Colapse of the Wave Funcion and the Schroedinger Equation

[wofsy]

I am trying to understand why the collapse of the wave function can not be a solution to the Shroedinger equation.

Certainly there are systems that evolve into eigenstates. For instance, in a two state system with constant Hamiltonian there are initial conditions from which the amplitudes oscillate back and forth and pass through eigen states periodically.

In the Stern-Gerlach apparatus a magnetic field separates particles into spin directions.Blocking one of the directions with a barrier selects for the other spin eigen state. But all of this seems to be a solution of the Shroedinger equation.

 

[conway]

This is a good question.

I agree that the Stern-Gerlach experiment is consistent with the normal time-evolution of an electron beam as governed by the Schroedinger equation. The beam naturally divides into two branches.

It gets trickier if you try to analyze it in terms of what happens "one electron at a time". People think you need to invoke a collapse of the wave function, but they tend to forget that it's not always so easy to know when you have exactly one electron.

 

[ZapperZ]

It gets trickier if you try to analyze it in terms of what happens "one electron at a time". People think you need to invoke a collapse of the wave function, but they tend to forget that it's not always so easy to know when you have exactly one electron.

Actually, we do! We know it well enough that we are making use of it!

http://physicsworld.com/cws/article/news/25159
http://physicsworld.com/cws/article/news/31720
http://physicsworld.com/cws/article/print/129

Detecting the state of one, single electron is no longer a big deal.

Zz.

 

[meopemuk]

I am trying to understand why the collapse of the wave function can not be a solution to the Shroedinger equation.

Collapse of the wave function is purely random, and there is no dynamical equation that can describe it.

This situation is very similar to the "collapse" of probability distributions in classical statistical mechanics. For example, such a classical "collapse" occurs when you throw a die on the table. Of course, the difference is that in the classical case there is a theory (classical mechanics) that is more fundamental than statistical mechanics. This more fundamental theory tells the die's probability distribution how to "collapse".

In the quantum case there is no more fundamental theory than quantum mechanics (unless you subscribe to "hidden variables"). So, the collapse is a random unpredictable event. Nature is not always predictable. That's the main lesson of quantum mechanics.

 

[zenith8]

In the quantum case there is no more fundamental theory than quantum mechanics (unless you subscribe to "hidden variables"). So, the collapse is a random unpredictable event. Nature is not always predictable. That's the main lesson of quantum mechanics.

Remember though that 'subscribing to "hidden variables"' - though you make it sound so disreputable - merely involves acknowledging that electrons exist when no-one is looking at them. Which, let's be honest, is hardly the moral equivalent to believing that the Moon is made of green cheese, and there's no actual fundamental reason to imagine they don't other than that Bohr told you so.

Now of course, the Schroedinger evolution of the wave function in time gives a linear superposition of all possibilities for ever, and when correlated with the measuring apparatus, you get a macroscopic superposition of quantum states, which is not what one sees. If you believe that electrons exist as well - as in the pilot-wave theory/Bohm interpretation - then they serve merely to pick out which branch (say in an SG experiment) is actually observed. Just because the electron trajectory dynamically ends up there - the only random bit is where the electron starts out. And so QM is just like statistical mechanics but with a non-classical underlying dynamics. Hence no more collapse problem.

The collapse hypothesis only gains physical content if actual coordinates for the collapsed system are posited. Schroedinger once thought that a cat was a big enough pointer to get that point across. More fool him..

 

[meopemuk]

acknowledging that electrons exist when no-one is looking at them.

This is exactly the central philosophical message of quantum mechanics: It does not make sense to talk about how physical systems look "when no-one is looking at them". Physics is a science about observations. Things that cannot be observed even in principle (ghosts, angels, electrons in the absence of a measuring device, etc.) should be left to theologians.

 

[Hurkyl]

Remember though that 'subscribing to "hidden variables"' - though you make it sound so disreputable - merely involves acknowledging that electrons exist when no-one is looking at them. Which, let's be honest, is hardly the moral equivalent to believing that the Moon is made of green cheese, and there's no actual fundamental reason to imagine they don't other than that Bohr told you so.

No, "hidden variables" means that the state of a system isn't completely described by the wave-function -- there are additional (hidden) variables that affect the results of experiments.

Bell's theorem puts some rather severe constraints on what properties such hidden variables can have.

 

[wofsy]

Collapse of the wave function is purely random, and there is no dynamical equation that can describe it.

This situation is very similar to the "collapse" of probability distributions in classical statistical mechanics. For example, such a classical "collapse" occurs when you throw a die on the table. Of course, the difference is that in the classical case there is a theory (classical mechanics) that is more fundamental than statistical mechanics. This more fundamental theory tells the die's probability distribution how to "collapse".

In the quantum case there is no more fundamental theory than quantum mechanics (unless you subscribe to "hidden variables"). So, the collapse is a random unpredictable event. Nature is not always predictable. That's the main lesson of quantum mechanics.

I still don't see why the collapse can't be a solution of the Shroedinger equation.

Further, in the Bohm formulation everything is deterministic.

 

[wofsy]

This is exactly the central philosophical message of quantum mechanics: It does not make sense to talk about how physical systems look "when no-one is looking at them". Physics is a science about observations. Things that cannot be observed even in principle (ghosts, angels, electrons in the absence of a measuring device, etc.) should be left to theologians.

I don't see your point. Certainly i can put a measuring device on a star and measure collapses. No observer is necessary.

 

[meopemuk]

And so QM is just like statistical mechanics but with a non-classical underlying dynamics.

This is a respectable hypothesis. However, it hasn't produced a single verifiable prediction in many decades of its existence. It would be a different matter if such an "underlying dynamics" could predict (at least, with some non-trivial accuracy) the timings of clicks in the Geiger counter or positions of flashes on the double-slit experiment screen. So far, the "hidden variable" hypothesis hasn't moved beyond philosophical bla-bla. This gives more credence to the idea that individual quantum events are truly random and do not obey any dynamical law.

 

[Hurkyl]

I still don't see why the collapse can't be a solution of the Shroedinger equation.

Collapse is not unitary -- in particular, it is not a mathematically invertible operation.

Time evolution according to the Schrödinger equation is unitary.

Therefore, collapse cannot occur in a system evolving according to the Schrödinger equation.



(However, do note that unitary evolution can lead to decoherence)

 

[wofsy]

Collapse is not unitary -- in particular, it is not a mathematically invertible operation.

Time evolution according to the Schrödinger equation is unitary.

Therefore, collapse cannot occur in a system evolving according to the Schrödinger equation.



(However, do note that unitary evolution can lead to decoherence)

Explain what unitary evolution is.

It seems you assuming what you want to prove. Why can't you have a complicated set of potentials that produce what looks like a collapse. this would redefine collapse or better put would get rid of it.

 

[meopemuk]

Further, in the Bohm formulation everything is deterministic.

Can the "Bohm formulation" predict positions of electron flashes on the double-slit experiment screen? Can it predict times of clicks in the Geiger counter? I guess not. So, it is not helpful in understanding physics. We can just as well assume that these events are completely random.

 

[Hurkyl]

Why can't you have a complicated set of potentials that produce what looks like a collapse.

Never said you couldn't. Do note that I mentioned decoherence...

 

[meopemuk]

I don't see your point. Certainly i can put a measuring device on a star and measure collapses. No observer is necessary.

An "observation" or "measurement" is complete only after you obtained the information about its results. If you placed a measuring device on a distant planet, and you have no means to communicate with it, then there is no measurement, no collapse, no nothing. Your knowledge about the physical world has not advanced a bit. Again, you have a situation "when no-one is looking". Such situations are of no interest to physics.

 

[wofsy]

Can the "Bohm formulation" predict positions of electron flashes on the double-slit experiment screen? Can it predict times of clicks in the Geiger counter? I guess not. So, it is not helpful in understanding physics. We can just as well assume that these events are completely random.

Yeah bit the same argument can be made against classical mechanics. In all but the simplist configuations outcomes are not predicatble. In mechanical chaos arbitrarily small changes in intitial conditions lead to arbitrarily large differences in outcomes. So your argument says that except in simple cases all of Physics is useless.

 

[meopemuk]

Yeah bit the same argument can be made against classical mechanics. In all but the simplist configuations outcomes are not predicatble. In mechanical chaos arbitrarily small changes in intitial conditions lead to arbitrarily large differences in outcomes. So your argument says that except in simple cases all of Physics is useless.

I agree that predictions of classical mechanics are not exact. However, they are accurate enough to build machines, fly spaceships, and predict orbits of planets.

In the case of quantum events (like flashes on the screen or Geiger counter clicks), the best we can do is to predict probabilities. That's what quantum mechanics does brilliantly.

You can believe that there is an underlying "hidden variable" theory behind quantum mechanics, and that this theory would allow us to go beyond probabilities. However, if we judge physical theories by their agreement with experiment, then "hidden variables" was a spectacular failure. So far it couldn't predict even a single experimental number.

 

[crazy_photon]

An "observation" or "measurement" is complete only after you obtained the information about its results. If you placed a measuring device on a distant planet, and you have no means to communicate with it, then there is no measurement, no collapse, no nothing. Your knowledge about the physical world has not advanced a bit. Again, you have a situation "when no-one is looking". Such situations are of no interest to physics.

sorry to barge in the middle of the discussion, but i also have to say that i don't see your point as well. of course any theory in physics needs measurement to verify its predictions, but once we have such theory i don't see why i can't be describing things 'without looking'.

maybe an example can help? if i (and you in your lab) independently measure same diffraction pattern electrons have produced after the slit(s) (and we were careful to keep experimental conditions the same) and we construct a theory that describes that pattern then (until someone has measured otherwise) why do i need to keep looking at that electron?

 

[meopemuk]

sorry to barge in the middle of the discussion, but i also have to say that i don't see your point as well. of course any theory in physics needs measurement to verify its predictions, but once we have such theory i don't see why i can't be describing things 'without looking'.

maybe an example can help? if i (and you in your lab) independently measure same diffraction pattern electrons have produced after the slit(s) (and we were careful to keep experimental conditions the same) and we construct a theory that describes that pattern then (until someone has measured otherwise) why do i need to keep looking at that electron?

My point is that physical theory must give predictions about observable things (like the shape of the diffraction pattern). However, it is not obliged to tell you about things that are not being observed. In your example you shouldn't ask "which slit the electron passed through?" Whatever the answer may be, there is no way to verify the validity of that answer (within your described experimental setup).

 

[crazy_photon]

My point is that physical theory must give predictions about observable things (like the shape of the diffraction pattern). However, it is not obliged to tell you about things that are not being observed. In your example you shouldn't ask "which slit the electron passed through?" Whatever the answer may be, there is no way to verify the validity of that answer (within your described experimental setup).

OK, agree 100%, i guess what triggered my response was the language issue, i would say 'things that are not observable' as opposed to 'things that are not observed'. anyway, i see misunderstood your point based on that (language issue).

 

[Fredrik]

Explain what unitary evolution is.

It's when the state changes with time according to $\psi(t)=U(t)\psi$ where U(t) is a unitary operator for all t. An operator U is unitary if $U^\dagger U=UU^\dagger=1$. The Schrödinger equation says that U(t)=$e^{-iHt}$, where H is the Hamiltionian. H must be Hermitian ($H^\dagger=H$), otherwise it wouldn't have real eigenvalues. It's easy to verify that U(t) is unitary for all t if H is hermitian.

If you're thinking, "hey, why not just let H be non-hermitian", that sort of thing is ruled out by a theorem that Wigner proved a long time ago. His theorem implies that U(t) must be unitary for all t, unless time translation invariance isn't really a symmetry of spacetime. We know that it's at least an approximate symmetry because without it, the concept of energy as we know it wouldn't exist.

Why can't you have a complicated set of potentials that produce what looks like a collapse.

A potential is just a part of H, which must be hermitian, and that makes U(t) unitary for all t.

 

[wofsy]

I agree that predictions of classical mechanics are not exact. However, they are accurate enough to build machines, fly spaceships, and predict orbits of planets.

In the case of quantum events (like flashes on the screen or Geiger counter clicks), the best we can do is to predict probabilities. That's what quantum mechanics does brilliantly.

You can believe that there is an underlying "hidden variable" theory behind quantum mechanics, and that this theory would allow us to go beyond probabilities. However, if we judge physical theories by their agreement with experiment, then "hidden variables" was a spectacular failure. So far it couldn't predict even a single experimental number.

in mechanical chaos predictions are probabalistic at best. In some situations you can't even do that.

 

[wofsy]

It's when the state changes with time according to $\psi(t)=U(t)\psi$ where U(t) is a unitary operator for all t. An operator U is unitary if $U^\dagger U=UU^\dagger=1$. The Schrödinger equation says that U(t)=$e^{-iHt}$, where H is the Hamiltionian. H must be Hermitian ($H^\dagger=H$), otherwise it wouldn't have real eigenvalues. It's easy to verify that U(t) is unitary for all t if H is hermitian.

If you're thinking, "hey, why not just let H be non-hermitian", that sort of thing is ruled out by a theorem that Wigner proved a long time ago. His theorem implies that U(t) must be unitary for all t, unless time translation invariance isn't really a symmetry of spacetime. We know that it's at least an approximate symmetry because without it, the concept of energy as we know it wouldn't exist.


A potential is just a part of H, which must be hermitian, and that makes U(t) unitary for all t.

so what? in my original example of the Stern-Gerlach experiment an eigen state of spin is isolated using a solution to the Shroedinger equation.

 

[wofsy]

here's one point that may step this dialogue forward.
A solution to the Shroedinger equation must be smooth except perhaps at the initial condition.This is a mathematical theorem. (The same truth applies to the heat equation and in fact the Shroedinger equation is a complex heat heat equation. It is not really a wave equation.) If by a collapse one means a discontinuity in the evolution then this can not be a solution of the Shroedinger equation.

 

[Hurkyl]

so what? in my original example of the Stern-Gerlach experiment an eigen state of spin is isolated using a solution to the Shroedinger equation.

You're referring to your opening post?


I think you're missing a fundamental thing. A "solution to the Schrödinger equation" is not a quantum state -- it is a function that assigns a quantum state to every possible value value of time. Schrödinger's equation doesn't tell you what the quantum states are¹ -- it tells you how they change over time.


And, by the way, this is wrong:

in a two state system with constant Hamiltonian there are initial conditions from which the amplitudes oscillate back and forth and pass through eigen states periodically.

in any system with a constant Hamiltonian, a solution to the Schrödinger equation is either:
(1) Always in an eigenstate, never changing which eigenstate it's in
(2) Never in an eigenstate


1: Nitpick: okay, I think it does say that quantum states have to have twice differentiable wavefunctions

 

[meopemuk]

If by a collapse one means a discontinuity in the evolution then this can not be a solution of the Shroedinger equation.

You are absolutely right. The collapse of the wave function cannot be described by the Schroedinger equation. Both unitary evolution and the collapse are two important parts of the quantum formalism. You cannot get rid of the collapse without undermining the logical structure of QM.

Perhaps it is useful to remember that quantum formalism is just an abstract mathematical model of reality, rather than reality itself. Wave functions and Hilbert spaces cannot be found anywhere in nature. The unitary evolution and the wave function collapse are not physical processes.

 

[conway]

You're referring to your opening post?


And, by the way, this is wrong:

in any system with a constant Hamiltonian, a solution to the Schrödinger equation is either:
(1) Always in an eigenstate, never changing which eigenstate it's in
(2) Never in an eigenstate

But you knew what he meant, didn't you?

 

[Bob_for_short]

...Collapse of the wave function is purely random, and there is no dynamical equation that can describe it.

...The collapse of the wave function cannot be described by the Schroedinger equation. ...

Nature is not always predictable. That's the main lesson of quantum mechanics.

Collapse is not unitary -- in particular, it is not a mathematically invertible operation.

Time evolution according to the Schrödinger equation is unitary.

If by a collapse one means a discontinuity in the evolution then this can not be a solution of the Shroedinger equation.

I like the meopemuk image attached below. It explains everything about collapse.

Normally one thinks that the electron is localized within its trace (localized, collapsed wave function). Before collapsing the wave function is more spread, is it not? The particle energy is the volume integral. Why do we think that the wave function shrinks? It can be as spread as before. Let me explain in simple terms: I take a chalk and draw a line on a blackboard. Is it reasonable to think that the total energy is localized at the point of touching the blackboard? No. It is my energy, spread over entire me. And the blackboard, it is also a solid and spread object. It cannot be reduced to a touching point solely. A small and separated piece of the blackboard will not stay still but will move if I try to draw a line on it. So the line is a product of local interaction of spread objects, the energy of each is a volume integral over space much larger than the line size. I am afraid there is no collapse in your meaning. A thin line is a result of collective behavior - of big mine and of the big blackboard's.

Now, Hamiltonian of a detector is a multi-particle Hamiltonian. There are so many degrees of freedom that any process of excitation relaxation is irreversible.

The "discontinuity" of evolution is dictated with "discontinuity" of the total Hamiltonian: in front of the detector there is nothing (V=0), and withing the detector there are potential and kinetic terms of the total Hamiltonian. The initially prepared energy of a projectile is gradually distributed over infinite number of degrees of freedom of the detector initially prepared at different space (its wave function occupies different space), so the time evolution is unitary but irreversible.

The local interaction point is not predictable, indeed, but we must go farther and say: Nature is never predictable. That's the main lesson of quantum mechanics which deals with separate events, not with inclusive picture. The QM inclusive picture (an average picture) is indeed deterministic but poor - it is CM picture (see "Atom as a "dressed" nucleus" by Vladimir Kalitvianski).

Bob_for_short.

 

[Fredrik]

so what? in my original example of the Stern-Gerlach experiment an eigen state of spin is isolated using a solution to the Shroedinger equation.

If a single silver atom passes through a Stern-Gerlach apparatus, its state (which I will write in the form |position,spin>) changes from |center,s> to a superposition of |right,+> and |left,->. This is not a measurement. It's just an interaction that produces a correlation between spin eigenstates and position eigenstates*, which enables you to determine the spin by measuring the position. A measurement of the spin is an interaction which produces a correlation between the spin eigenstates and macroscopically distinguishable states of a system that for all practical purposes can be treated as classical.

*) They aren't really position eigenstates, but they are as close as you can get. They are superpositions of narrow ranges of position eigenstates.

For example, you can measure the spin state of a silver atom by letting it pass through a Stern-Gerlach apparatus and allowing it to be detected by one of two detectors that you have put in the two relevant positions. Each detector can always be described as either having detected a particle or not having detected a particle.

So what the Stern-Gerlach apparatus does isn't a measurement. It's a state preparation. This concept is more clear if we consider an ensemble of systems instead of an individual system. If we send a beam of silver atoms into the apparatus, it gets split in two. The spin state of the initial beam is represented by the statistical operator (a.k.a. density operator, density matrix or mixed state) |+><+|+|-><-|. The state of the beam that comes out on the right is represented by the pure state |+><+|. What this means, at least for those of us who think of both state vectors and statistical operators as representing the probabilities of different measurement results (rather than as representing objective properties of physical systems) is that the "reduction of the state vector", "collapse of the wavefunction", or whatever you'd like to call it, isn't even an interaction that the system is involved in. It's a selection of a subset of the ensemble, on which we indend to perform measurements.

Suppose that we only send one atom through, and that its initial state is |+>+|->. I like to think of a state vector as representing the probabilities of different results (and not as an objective property of the system), but this is equivalent to saying that the state vector represents an objective property of an ensemble of identically prepared systems. In other words, we have to imagine a large number of identical experiments being performed. The set of all silver atoms in all those experiments is an ensemble, and the "collapse of the wavefunction" is a selection of a subset of that ensemble.

By the way, what I just described here is also described in the excellent book "Lectures on quantum theory: mathematical and structural foundations" by Chris Isham, which I read about a week ago. I had not understood the importance of distinguishing between state preparation and measurement until I read it.

 

[wofsy]

You're referring to your opening post?


I think you're missing a fundamental thing. A "solution to the Schrödinger equation" is not a quantum state -- it is a function that assigns a quantum state to every possible value value of time. Schrödinger's equation doesn't tell you what the quantum states are¹ -- it tells you how they change over time.


And, by the way, this is wrong:

in any system with a constant Hamiltonian, a solution to the Schrödinger equation is either:
(1) Always in an eigenstate, never changing which eigenstate it's in
(2) Never in an eigenstate


1: Nitpick: okay, I think it does say that quantum states have to have twice differentiable wavefunctions

thanks for the correction.

a collapsed wave function is still a wave function. I am not missing the point.

 

[wofsy]

If a single silver atom passes through a Stern-Gerlach apparatus, its state (which I will write in the form |position,spin>) changes from |center,s> to a superposition of |right,+> and |left,->. This is not a measurement. It's just an interaction that produces a correlation between spin eigenstates and position eigenstates*, which enables you to determine the spin by measuring the position. A measurement of the spin is an interaction which produces a correlation between the spin eigenstates and macroscopically distinguishable states of a system that for all practical purposes can be treated as classical.

*) They aren't really position eigenstates, but they are as close as you can get. They are superpositions of narrow ranges of position eigenstates.

For example, you can measure the spin state of a silver atom by letting it pass through a Stern-Gerlach apparatus and allowing it to be detected by one of two detectors that you have put in the two relevant positions. Each detector can always be described as either having detected a particle or not having detected a particle.

So what the Stern-Gerlach apparatus does isn't a measurement. It's a state preparation. This concept is more clear if we consider an ensemble of systems instead of an individual system. If we send a beam of silver atoms into the apparatus, it gets split in two. The spin state of the initial beam is represented by the statistical operator (a.k.a. density operator, density matrix or mixed state) |+><+|+|-><-|. The state of the beam that comes out on the right is represented by the pure state |+><+|. What this means, at least for those of us who think of both state vectors and statistical operators as representing the probabilities of different measurement results (rather than as representing objective properties of physical systems) is that the "reduction of the state vector", "collapse of the wavefunction", or whatever you'd like to call it, isn't even an interaction that the system is involved in. It's a selection of a subset of the ensemble, on which we indend to perform measurements.

Suppose that we only send one atom through, and that its initial state is |+>+|->. I like to think of a state vector as representing the probabilities of different results (and not as an objective property of the system), but this is equivalent to saying that the state vector represents an objective property of an ensemble of identically prepared systems. In other words, we have to imagine a large number of identical experiments being performed. The set of all silver atoms in all those experiments is an ensemble, and the "collapse of the wavefunction" is a selection of a subset of that ensemble.

By the way, what I just described here is also described in the excellent book "Lectures on quantum theory: mathematical and structural foundations" by Chris Isham, which I read about a week ago. I had not understood the importance of distinguishing between state preparation and measurement until I read it.

OK. I understand this. However if I put a barrier inside the Stern-Gerlach apparatus that blocks the left trajectory then I have collapsed the wave function of any particle that exits the apparatus. The barrier is perfectly consistent with the Shroedinger equation. So without measurement using a detector I can collapse the wave function.

 

[wofsy]

I do not feel that this thread has gone in the right direction. I understand the what the postulates of quantum mechanics say. But these postulates didn't fall from the sky. They were arrived at after soul searching reflection. It is clear that the collapse of the wave function, the salient effect of a measurement, was proved to be outside of the possible solutions of the Shroedinger equation even with arbitrarily complex potentials added in.

That was my question. This must be a purely mathematical theorem.

But also it is clear that physicists decided that total collapse was what actually happened rather than approximate collapse and it was only later that some people proposed that approximate collapse could occur spontaneously. But this is modeled as a random even independent of the shroedinger equation. So it must also be that approximate collapse is not possible in the Shroedinger equation either.

In my feeble examples I tried to construct collapses that were in fact solutions of the Shroedinger equation but those examples are stabs and it doesn't really matter if they are right or wrong.

What is important is the theorem that says that collapses and approximate collapses are not solutions of the Shroedinger equation. Any ideas?

 

[Fredrik]

OK. I understand this. However if I put a barrier inside the Stern-Gerlach apparatus that blocks the left trajectory then I have collapsed the wave function of any particle that exits the apparatus. The barrier is perfectly consistent with the Shroedinger equation. So without measurement using a detector I can collapse the wave function.

When a silver atom is sent through the SG apparatus, its state is changed according to

$$|\mbox{center}\rangle|s\rangle\rightarrow|\mbox{right}\rangle|+\rangle+|\mbox{left}\rangle|-\rangle$$

where

$$|s\rangle=|+\rangle+|-\rangle$$

This is a unitary process, meaning that the state vector on the right is just

$$e^{-iHt}|\mbox{center}\rangle|s\rangle$$

where the Hamiltonian is defined by the SG apparatus. This process is not a measurement. It's just a state preparation. A state preparation of an ensemble E is an interaction that that partitions E into smaller ensembles E_i that have the property that a subsequent measurement of some observable A on E_i would yield the result a_i with certainty. A measurement is an interaction that also entangles the states that the E_i are in with macroscopically distinguishable states of a system that's approximately classical.

The interaction that prepares the state can be a measurement, but it doesn't have to be. In the Stern-Gerlach experiment, it isn't. If it was, the position of the silver atom would be either "right" or "left". It wouldn't be a superposition of both.

I said before that the collapse of the wavefunction is just a selection of one of the E_i. I think I'm going to have to take that back. A collapse is a transition from a pure state to a mixed state, but we don't get a mixed state just by choosing one of the beams.

You're suggesting that we insert a barrier that prevents the right beam from leaving the SG apparatus. I'm not sure what the effect of the barrier would be. Does it or does it not entangle the spin eigenstates with macroscopically distinguishable states of the barrier? I don't know. If it does, then it has collapsed the wavefunction, but this isn't a collapse without a detector. In this case the barrier is the detector. It's just a really bad one, in the sense that doesn't make it easy for you to find out what the result was.

 

[Fredrik]

That was my question. This must be a purely mathematical theorem.
...
What is important is the theorem that says that collapses and approximate collapses are not solutions of the Shroedinger equation. Any ideas?

If the system has been prepared in state $|\psi\rangle$, the SE says that the corresponding statistical operator (density matrix) evolves according to

$$|\psi\rangle\langle\psi|\rightarrow e^{-iHt}|\psi\rangle\langle\psi|e^{iHt}$$

The operator on the right is a pure state, i.e. it's the projection operator of a 1-dimensional subspace. A measurement on the other hand, changes the state according to

$$|\psi\rangle\langle\psi|\rightarrow\sum_aP_a|\psi\rangle\langle\psi|P_a$$

where $P_a=|a\rangle\langle a|$ is the projection operator of the 1-dimensional subspace that contains the eigenvector with eigenvalue a. Here the operator on the right isn't a pure state.

 

[zenith8]

I do not feel that this thread has gone in the right direction. I understand the what the postulates of quantum mechanics say. But these postulates didn't fall from the sky. They were arrived at after soul searching reflection. It is clear that the collapse of the wave function, the salient effect of a measurement, was proved to be outside of the possible solutions of the Shroedinger equation even with arbitrarily complex potentials added in.

That was my question. This must be a purely mathematical theorem.

But also it is clear that physicists decided that total collapse was what actually happened rather than approximate collapse and it was only later that some people proposed that approximate collapse could occur spontaneously. But this is modeled as a random even independent of the shroedinger equation. So it must also be that approximate collapse is not possible in the Shroedinger equation either.

In my feeble examples I tried to construct collapses that were in fact solutions of the Shroedinger equation but those examples are stabs and it doesn't really matter if they are right or wrong.

What is important is the theorem that says that collapses and approximate collapses are not solutions of the Shroedinger equation. Any ideas?

Let us first assume the wave function is a real field (a 'real wave' if you like). Of course you can say that it isn't and that QM is just a way of cataloguing observations from an unknown underlying 'mechanism' which one should refuse to speculate about - but (a) that doesn't get you anywhere and is a bit boring, (b) you then have no way of explaining how you get a perfectly standard interference pattern in e.g. a two-slit experiment. What exactly is interfering with what, if it isn't a real wave passing through the slits? To say that 'nothing passes through the slits' as is often done is simply silly - clearly something does and to say otherwise is merely to play with words (imperfectly quoting Deutsch).

So OK - the wave function objectively exists - then the Schroedinger evolution in time will give you a wave representing all possible outcomes of the experiment (in your example, a finite lump of wave in both branches of the SG apparatus). When this is coupled to a macroscopic apparatus (e.g. detect the position of your electron/silver atom with a phosphorescent screen) then the Schroedinger time evolution predicts a macroscopic superposition of everything that could happen, which is not what you see.

The standard way of getting round this is to say (effectively) "Ah well, when we observe it, the wave function er.. stops evolving according to the time-dependent Schroedinger equation, and er.. does something else - it collapses.". This is not really a solution - though often presented as such - it is merely stating that what you see is different from what the Schroedinger equation predicts (what do you need to do to make this happen, exactly? How is this different from just any ordinary many-body interaction?) .

At this point someone will bring in 'decoherence' to save the day - however this merely says that the different 'branches' of the wave function will tend to become orthogonal in time (i.e. they cease to have a finite overlap integral) thus you can't subsequently bring the different branches to interference. But so what? All the different branches continue to exist.

Then in desperation, someone will say "Ah well, you see, er.. every branch of the wave function then forms er.. its own separate parallel universe and disappears off into that". He ought to be carted off in a straightjacket at this point, but somehow manages through sheer bloody mindedness to convince himself that this is the only way to understand QM. Well OK, you can do this if you want, but it really should be absolutely the last resort in the absence of any simpler and much less bizarre explanation.

In reality you have - or should have - just two genuine options to explain what you see (I imperfecty quote Bell):

(1) the wave function is not all there is (--> hidden variables, with pilot-wave/Bohm theory being the only mainstream example),

or

(2) The Schroedinger equation - note the c - is wrong (--> objective collapse models - GRW etc.).

Which of these two you prefer is up to you. I like the former because it's just simple, and all the usual paradoxes disappear. Just say that particles (electrons or whatever) exist and are guided by the wave. The wave evolves according to the Schroedinger equation. The particles follow trajectories derived from the probability current (the guidance equation). The particle deterministically ends up in one branch or the other. The wave function doesn't collapse at all, but it 'effectively' does because the particles end up being guided only by one of the branches (remember decoherence has made them not overlap).

In the two-slit experiment, the particle goes through one slit, the wave goes through both. An interference pattern develops in the wave, which affects the trajectories of the particles and guides them into clumps. In the SG experiment, the wave separates into two branches, the particles ends up in one branch or the other at random (depending on its starting position) - note we're not measuring a 'property of the particle called spin' here - which is interesting.

Note that the answer to most questions that people put on the Quantum Physics forum depend on a careful understanding of the various interpretations of QM. So which ever moderator keeps shoving discussions of them into the Philosophy section (e.g. this morning's Interpretation Poll - whose originator has made a similar complaint) should in my opinion stop doing so. How does stopping people discussing the meaning of a theory contribute to its understanding?

For what it's worth, both the above options (1) and (2) make specific experimentally testable predictions (albeit with a large degree of practical difficulty) and as such should be taken to be physics - not philosophy. To say otherwise and maintain that all discussion of 'mechanism' is meaningless is merely to maintain the discredited Bohrian attitude of the 1960s and before. And you don't want to do that, do you? Keep up! Things are moving on..