Colapse of the Wave Funcion and the Schroedinger Equation

1. May 30, 2009

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.

2. May 30, 2009

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.

3. May 30, 2009

ZapperZ

Staff Emeritus
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.

Last edited by a moderator: Apr 24, 2017
4. May 30, 2009

meopemuk

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.

5. May 30, 2009

zenith8

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..

6. May 30, 2009

meopemuk

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.

7. May 30, 2009

Hurkyl

Staff Emeritus
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.

8. May 30, 2009

wofsy

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.

9. May 30, 2009

wofsy

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

10. May 30, 2009

meopemuk

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.

11. May 30, 2009

Hurkyl

Staff Emeritus
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)

12. May 30, 2009

wofsy

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.

13. May 30, 2009

meopemuk

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.

14. May 30, 2009

Hurkyl

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

15. May 30, 2009

meopemuk

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.

16. May 30, 2009

wofsy

Yeah bit the same arguement 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.

17. May 31, 2009

meopemuk

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.

18. May 31, 2009

crazy_photon

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?

19. May 31, 2009

meopemuk

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).

20. May 31, 2009

crazy_photon

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).

21. May 31, 2009

Fredrik

Staff Emeritus
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.

22. May 31, 2009

wofsy

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

23. May 31, 2009

wofsy

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.

24. May 31, 2009

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.

25. May 31, 2009

Hurkyl

Staff Emeritus
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