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.