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For example, in velocity, when we say $v=\frac{dx}{dt}$, there is no proof for it. Its almost like an axiom. Something taken to be true, without a proof. How do I know that for every $x=f(t)$, $v=f'(t)$?

Also, how can I say that the position of an object as a function of time, say an electron, is $x=f(t)$ (this comes when I'm studying electromagnetism and we have to chart the course that a charged particle takes in a constant electric field) when QM says that an object's position cannot be known without uncertainty?

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    $\begingroup$ There is no "proof" for the equation of velocity because it is a definition. Velocity is defined as the time-rate-of-change of position, so there's nothing to prove. I'm not sure what you're trying to ask here exactly... $\endgroup$ – tpg2114 Jul 20 at 2:25
  • $\begingroup$ There is a classical limit in which quantum mechanics can be ignored. (Just look around you!) You are studying the classical limit of electromagnetism, not QED. $\endgroup$ – G. Smith Jul 20 at 2:28
  • $\begingroup$ Be careful. Definitions are not axioms. $\endgroup$ – Aaron Stevens Jul 20 at 5:20
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As for your first question, velocity is defined to be the first derivative of position with respect to time. It must be true because it is defined to be true. Likewise, acceleration is defined to be the second derivative of position.

Those need no proof. One never needs a proof for a definition. Now where it starts to get interesting is when we add $F=ma$ into the mix. This equation brings in some physical implications. It's possible to create tests to see whether $F=ma$ is consistent with the real world around us (and, with little surprise, we find that it does indeed work. There's a reason it's a famous way of modeling the world!)

As for quantum uncertainty, there's an important phrase to help with that which I wish was taught earlier in our education:

All models are wrong. Some are useful.

That quote, attributed to George Box, is the key to understanding why science's models are applicable to the real world. Every one of them is wrong. They all miss some detail somewhere. There's even some fun theories out there which suggest this must always be true. There must always be something missing.

However, when we start talking practically, these models have amazing predictive power. You can look at a situation, analyze it, and predict with great certainty what will happen.

In the case of calculating positions and QM uncertainty, we have to apply statistics. While a particle's position is uncertain, the expectation of an object's position (read: the average position if you measured it) is relatively constant, and its standard deviation is very small. You might measure one box as being $1.347m$ away from another box. That's a pretty good measurement. Now due to uncertainty, we know that there's some statistical error to worry about, but it will be less than $0.00000000000000001m$ (I'm pretty sure it should actually be more than 30 digits long, but that depends on your measurement, so I'm giving it a lot of benefit of the doubt, assuming the effects of uncertainty are even larger than they should be).

Now practically speaking, you are not going to succeed at measuring these objects to an accuracy where you can observe this quantum fiddly-ness. So we model it without such uncertainty. We know this model is wrong, because it doesn't include uncertainty, but it is useful. For all except particle-physics level predictions, it's a useful way to predict what happens.

Your question is specifically about electrons, which are certainly small. In practice, you'll have to rely on your teachers to help you understand when it is reasonable to model electrons as simple classical particles and when you need to bring in quantum mechanics. As an example, you'll learn about a diode, which lets electrons flow in only one directions. In most of its operating regime, classical electromagnetism is sufficient to explain its properties. However, if you reverse-bias them (putting a strong voltage in the direction it doesn't flow), a strange effect called "avalanche breakdown" occurs, letting current flow when the classical models say the current can't flow. The reason? At high enough voltages, the layer which prevents electrons from flowing gets thin enough that quantum mechanical effects start to matter -- we start seeing electrons tunnel through the barrier.

How do you know when you can use one model vs the other? Well, the end all answer is empirical testing -- go try it and see. But your teacher should help you understand equations which show what voltages are needed to start having tunneling currents that are not ignorable.

In your career, you'll make more assumptions. And sometimes the results wont be useful. A classic example is collisions. When you start modeling collisions, you'll assume they happen instantaneously. If you start trying to do things like design protective gear to save your life in a collision, you'll start to find that model isn't useful. In reality, collisions are complex beasts that take time. If you're designing safety gear, you need better models.

Another example is aircraft flight. Typically you get to assume that air moves like a wave. However, as you get close to supersonic speeds, that assumption starts to fall apart. The way it falls apart looks like a shock wave.

So expect your career to be full of models that are wrong. But from experience, many of these models are useful. You merely have to know when it is reasonable to apply them.

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