Accuracy of Classical Mechanics

[member 529879]

I don't know very much about quantum mechanics, but if I'm correct, the future can not be predicted with certainty according to quantum mechanics. If this is true, how can we have formulas in classical mechanics that do predict the future with certainty?

 

[Doug Huffman]

Classical mechanics are limited to subsystems in a box, to free body cartoons, to ensure accuracy.

 

[atyy]

Classical mechanics is only an approximation to quantum mechanics. Quantum mechanics tells us how accurate the classical approximation is.

 

[member 529879]

Okay that makes sense, would a simple formula like f = ma not be 100% accurate then? Can somebody give an example of how a formula like this can have small inaccuracies?

 

[MisterX]

Even within classical physics, formulas can be replaced by others considered more accurate or appropriate. For example another version of $$F = ma$$ is
$$F = \frac{d}{dt}[ \gamma(v) m v ]$$ Where $\gamma(v)$ is the Lorentz factor. This is a relativistic (not quantum) change to the formula that is a lot more appropriate for $v$ close to the speed of light.

 

[Nugatory]

I don't know very much about quantum mechanics, but if I'm correct, the future can not be predicted with certainty according to quantum mechanics. If this is true, how can we have formulas in classical mechanics that do predict the future with certainty?

If we actually tried to use the formulas of classical mechanics to predict the future with 100% accuracy, we would fail. This is not because these formulas are wrong - they aren't. $F=ma$, for example, is absolutely 100% accurate (although relativity requires that we word it differently and more carefully). Quantum mechanics hasn't done anything to change this. What quantum mechanics has done is to introduce a fundamental randomness in the behavior of the smallest particles.

Consider two identical particles moving at the same speed, approaching each other and colliding elastically. They can rebound in any direction and classical laws such as conservation of energy and momentum will be exactly satisfied as long as they end up moving in opposite directions with the same speed (and this is what we always observe).

Classical physics says that if we just knew enough about the exact trajectories of the incoming particles and how they interacted, we could predict the rebound angle for any collision. Quantum mechanics says that all we can do is predict the probability of getting various rebound angles. That's fine for making statistical predictions ("odds are it ill trun out this way instead of that way") but it's not a 100% prediction of one particular future.

 

[Apogee]

Okay that makes sense, would a simple formula like f = ma not be 100% accurate then? Can somebody give an example of how a formula like this can have small inaccuracies?

F=ma is more of a definition than anything else, so it would be 100% accurate. Newton's Second Law effectively describes the way that we think of force. It isn't exactly something that is derived.

I don't know very much about quantum mechanics, but if I'm correct, the future can not be predicted with certainty according to quantum mechanics. If this is true, how can we have formulas in classical mechanics that do predict the future with certainty?

Classical mechanics is not absolutely certain. CM is formulated from assumptions, for example Newton's Laws. If we assume that these assumptions hold, then in that little box, it is exact. When applied to real world, we could not possibly account for every little detail. So, we make enough assumptions to receive an accurate approximation of what is really going on. In our little theoretical world we've constructed, everything is exact. But for real world purposes these are approximations.

I'm no expert, though I realize I sound extremely pretentious trying to clarify that hahahaha. From what I understand about quantum mechanics, there is an inherent limitation in measurement, preventing our calculations really being precise. Once we get to a certain degree of accuracy, the very act of observing the system already changes the calculations. So, you are correct: the future cannot be predicted with absolute certainty. However, if we ignore Heisenberg Uncertainty constraints on measurement precision, potentially collapsing quantum wave functions and changing the system at hand, etc., as we established above, we do not really have formulas in classical mechanics that predict the future with certainty. Those fun "a ball slides along a frictionless table" problems are merely approximations of real world events.

 

[bhobba]

Classical mechanics is not absolutely certain. CM is formulated from assumptions

To be precise the assumptions are the Principle Of Least Action (PLA) and the Galilean Principle of Relativity ie time is an absolute. The detail can be found in Landau - Mechanics.

QM is based on the two axioms detailed here:
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

Plus, just like the classical case, the Galilean Principle of Relativity - the detail can be found in Ballentine - Quantum Mechanics - A Modern development.

Also from those two axioms you can derive the PLA.

So, at rock bottom, non-relativistic mechanics, either classical or quantum, is based on absolute time. And since that assumption basically defines the non-relativistic domain we see QM is the rock bottom essence.

Thanks
Bill

 

[kith]

Okay that makes sense, would a simple formula like f = ma not be 100% accurate then? Can somebody give an example of how a formula like this can have small inaccuracies?

In QM, something very similar to $F=ma$ is true for expectation values.

Observables like $F$ and $a$ don't have sharp values in QM but obey a probability distribution like the famous bell-shaped Gaussian distribution. For a Gaussian distribution, the expectation value is the peak of the curve and it tells you what you should expect if you measure the observable many times and calculate the mean value. Another important parameter of the distribution is the width of the curve at a certain height (often at half the height of the peak). This is a measure for how you should expect the individual measurement outcomes to spread, i.e. how far they will be from the mean value.

Now in QM, $F=ma$ is a law which is valid for the expectation values of $F$ and $a$. In general, the formalism of QM doesn't tell you what happens in an individual run of an experiment but only what happens "on average". Only if the spread of the probability distribution was zero, it would be meaningful to talk about what happens in a single run of the experiment (because it would be equal to what happens on average).

In classical mechanics, there's no lower bound to the spread, so it can be zero in principle and F=ma can be viewed as an exact law which tells you what "really goes on" during a single run of the experiment. (However note that this is idealized and that F and a can't be known exactly because of the limited resolution of your experimental apparatus.)

In QM, there's a fundamental lower bound to the spread which is given by Heisenberg's uncertainty principle (HUP). But this bound involves Planck's constant, which is incredibly small compared to everyday scales. So although the spread cannot be zero, it can be very small compared to the expectation value. And if it is smaller than the resolution of your measurement apparatus, the predictions of QM and classical mechanics are exactly the same.

(There are a few technical inaccuracies in what I wrote above but I think they are justified in order to make the central point more clear.)

 

[Apogee]

To be precise the assumptions are the Principle Of Least Action (PLA) and the Galilean Principle of Relativity ie time is an absolute. The detail can be found in Landau - Mechanics.

QM is based on the two axioms detailed here:
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

Plus, just like the classical case, the Galilean Principle of Relativity - the detail can be found in Ballentine - Quantum Mechanics - A Modern development.

Also from those two axioms you can derive the PLA.

So, at rock bottom, non-relativistic mechanics, either classical or quantum, is based on absolute time. And since that assumption basically defines the non-relativistic domain we see QM is the rock bottom essence.

Thanks
Bill

I'm assuming the Galilean Principle of Relativity is relaxed in Special Relativity, for Galilean relativity assumes that time is universal, whereas in Special Relativity, time depends on your frame reference, velocity, etc. For example, time slows down considerably when traveling at a sufficiently fast rate, i.e. close to the speed of light. Therefore, if this is allowed to occur, the notion of universal time must be relaxed. I could be thinking of this incorrectly though.

 

[Apogee]

In QM, something very similar to $F=ma$ is true for expectation values.

Observables like $F$ and $a$ don't have sharp values in QM but obey a probability distribution like the famous bell-shaped Gaussian distribution. For a Gaussian distribution, the expectation value is the peak of the curve and it tells you what you should expect if you measure the observable many times and calculate the mean value. Another important parameter of the distribution is the width of the curve at a certain height (often at half the height of the peak). This is a measure for how you should expect the individual measurement outcomes to spread, i.e. how far they will be from the mean value.

Now in QM, $F=ma$ is a law which is valid for the expectation values of $F$ and $a$. In general, the formalism of QM doesn't tell you what happens in an individual run of an experiment but only what happens "on average". Only if the spread of the probability distribution was zero, it would be meaningful to talk about what happens in a single run of the experiment (because it would be equal to what happens on average).

In classical mechanics, there's no lower bound to the spread, so it can be zero in principle and F=ma can be viewed as an exact law which tells you what "really goes on" during a single run of the experiment. (However note that this is idealized and that F and a can't be known exactly because of the limited resolution of your experimental apparatus.)

In QM, there's a fundamental lower bound to the spread which is given by Heisenberg's uncertainty principle (HUP). But this bound involves Planck's constant, which is incredibly small compared to everyday scales. So although the spread cannot be zero, it can be very small compared to the expectation value. And if it is smaller than the resolution of your measurement apparatus, the predictions of QM and classical mechanics are exactly the same.

(There are a few technical inaccuracies in what I wrote above but I think they are justified in order to make the central point more clear.)

Why is this the case in quantum mechanics though? I don't too much about QM, but I do know that it is heavily based on probability distributions, unlike CM. What fundamental assumptions in QM lead us to draw these conclusions?

 

[bhobba]

I'm assuming the Galilean Principle of Relativity is relaxed in Special Relativity, for Galilean relativity assumes that time is universal, whereas in Special Relativity, time depends on your frame reference, velocity, etc. For example, time slows down considerably when traveling at a sufficiently fast rate, i.e. close to the speed of light. Therefore, if this is allowed to occur, the notion of universal time must be relaxed. I could be thinking of this incorrectly though.

That is correct. Galilean relativity is only valid as a very good approximation for small velocities.

If you move beyond that you get Quantum Field Theory or Classical Relativistic Mechanics depending on if you are dealing with quantum or classical phenomena.

What I gave in my post is the deep advanced view you do not normally encounter until you read graduate level texts like Landau and Ballentine - but they are accessible after a course in multi-variable calculus or an intermediate text in QM such as Griffiths respectfully.

Thanks
Bill

 

[kith]

Why is this the case in quantum mechanics though? I don't too much about QM, but I do know that it is heavily based on probability distributions, unlike CM. What fundamental assumptions in QM lead us to draw these conclusions?

The central point for this is that in QM, the probability distributions cannot get arbitrarily narrow (at least not simultaneously for all observables). This is dictated by Heisenberg's uncertainty principle (HUP).

Now the why question is difficult because ultimately, the answer will be "because of an axiom of the theory". We could simply take the HUP as an axiom, thus recognizing it as a fundamental fact about nature which cannot be explained by more fundamental things.

Or we could use the traditional axioms of QM that the states of a quantum system correspond to vectors in a complex Hilbert space and that the observables correspond to self-adjoint operators, acting on this space. The general HUP then follows from simple linear algebra, and the HUP for $x$ and $p$ from the canonical commutation relations which is an additional axiom.

Or we could use a wave mechanics approach and notice that the width of a distribution is inversely related to the width of its Fourier transform. If we take $\vec p = \hbar \vec k$ as an axiom, this gives us the HUP for $x$ and $p$.

We may stop here and accept the new axioms which we have just used to derive the HUP, but we may also ask if there's a more intuitive explanation for them. For this, I recommend two things:
1) This lecture by Scott Aaronson, which explores the state and probability axioms by looking at possible generalizations of probability theory (elementary)
2) Chapter 3 in Ballentine's book "Modern Quantum Mechanics", which derives the canonical commutation relations of the observables from space-time symmetry (advanced)