Can uncertainty be quantified?

[mh8782]

I hope this is in the correct forum. Apologies, if not. In the field of science, there's a lot of work on quantifying uncertainty. Debates abound over whether one should use statistics, or whether fuzzy logic methods are possible?

I'm going to say that uncertainty is defined as that which we don't know. Therefore, how can uncertainty be quantified? Is it possible to measure how much we don't know?

[cpollett]

Well, given a state space and an energy level, the entropy says something like the average number of bits we'd need to completely fix the state.
So it typically quantifies how much we don't know. I suppose you could
combine this with the energy uncertainty relation
\Delta E \Delta t \geq \hbar /2
to quantify how uncertain you are in the number of bits needed to completely fix the state. I am no physicist though, so maybe someone should say something more precise.

[primitivefuture]

It has been scientifically proven that the universe is expanding. Can you find the rate of expansion?? NO

[Preno]

Ever heard 'probability'? :huh:

[cpollett]

I thought I'd babble a little bit more about my intuitions on what energy and entropy are. Suppose we have a n particle system where the only properties we have for each particle is that it can be up or down. So we can identify the system with some bit string. For instance, if n=10, we might imagine having a string
1001001100.
Here 1 means up 0 means down. The energy of the system is roughly telling us at most how many bits of our string we might be changing in a given amount of time. For instance, if we say we can flip up to 3 bits randomly at a time in the above, there are lots of different bits strings we could be depending on which bits we choose to flip. We might flip

10X10X1X00 ------Here X are the locations we might flip
or we might flip
X00X0X1100
or
...
So there are lots of different possible bit strings we might be in. The entropy quantifies how many bits of information we need to fix which state we are in. Basically in the case above we need to know where the X's are and how they were flipped. Each X takes log of the length of the bit string bits to fix. So the entropy of the system is roughly the number of bits which can be changed x

So why is energy roughly the number of bits that can be flipped in a given time? Now for some somewhat flaky intuition. In quantum mechanics, an energy corresponds to an eigenvalue of the Hamiltonian of the system. The corresponding eigenvector for the eigenvalue is the state one would be in if one had this energy value. You might have seen pictures of these eigenvectors for a square well, where you get one arc for the wavefunction in the lowest eigenenergy, something with two bumps for the next level energy level, etc. The oscillation of the state vector whether or not we are at an eigenergergy is gotten roughly by multiplying the matrix exponential of the Hamiltonian against the vector. If the energy is s, the state vector will be the s bump version and the wave functions amplitudes will oscillate around these s bumps. Now as the energy uncertainty principle shows over a given time interval we cannot be exactly certain of the energy of the system. So we might be in a system with s bumps or maybe s-1 or s-2 or s+1 or s+2. So the number of places we might be "screwed up" in our knowledge of the wave function is roughly s and these places change with time.

As I said before I am no physicist, so someone who is, please correct me if the above is not a quasi-accurate intuition of what's going on.

[mh8782]

Ever heard 'probability'? :huh:

Of course I've heard of probability. However, probability is one methodology, but relies on certain assumptions. One of those is that you already know what the possible outcomes are. This may be a more scientific topic. I'm specifically thinking about problems such as climate change (or other scientific modelling).

Let's imagine a prediction that the temperature will increase by 2C by the end of 2100. Is it possible to describe the associated uncertainty, seeing that there's so much one doesn't know. (Please note, I'm not a climate change skeptic, I'm just picking it as an example - it could be a forecast of the population of a given species on an island).

[Preno]

Well, you need to specify what 'uncertainty' you are talking about, exactly, because 'uncertainty' is a rather vague notion. Your example seems to point to probability. If that's not the kind of uncertainty you had in mind, then you should clarify what other kind you did have in mind.

The fact that you cannot actually enumerate all the possible states of Earth by the end of 2100 doesn't mean that you can't speak about the possible states of Earth by the end of 2100, and the fact that you cannot in practice quantify the exact probability of such a temperature increase doesn't mean that probability doesn't apply here.