Los Frupanishads

[nurbldoff]

If we had complete knowledge of the conditions, why wouldn't we be able to predict the motion?

I guess we would, if we had a perfect model and unlimited computing power. But this knowledge seems to be fundamentally unattainable.

[Bebek Sincap Ratatosk]

If we had complete knowledge of the conditions, why wouldn't we be able to predict the motion?

I guess we would, if we had a perfect model and unlimited computing power. But this knowledge seems to be fundamentally unattainable.

Nope, Heisenberg's Principal would show up and give all the models afterschool detention.

[LMNO]

I have a rant about that, by the way.

Hiesenberg used incredibly crappy metaphors in describing what's going on.

I really should just save these things and repost where needed.

The "uncertainty principle"

The Schrödinger field pattern in position space determines where a detection event is likely to be found, and its pattern in wavelength space determines the momentum we associate with the object causing the event. If the events are localized in a small region, the wave pattern will be localized but consequently it will contain many elementary waves – its momentum will not be well-defined.

Conversely, if the momentum detector clicks only for a narrow range of momentum values, the wavelength is well-defined, and the wave pattern must extend over many cycles – its location in space is not well-defined. You can have waves with well-defined position or well-defined momentum, but not both at once. This is the true meaning of the uncertainty relation first enunciated in 1927 by Heisenberg.

To be precise, the relation states that among many measurements on a large set of systems each described by the same wave function the range of momentum values times the range of position values is greater than or equal to Planck's constant times a certain number.

The number depends on how you define the ranges or spreads, commonly written Δq and Δp, where q and p stand for position and momentum. For the most common statistical definition (the standard deviation of the measurements) the constant is 1/4π. Thus the algebraic formula: ΔqΔp ≥ h/4π. A similar relation exists between the range of measured energy values ΔE and the interval of time Δt during which the system is allowed to evolve undisturbed.

The "Heisenberg uncertainty relation" emerged in an atmosphere of confusion from which it has never quite escaped. Much of the fault lies with Heisenberg himself who was not content with setting forth the bare theory, more or less along the lines I have described above (but in mathematical language), he also tried to make the result more comprehensible with suggestive physical arguments.

For example, he implied that the uncertainty has its origin in the inevitable disturbance caused by the measurement process (which is not inherently a quantum concept). Bohr objected to these explanatory efforts, convinced that the matter was deeper than Heisenberg made it out to be.

As I see it, most problems of interpretation are resolved by the simple fact that the microscopic theory does not refer to any physical waves or particles. It refers to well-defined detectors and unambiguous events of detection. Accounts that ascribe position to particles and momentum to waves apply macroscopic language inappropriately to microscopic nature.

You can set a detector to register an event with well-defined momentum, or you can set it to record an event with well-defined position. That does not entitle you to say that the event is caused by a "wave" or by a "particle."

[Cramulus]

Douglas Hofstadter has an essay about how the Uncertainty Principle is really widely misunderstood.

Basically he underscores that quantum mechanical reality does not correspond to macroscopic reality. That Heisenberg was talking about quanta, not systems on the level which we can personally observe. The metaphor usually carries, but should not be seen as a law or nothin'.

[nurbldoff]

LMNO: Right. But in most actual cases, the limits of measurement are much larger than Heisenberg's. Instruments have limited resolution, there are disturbances, interference, etc. And no system (OK, maybe the universe as a whole) is completely isolated from everything else.

Cramulus: certainly. Looking at the uncertaity relation, it just tells us that the uncertainty in position times the uncertainty in momentum (=velocity*mass) is larger than or on the order of Plancks contant. Planck's constant is

really small

. That means that these uncertainties are generally of no consequence in a macroscopic system (see above).

But really, Heisenberg has nothing to do directly with chaos theory. Chaotic behaviour is in the

model

.