Can anyone elucidate on this please? Schrödinger

[Interbane]

That's what's so interesting about it Mary, the quote I provided is from an explanation of the uncertainty principle that shows my logic to be wrong.

[johnson1010]

The cat is indicative of our limits, not the nature of reality.

It is impossible to observe quantum level events accurately because our instruments are still all thumbs. When you blast these particles with energy to get a look at them you change them. Like turning a fan on a desk of paperwork to get a good look at it.

This is similar to the event horizon problem. If the sun were to disappear right now we would not notice it’s absence for 8 minutes (assuming gravity cannot exceed the speed of light), as that is how long it takes for its light to reach us. So, we won’t know if the sun is still there, RIGHT NOW, until 8 minutes into the future. So, like the cat, the sun is both existent and non-existent simultaneously until the event horizon reaches us. Whether or not we observe it, however, does not change the status of the event.

This is the whole “tree falls in the woods with no one to hear it” thing all over again.

Thinking of quantum packets, when you look at a possible area for the particle to be based on your best predictive analysis you are saying that the particle must reside in these places based on percentages. You could say there is a 1 percent chance it will be in part A, and a 1 percent chance it will be in part B, but a 99.99 percent chance it lies somewhere between the two.

[MaryLupin]

That's what's so interesting about it Mary, the quote I provided is from an explanation of the uncertainty principle that shows my logic to be wrong.

Isn't it wonderful! So much of what we (humans) take to be inviolable, like logic, appears to be only applicable at one specific organizational level of reality. That creates the most wonderfully absurd sense of happiness in me.

[Randall R. Young]

Schrodinger's thought experiment was actually designed to point up a problem they were all having at the time. Namely, when--exactly--does a "probability wave" collapse into real history? Under the Copenhagen interpretation, this matter never got settled. (Reference "Wigner's Friend" thought experiment.)

In fact, the Heisenberg Uncertainty Principle does have exceptionally good evidence to demonstrate its necessity. It simply isn't true that particles even have both of these properties at the same time, for every scale you might be viewing them at. When quantum effects are important, the very idea of them having both perfect position and momentum becomes incoherent, and even God can't know both at once, below the level of precision specified by Heisenberg.

I can explain this by an analogy with musical notes, if you'd like details.

[johnson1010]

Hit us up with this musical note explanation.

That is a fun reference point.

[Randall R. Young]

OK, here goes!

I am going to see if I can write a few simple songs. But I want to write them so as to completely specify the notes/"pitches" of the song, and the rhythms/durations of each note. Is this possible?

To keep it simple, I will make my songs entirely out of Middle A quarter notes ("A"), alternating with quarter rests ("_"), thusly:

>>>>>>>>>>>>>>>>>>>>>>>>>
Quantum Etude #1, by R. Young

mm60: (Tempo 60 quarter notes per minute, or 1 quarter note per second)

A_A_|A_A_|A_A_|A_A_|.... (repeat and fade)

Now--boring though this etude may be--everyone who hears it will agree that each note has a well defined pitch and duration. But let's try a slight variation. Let's do an accelerando; same notes & rests, but let's let the tempo increase as it goes.

>>>>>>>>>>>>>>>>>>>>>>>>>
Quantum Etude #2, by R. Young

mm60:

accel.
A_A_|A_A_|A_A_|A_A_|...(repeat and fade)

(Notice that there is no end to the accelerando, since I haven't said when to stop getting faster. How far can we take this? (Luckily, in this day & age, we can allow our computers to play things at arbitrarily fast tempos, so we don't need to rely on imprecise human musicians. Thus, we can take this accelerando to ridiculous extremes.)

If we let the tempo keep doubling every bar, let's say, then in a short time we will encounter a slight problem. My notes stop having pitches! This is because a note needs to have at least a few waves in a row to be perceived as notes, rater than click noises.

The first note, at mm60 had 440 complete waves, then stopped. After one doubling of the tempo, a middle A only has 220 waves. If we tabulate the durations of our "notes" at each doubling of mm, we would get these numbers of waves per note:
>>>>>>>>>>>>>>
440
220
110
55
27.5 (Half waves, and fractional waves, add non-harmonic components from this point, on.)
13.75
6.875
3.4375 (very, very click-y, to the ear.)
1.71875
0.859375 (no pitch is defined from here on, even theoretically, since pitch relies on the repetition of a waveform.)
0.4296875
etc.
>>>>>>>>>>>>>>>

With this background, let me draw the analogy with Heisenberg's Uncertainty Principle of position/momentum (there's more than one type of UP):

note <=> particle (or "wave packet");
position in "score time" (aka "rhythmic value") <=> position in space / "x";
pitch <=> momentum / "p";
pitch / rhythm are specified orthogonally on the page, which makes them <=> orthogonal pairs;

The takeaway is that I cannot perfectly specify both a rhythm and a pitch beyond a certain accuracy. Notes which are too short stop having pitch at all. And pitch cannot be specified perfectly, either, since the waves have some finite onset, otherwise known as "attack". Physically, this attack time can never be precisely 0 seconds long, since that would require our speakers and eardrums to move infinitely fast. Many other limiting factors enter into this limit, such as the springiness of any transmitting medium, like air molecules, which have momentum.

Also, the same problem occurs in a 'complimentary' fashion: If I try too hard to specify my rhythmic values, as for instance, if I write 1,000 snare drums per quarter note at mm60--or about 9 flags--I lose any hope of perceiving the events as notes, and what we hear instead is a 1,000 hz wave; Notes at this rhythmic precision stop being individual 'events', and become a "frequency" which the ear processes as a single note.

(I would write this musical example out as "Quantum Etude #3", but it would take up more space than I think is allowed!)

Lastly, to complete the analogy:
(smear in rhythm) * (smear in pitch) ≥ (constant) <=> ΔxΔp ≥ ℏ/2π;

If pressed, I think I could calculate a value for this musical constant.

[Randall R. Young]

If I can figure out how to include some music clips, I'll post them here. They may make this example a lot more intuitive.

[Dexter]

I
Heisenberg's Uncertainty Principle is similar in that the limitation or error is human. Because we cannot know both a particle's position and momentum precisely at the same time does not mean a particle doesn't have position and momentum, it merely means the only methods of measurement to deduce these quantities is severely limited. Of course, this is where quantum mechanics trumps human intuition;

"In fact, states with both definite position and momentum just do not exist in quantum mechanics, so it is not the measurement equipment that is at fault."

But then perhaps our theories aren't quite up to the task? I read the quantum mechanics articles in most of the science magazines each month, and still think it's the most surreal of our sciences. Discoveries in this field are fascinating.

I don't know enough about it to say anything definitive, but as your quote indicates, I was under the same impression -- that we can't just say it's a limitation of measurement. That would be too easy -- it wouldn't really be that weird then. But that can't explain some of the experimental results like the double slit and quantum entanglement. I am partial to your thought that we need a better theory though. (See the Feynman quote in my sig!)

[Randall R. Young]

One of the points I had hoped to bring out in my musical analogy was that a waveform is not something which can have a definite point-like position in the first place. It doesn't even have a precisely defined center, since the ends of it necessarily grade in. This is equally true of quantum wave packets.

In a similar way, their momentum is ill-defined, as well. Since the momentum is defined as mass*velocity, and velocity is Δp/time, the lack of available positional precision implies that the momentum is at least as unknowable as the position is. At best, all you get is some sort of average value.

As long as we have this waviness going on, no new theory can ever make the classical "wishful thinking" of Einstein back into a reality. The "particles" just aren't partical-y enough! QM therefore likes to call them "wavicles", which is exactly how I think of musical notes. When a classical composer conceptualizes the notes on his score, he thinks of them as having way more of a precision than is in fact available. He neglects that both the frequency of them (pitch) and the time of them (rhythm) are inherently vague. For most "human scale"/human playable notes, these factors can be neglected, but if you ever have occasion to actually edit music at the "sample-level" resolution, you will quickly realize that the smears are omnipresent, both in time and pitch. An accurate, knowledgeable composer will understand these problems innately (if not consciously), as for instance, when he writes for pizzicato strings, and is expecting the section not to hit all these pizz's at exactly the same instant. If that were to happen, it would just sound like one really loud pluck, with all the wavefronts in much too close a phase to be the same thing at all.

[Penelope]

Randal, I do want to thank you for your posts. I do think there is a curious link between music and mathematics and here the link goes even further in helping to explain something about quantum theory.

I am going to print off your post about musical notes in order to study it more closely. But I can see where you are leading with it.

Excellent. Many, many thanks.