Okay, then describe an instrument that could return a value from a continuous set, without any approximations.
Then you misread my post #4.
That’s wrong. Take a free particle, measure its energy to within a certain accuracy. Call this energy E[sub]0[/sub]. Measure an equivalent particle again. Call that measurement (assumed yielding a higher energy without loss of generality) E[sub]100[/sub]. Partition the interval between E[sub]0[/sub] and E[sub]100[/sub] in sub-intervals of uniform size. This gives you a scale.
Now, discrete measurements mean that on that scale, up to measurement accuracy, you only ever measure values that, for instance, correspond to an integer multiple of E[sub]0[/sub]. Continuous measurements mean that, up to measurement accuracy, every value on the scale is possible. There are limits on the accuracy of any given measurement, of course. Those have, however, nothing to do with ‘the quantized nature of the universe’, and can be made arbitrarily small through simply repeating the measurement many times.
The discrete and the continuous case are observably different: in the many measurements limit, in the discrete case, you will observe a series of sharper growing peaks at the discrete expectation values; in the continuous case, that same limit takes you to an equidistribution.
Of course, for any finite amount of measurements, you can postulate an underlying discrete distribution; but adding more measurements will, according to quantum mechanics, falsify this hypothesis in the continuous case every time. If what you claim were right, then at some point, we should always see a discrete distribution emerge; that’s however not what quantum mechanics predicts.
See, we’re actually in agreement–as you say, each measurement has a discrete distribution. It’s only in the limit of a large number of measurements that something approaching a continuous distribution is possible.
I agree with Chronos, and was originally going to make the same point that this is more philosophy than science but felt I didn’t know enough about the various interpretations to say anything meaningful. Even the most recent discussion has more of a philosophic bent to it, and ponders the basic question about whether reality exists independent of observers. We have a model of a continuous wave function that we claim represents a physical system, and our theories and observations say that the model is accurate, but if reality does not exist outside of our measurements then all we have is a model of how our measurements will turn out. Since we then have to specify a physical measurement device to complete our model, we only have a discrete number of possible observables.
Taking the viewpoint that Bell’s inequality violations imply no objective reality beyond what was measured is, from what I understand, a perfectly consistent philosophy. It’s a major logical impediment to doing most scientific research and I don’t think that people actually believe it, but I see it as truly a philosophic issue.
This has, however, nothing to do with quantum mechanics – it’s just as well the case classically. Also, a measurement has no distribution at all – it’s just a data point. And what you’re saying now is something very different from your original assertion that all quantum events are discrete, and that an electron will have one of a finite number of states.
Yes. Science is precisely our informed predictions of our observations.
I’m not sure what the impediment is. We can build models that allow us to more easily understand our observations. A model is not reality, but if it predicts our observations well, it can inform future research.
I’m talking about the distribution of possible results, not the particular result observed. A classical instrument draws a result from continuous distribution. A quantum instrument draws from a discrete distribution–isn’t that what you meant by “continuous measurements mean that, up to measurement accuracy, every value on the scale is possible”? You can only distinguish the “continuous” case from the “discrete” case by making many measurements and then looking at what values (drawn from the discrete scale) were actually found.
Thus, while a continuous set of states may be convenient to use in calculations (especially when the measuring device is not yet defined), the actual measurements will always be drawn from a discrete set of values, determined by the particular device used. The measurements will be consistent with the continuous set of states, yet only countable number of values can possibly be observed.
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Here’s a simple analogy. Let’s say we have a pitching machine that tosses out balls onto a field. Where the ball lands matches some bell-shaped distribution. We want to find the distribution of distance thrown.
A classical device to measure would be to use a tape from the machine to the dent the ball makes in the turf. After many tosses, we have continuous sampling of the distribution.
A quantum device to measure would be to put sticky buckets on the field and see which bucket the ball lands in. After many tosses, we have a discrete sampling of the distribution.
In both cases, the underlying distribution is continuous, and with enough tosses we can be reasonably sure of it. The difference between the ball in this example and a quantum particle is that for the former it’s meaningful to talk of the ball’s flight irrespective of how we measure the distance. For the latter, in general we can’t talk about the particle’s motion without knowing how we’re going to measure the distance. They are not independent.
No; quantum distributions, for the free particle for example, can be just as continuous as classical ones. Both instruments can only measure to finite accuracy, but that has nothing to do with any qm/classical divide.
The scale I gave is a continuous one. Every value x*E[sub]0[/sub], where x is a real number, is possible. Perhaps in your analogy think about reshuffling the buckets for each throw: each measurement has a finite accuracy (the bucket width), but for many measurements, arbitrarily small differences can be resolved.
It’s not continuous. The fact that any value could be observed in congregate does not remove the fact that each observation has only a discrete set of possible values.
Actually it does. It means that the wave function is continuous. You don’t seem to be subscribing to orthodox quantum mechanics.
No one is saying that the wave function is not continuous. We’re talking about whether the set of possible observables is discrete or continuous.
Where are you getting your ideas from Pleonast?
Measuring devices are a red herring as quantum mechanics says nada about the design, limitations, etc of measuring devices. This is infact part of a famous problem in quantum mechanics called the quantum mechanical measuring problem (i.e. the failure of quantum mechanics to define ‘a measurement’ despite it being an importnat concept in the theory.
Secondly what is inherently quantum about ‘sticky buckets’ and inherently unquantum about tape?
As I said, the fact that there are clearly solutions in QM representing physical situations where ceratin important QM observables are represented by operators with continious eigenvalue spectra shows, trivially, that this is a false assertion.
The fact is that each observation has a continuum of possible values. The Hamiltonian’s eigenvalues form a continuum of possible energy values, each of which may be observed in an experiment.
Five years of grad school and a PhD in Physics, field of superconductivity.
Measurements are all we have to check the validity of our theories. And in quantum mechanics, one can’t simply ignore the device used to make the measurement. The quantization of the instrument may be fine enough that we can approximate it as continuous, but fact remains that the instrument can only return one of a countable number of values.
Eh, the analogy either works for you or not.
You are referring to a state independent of the device used to observe it. It is a useful shorthand, but the state and device used to observe it are not independent.
In any particular experiment, not all values are potentially observable.
Do you understand the definition of the wave function? You realize that if the wave function is continuous, the set of possible observables is, by definition, continuous.
If the universe contained only one ruler, it is possible that physicists would attempt to come up with a less parsimonious form of quantum mechanics in which the wave function was always non-continuous in such a way as to exactly correspond with the ruler’s observational limitations. Besides being, well, hideous, this would be extraordinarily difficult – the Schrodinger equation would no longer apply, and some highly non-linear differential equations would be required to correctly describe the wave function’s evolution. In any case we don’t live in such a universe – we live in a universe where rulers are abundant – ready to sample at every place within a continuous wave function. Wave functions are continuous. Some experiments will sample from them non-continuous eigenvalues. So what? The set of all possible experiments completely covers the dense set of possible eigenvalues. All values along the continuum are potentially observable. This is reflected in the definition of a continuous wave function in quantum mechanics.
To be honest I’m not sure whether to believe that as the issues here are well below the level of grad school and you haven’t really put together a coherent argument.
Like I said quantum mechanical measurement problem. So infact for the most part we do ignore the device in question.
However even if we were to consider the measuring device a la many worlds you still haven’t provided any basis for your argument. Your essentially saying that the measuring device will delete the continuous nature of a QM observable which is a pretty bold statement.
Well there’s nothing to back the analogy up.
As I say, 1) we do ignore the device 2) There’s no reason to think that considering the state of the device-particle system would delete the continuous nature of the observable, unless you wish to provide a reason.
In any particular experiment, not all values are potentially observable.
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The classic particle in a box has a continuous wave function, but a discrete set of energy eigenvalues.
And that’s where you get off track. It can be convenient to ignore it, but the results observed will always depend on it.
Quantum measurements are always discrete. It’s impossible to construct a single device to make a single quantum measurement that could potentially return any value from a continuous set.
That means that someone using a many worlds interpretation of quantum mechanics does not need to worry about a continuum of worlds. Any observation that “splits” the world will always produce a countable number of worlds.
Ignoring the device is a short cut. At some point, if we want to measure that state, we’ll have to use a device that cannot return a continuum of values.
So? I think we can all agree that some eigenvalues are discrete, depending on boundary conditions. The position eigenvalues, in the example you have given, are continuous.
You continue to show that you don’t understand the MWI.
How do you reconcile this:
With this:
?