac's Principles ... page 13, paragraph 2, especially the italicised bit at the end.

But if you have a detector with two LEDs, one labeled "UP" and one labeled "DOWN", you'll see one or the other lit up.

 

[Mentz114]

But if you have a detector with two LEDs, one labeled "UP" and one labeled "DOWN", you'll see one or the other lit up.

Exactly as if the detector was in a statistical mixture $\alpha | UP\rangle + \beta | DOWN\rangle$.

 

[jaydnul]

Tell me if this is right. A particle is described by its wave function. The more it interacts with other particles, the more its wave function will start to favor certain outcomes and hinder other outcomes. If the interaction is large enough, the probability will narrow down to one possible outcome. So really the particle doesn't snap to a certain outcome, it transitions to it continuously.

If that is all correct, then my question is what state is the matter that we experience in macroscopic life in? The particles in the table in front of me, are they constantly narrowed down to a single state, are they constantly in a superposition of states, or are they sometimes in a superposition of states and sometimes in a single state? If the latter is true, then on average at what frequency are the particles in the table going from a superposition of states, then to a definite state, then evolving back into a superposition of states?

 

[Mentz114]

Tell me if this is right. A particle is described by its wave function. The more it interacts with other particles, the more its wave function will start to favor certain outcomes and hinder other outcomes. If the interaction is large enough, the probability will narrow down to one possible outcome. So really the particle doesn't snap to a certain outcome, it transitions to it continuously.

If that is all correct, then my question is what state is the matter that we experience in macroscopic life in? The particles in the table in front of me, are they constantly narrowed down to a single state, are they constantly in a superposition of states, or are they sometimes in a superposition of states and sometimes in a single state? If the latter is true, then on average at what frequency are the particles in the table going from a superposition of states, then to a definite state, then evolving back into a superposition of states?

A particle may be described by its Hamiltonian, from which we get $\mathbf{\hat{H}}\psi = E\psi$ which (if we're lucky) allows us to solve for $\psi$. The Hamiltonian formalism implies that a particle can only have one value for each dynamic variable.

Super-positions are statistical - it is not possible to define one without using probability. But probability is not stuff, it cannot be measured, it does not interact with matter.
So actual physical super-positions are not possible.

 

[stevendaryl]

Exactly as if the detector was in a statistical mixture $\alpha | UP\rangle + \beta | DOWN\rangle$.

That's what I was asking: What does it mean to be in a statistical mixture of the two LED states? Does it mean that either one or the other is on, but I just don't know which, until I take a look? If so, that's a different interpretation of a superposition than is used for microscopic superpositions. That's the whole point of Bell's proof, that it is not consistent to view the composite entangled two-particle spin state $\frac{1}{\sqrt{2}} (|u\rangle |d\rangle - |d\rangle |u\rangle)$ as meaning "Either the first particle is spin-up and the second is spin-down, or the first particle is spin-up and the second is spin-down, but we don't know which until we measure it."

So you're using a different interpretation of superpositions/mixtures for macroscopic objects than for microscopic objects.

 

[stevendaryl]

Tell me if this is right. A particle is described by its wave function. The more it interacts with other particles, the more its wave function will start to favor certain outcomes and hinder other outcomes. If the interaction is large enough, the probability will narrow down to one possible outcome.

No, that is not correct. Or at least, it's an incomplete explanation. Do you know the EPR experiment? You create a pair of anti-correlated particles. Alice measures the spin of one of the particles, and finds it to be spin-up along the z-axis. Then at that point, it is known with certainty that Bob, measuring the spin of the other particle along the z-axis, will find it to be spin-down. So Alice can predict that Bob will get a definite result before his measuring device even interacts with his particle. It doesn't seem plausible that Bob's definite result is a matter of some complicated interaction between Bob's device and his particle, because the outcome is known ahead of time. (Unless the whole interaction on Bob's end is all for show, like professional wrestling).

 

[PeterDonis]

Super-positions are statistical - it is not possible to define one without using probability.

This is not correct. Superpositions are just linear algebra. You can define them without any notion of probability whatever. And whether or not a particular pure state is a superposition is basis dependent; so "superposition" is not even an absolute property of a state.

I think you might be confusing "superposition" with "mixed state" (as opposed to "pure state").

 

[ftr]

This is a classical model, not a quantum model. It makes no sense to ask questions about a classical model that are only meaningful for a quantum model.

I am not sure what you are saying here, but in Zee's QFT in the first few chapters derives the coulomb potential and calls that the greatest achievement of 20th century.

 

[PeterDonis]

I am not sure what you are saying here

I am saying that a model in which electrons travel towards each other, then stop and move apart again, is a classical model. It's not a quantum model. In a quantum model electrons are not classical particles with definite trajectories.

in Zee's QFT in the first few chapters derives the coulomb potential

As a quantum field interaction potential, yes. Not as a classical potential between classical point particles.

 

[Mentz114]

This is not correct. Superpositions are just linear algebra. You can define them without any notion of probability whatever. And whether or not a particular pure state is a superposition is basis dependent; so "superposition" is not even an absolute property of a state.

I think you might be confusing "superposition" with "mixed state" (as opposed to "pure state").

You make my point for me. It is just linear algebra operating on amplitude vectors. The fact that super-position can be created by a change of basis shows that super-position has no physical significance. Just like transformation of Hamiltonian conjugates into new dynamical variables does not change the physics.

FAPP a super-position is a statistical mixture.

 

[ftr]

super-position has no physical significance.

That implies that there should be no "measurement problem".

FAPP a super-position is a statistical mixture

mixture of what exactly.

 

[ftr]

As a quantum field interaction potential, yes. Not as a classical potential between classical point particles.

my understanding is that the former is an explanation for the later, or am I wrong?

 

[Mentz114]

That implies that there should be no "measurement problem".mixture of what exactly.

Exactly. There is no measurement problem. Unitary evolution of probability (amplitude) distributions can only give more probabilities. We always get an outcome so clearly it is being reached by dynamics, not statistics.

 

[Mentz114]

That's what I was asking: What does it mean to be in a statistical mixture of the two LED states?

It means that each run in the ensemble produces a Up or Down, and the relative frequencies will converge to some probabiities.

So you're using a different interpretation of superpositions/mixtures for macroscopic objects than for microscopic objects.

Definately not - I regard all super-positions to be statistical mixtures, from baseballs to atoms

 

[bhobba]

Because...?

Decohrence transforms a superposition to a mixed state diagonalised in a certain basis, usually position.

Collapse is not the interpretation of a mixed state. In the basis it is diagonal in the diagonal elements give the probability of that outcome.

Now it can be interpreted in a number of ways of which collapse is just one.

In ignorance ensemble, my favorite interpretation, there is no collapse - somehow the improper mixed state becomes a proper one - exactly how - in that interpretation - blank-out.

Thanks
Bill

 

[bhobba]

Under what circumstances do we get this "infection" or "collapse"?

We don,t.

All that happens is a state, via decoherence, becomes a mixed state diagonal in a certain basis. One way to form such a mixed state is you take the states corresponding to the basis it is diagonal in and simply present them for observation with a certain probability. There is no mystery here - the outcome is there prior to observation - we just don't know which one. Such mixed states are called proper. Now another way is via decoherence. There is no way to tell the difference, but it is prepared differently and therein lies the rub. How does an improper mixed state become a proper one and/or does it even matter since you can't tell the difference. Each interpretation has its own take. For example in MW one simply defines each outcome as a different world and everything just keeps evolving. In classical Copenhagen they call the change from mixed to proper collapse. In ensemble it simple means you have subjected the state to a different preparation procedure and states and preparation procedures are synonymous. We simply don't know what state it results in.

Thanks
Bill

 

[bhobba]

That implies that there should be no "measurement problem".

If there is a measurement problem, and even what it is, the formalism is silent on. That's what interpretations are about.

Since the formalism is silent on it to answer the question - is there a measurement problem you need to specify an interpretation.

In ignorance ensemble its - how does an improper mixture become a proper one - but even there you will find some arguing it doesn't even matter.

So please rephrase your query - in such and such interpretation is there a measurement problem and what is it.

Thanks
Bill

 

[PeroK]

FAPP a super-position is a statistical mixture.

You are definitely misinterpreting superposition, albeit in a relatively subtle way. I think it would be worth rethinking this.

First, a superposition of states is precisely a linear combination of vectors. Only the terminology is different. There is no inherent statistical element. For a pure state, there is always a single (pure) state regardless of the basis, and regardless of whether the state is expressed as a superposition or not.

When it comes to the measurement of an observable, the state has a statistical interpretation. But, this applies regardless of the basis or any superposition. The statistics relate to the state, not to the superposition.

For example, you could have an energy eigenstate expressed as the superposition of two other states. A measurement of energy will definitely return a single value.

Or, you could have a superposition of energy eigenstates expressed a single state with no superposition. A measurement of energy would return a range of values with certain probabilities.

Now, true, if you intend to measure a certain observable, you can always express the state as a superposition of eigenstates of that observable. In this case, the probabilities and the superposition are aligned.

But, fundamentally, the probabilities are inherent in the state, and do not depend on any particular superposition.

 

[durant35]

Decohrence transforms a superposition to a mixed state diagonalised in a certain basis, usually position.

Collapse is not the interpretation of a mixed state. In the basis it is diagonal in the diagonal elements give the probability of that outcome.

Now it can be interpreted in a number of ways of which collapse is just one.

In ignorance ensemble, my favorite interpretation, there is no collapse - somehow the improper mixed state becomes a proper one - exactly how - in that interpretation - blank-out.

Thanks
Bill

Yep, I agree with you, collapse is just one way of interpreting it and I mentioned it in reference to OP's question where he invoked the word 'collapse'.

Anyway, thanks for the details in your post.

 

[stevendaryl]

It means that each run in the ensemble produces a Up or Down, and the relative frequencies will converge to some probabiities.

What does it mean to say that it "produces" an Up or a Down? I think that you're going down a path toward an interpretation that is actually inconsistent with Bell's theorem, if you are treating microscopic and macroscopic objects equivalently.

Let's take the simple example of single electrons, where we only consider the spin degrees of freedom. You have some process that produces the state $\alpha |u\rangle + \beta |d\rangle$. You want to say that the meaning of the amplitudes $\alpha$ and $\beta$ is that every run, you either produce an electron in state $|u\rangle$ (with probability $|\alpha|^2$) or in state $|d\rangle$ (with probability $|\beta|^2$). But that doesn't actually make any sense, for the following reason:

Instead of the basis $|u\rangle$ and $|d\rangle$, (spin-up and spin-down in the z-direction), we can write it in the basis $|u_x\rangle = \frac{1}{\sqrt{2}} (|u\rangle + |d\rangle)$, $|u_y\rangle = \frac{1}{\sqrt{2}} (|u\rangle - |d\rangle)$. Then the original superposition $\alpha |u\rangle + \beta |d\rangle$ can equally well be written in the form:

$\frac{\alpha + \beta}{\sqrt{2}} |u_x\rangle + \frac{\alpha - \beta}{\sqrt{2}} |d_x\rangle$

So using your reasoning, this should be interpreted as: "Every run, the electron is either spin-up in the x-direction, with probability $\frac{|\alpha + \beta|^2}{2}$, or spin-down in the x-direction, with probability $\frac{|\alpha - \beta|^2}{2}$".

So does that mean that every run, the electron is either:

1. spin-up in the x-direction and in the z-direction
2. spin-up in the x-direction and spin-down in the z-direction
3. spin-down in the x-direction and spin-up in the z-direction
4. spin-down in the x-direction and in the the z-direction

Electrons can't simultaneously have a definite value for spin in the x-direction and spin in the z-direction.

The interpretation of amplitudes as giving probabilities only makes sense after you've chosen a basis.

 

[stevendaryl]

You make my point for me. It is just linear algebra operating on amplitude vectors. The fact that super-position can be created by a change of basis shows that super-position has no physical significance. Just like transformation of Hamiltonian conjugates into new dynamical variables does not change the physics.

FAPP a super-position is a statistical mixture.

I think that you're getting things mixed up. It's not true that super-position has no physical significance. If you pass light through two slits and onto a screen, the amplitude of light on the screen is a superposition of the light due to one slit and the light due to the other slit. It certainly has a physical meaning; the superposition causes some dark regions of destructive interference and some light regions of constructive interference. The interpretation that light either passes through one slit (with such-and-such probability) or the other (with some other probability) is inconsistent with that interference pattern.

In contrast, a statistical mixture of two possibilities definitely can be interpreted as: either one is true, with a certain probability, or the other is true, with a different probability.

So I think what you said above is just not true.

 

[stevendaryl]

I think that you're getting things mixed up. It's not true that super-position has no physical significance.

There's a terminology problem, which is that it doesn't actually make sense to talk about something "being a superposition". The correct terminology would be "being a superposition of this state and that state". Calling something a superposition without mentioning what it's a superposition of doesn't make sense. It's sort of like calling the number 5 a "sum". It's the sum of 2 and 3, or of 1 and 4, etc, but calling it a sum doesn't convey any information.

In contrast, there is in quantum mechanics a test for whether something is a statistical mixture, which is independent of basis. If your system is described by a density matrix $\rho$, a pure state has the property that $\rho^2 = \rho$.

 

[Mentz114]

You are definitely misinterpreting superposition, albeit in a relatively subtle way. I think it would be worth rethinking this.

First, a superposition of states is precisely a linear combination of vectors. Only the terminology is different. There is no inherent statistical element. For a pure state, there is always a single (pure) state regardless of the basis, and regardless of whether the state is expressed as a superposition or not.

..
..

But, fundamentally, the probabilities are inherent in the state, and do not depend on any particular superposition.

Thanks for this. I'll think about it but I doubt it will change my views. In fact the last sentence just reinforces the fact that the 'state' you talk about is only a statistical statement - however it came about.

 

[Mentz114]

There's a terminology problem, which is that it doesn't actually make sense to talk about something "being a superposition". The correct terminology would be "being a superposition of this state and that state". Calling something a superposition without mentioning what it's a superposition of doesn't make sense. It's sort of like calling the number 5 a "sum". It's the sum of 2 and 3, or of 1 and 4, etc, but calling it a sum doesn't convey any information.

Sunstitute a state $\psi$ for '5' and you have my point. Expressing it as a sum conveys no useful information.

In contrast, there is in quantum mechanics a test for whether something is a statistical mixture, which is independent of basis. If your system is described by a density matrix $\rho$, a pure state has the property that $\rho^2 = \rho$.

The important thing is whether an experiment can detect this.

 

[stevendaryl]

The important thing is whether an experiment can detect this.

For microscopic objects, we can definitely demonstrate the difference. As I said already, an interference pattern is due to a superposition of two different sources of light (or particles).

For macroscopic objects, there is no way in practice to demonstrate interference between macroscopically distinguishable states, so the distinction between superpositions and mixtures can't be demonstrated.

However, it seems strange to me, or even inconsistent, to assume that macroscopic objects can only be in proper mixtures, while microscopic objects can be in superpositions. It doesn't make sense to me to have different rules for macroscopic systems.

 

[stevendaryl]

Thanks for this. I'll think about it but I doubt it will change my views. In fact the last sentence just reinforces the fact that the 'state' you talk about is only a statistical statement - however it came about.

I don't think that's a consistent interpretation of quantum probabilities. To say that "spin-up in the x-direction" means "50/50 probability of being spin-up or spin-down in the z-direction" doesn't make a lot of sense to me.

 

[Mentz114]

For microscopic objects, we can definitely demonstrate the difference. As I said already, an interference pattern is due to a superposition of two different sources of light (or particles).

Classical light is a wave and can form super-positions so it's not very relevant and is an exception where super-position actually happens.

For macroscopic objects, there is no way in practice to demonstrate interference between macroscopically distinguishable states, so the distinction between superpositions and mixtures can't be demonstrated.

OK.

However, it seems strange to me, or even inconsistent, to assume that macroscopic objects can only be in proper mixtures, while microscopic objects can be in superpositions. It doesn't make sense to me to have different rules for macroscopic systems.

If you insist on using the same rules for macro/micro contexts then you should use only statistical mechanics when doing a classical problem.
Also, you should never assume the position has a value etc.

I don't think that's a consistent interpretation of quantum probabilities. To say that "spin-up in the x-direction" means "50/50 probability of being spin-up or spin-down in the z-direction" doesn't make a lot of sense to me.

Nor me. But you said it. Maybe a typo ?

 

[Nugatory]

Nor me. But you said it. Maybe a typo ?

No matter who said it, it sounds to me more like one of the innumerable pitfalls of using natural language than anything else. '"Spin-up in the x-direction" means "50/50 probability of being spin-up or spin-down in the z-direction"' doesn't make a lot of sense to me either because of the problem with the word "being"... But "A particle prepared in the state 'spin-up in the x-direction' will, if measured in the z-direction, yield spin up or spin down with 50/50 probability" works a lot better.

 

[stevendaryl]

Classical light is a wave and can form super-positions so it's not very relevant and is an exception where super-position actually happens.

But you would agree that a classical notion of superposition is not statistical. Classically, light doesn't have a certain probability of going through one slit or another. Superposition is NOT about statistics, classically or quantum-mechanically.

 

[stevendaryl]

If you insist on using the same rules for macro/micro contexts then you should use only statistical mechanics when doing a classical problem. Also, you should never assume the position has a value etc.

I really don't understand what you're talking about. At the microscopic level, superpositions are NOT about statistics. So the mystery is how do they become about statistics at the macroscopic level?

 

[stevendaryl]

I said:

I don't think that's a consistent interpretation of quantum probabilities. To say that "spin-up in the x-direction" means "50/50 probability of being spin-up or spin-down in the z-direction" doesn't make a lot of sense to me.

@Mentz114 responded:

Nor me. But you said it. Maybe a typo ?

Then I have no idea what you mean by saying that superpositions are statistical. You previously said (about what it might mean for a macroscopic detector be in a superposition of an "Up" state and a "Down" state):

It means that each run in the ensemble produces a Up or Down, and the relative frequencies will converge to some probabiities.

It seems to me that you are saying two things that seem mutually inconsistent:

1. For a macroscopic device, being in a superposition of an "Up" state and a "Down" state means that for every run, it's either in the "Up" state or the "Down" state, with probabilities given by the relevant coefficients.
2. For a single particle, being in a superposition of "spin-up in the z-direction" and "spin-down in the z-direction" does NOT mean that for every run, it's either in the "spin-up in the z-direction" state or "spin-down in the z-direction" state, with certain probability give by the relevant coefficients.

You are explain macroscopic superpositions as being statistical, but microscopic superpositions are NOT statistical.

 

[stevendaryl]

No matter who said it, it sounds to me more like one of the innumerable pitfalls of using natural language than anything else. '"Spin-up in the x-direction" means "50/50 probability of being spin-up or spin-down in the z-direction"' doesn't make a lot of sense to me either because of the problem with the word "being"... But "A particle prepared in the state 'spin-up in the x-direction' will, if measured in the z-direction, yield spin up or spin down with 50/50 probability" works a lot better.

I said it because that was how I interpreted @Mentz114 when he said that the meaning of a superposition was statistical. There is nothing statistical about spin-up in the x-direction being a superposition of spin-up in the z-direction and spin-down in the z-direction.

 

[PeterDonis]

The fact that super-position can be created by a change of basis shows that super-position has no physical significance.

This is true, but it does not imply this:

FAPP a super-position is a statistical mixture.

You can do experiments that distinguish pure states from mixed states. A superposition is a pure state (more precisely, a pure state can be said to be a superposition with an appropriate choice of basis). It is not a mixed state, which is what the term "statistical mixture" denotes.

 

[stevendaryl]

You can do experiments that distinguish pure states from mixed states. A superposition is a pure state (more precisely, a pure state can be said to be a superposition with an appropriate choice of basis). It is not a mixed state, which is what the term "statistical mixture" denotes.

Yes, but the miracle of decoherence, if you want to consider it that, is that a superposition involving an astronomical number of degrees of freedom is FAPP indistinguishable from a statistical mixture. So once a system has interacted sufficiently with the environment, it can be treated as if it were in a statistical mixture.

 

[Mentz114]

It seems to me that you are saying two things that seem mutually inconsistent:

1. For a macroscopic device, being in a superposition of an "Up" state and a "Down" state means that for every run, it's either in the "Up" state or the "Down" state, with probabilities given by the relevant coefficients.
2. For a single particle, being in a superposition of "spin-up in the z-direction" and "spin-down in the z-direction" does NOT mean that for every run, it's either in the "spin-up in the z-direction" state or "spin-down in the z-direction" state, with certain probability give by the relevant coefficients.

You explain macroscopic superpositions as being statistical, but microscopic superpositions are NOT statistical.

I don't understand.

To me - a single particle being in a superposition of "spin-up in the z-direction" and "spin-down in the z-direction" means EXACTLY and ONLY that it will be detected in either of those states with probability 1/2.

I guess I'm missing the point.

Post #26 by VanHees71 in the other topic https://www.physicsforums.com/threads/are-superposition-states-observable.909502 cleared some things for me - but not in a way you would necessarilly agree with.