What’s the underlying principle on why you can’t know the position and momentum of a particle at the same time? Is there an explanation? I don’t have any knowledge of calculus or the math behind anything so I’m asking here, thanks!
In: 57
It’s less that you can’t know, and more that the particle doesn’t have a well defined position and momentum at the same time.
The uncertainty principle applies to more than just position/momentum and fundamental particles. It’s really a property of waves and it makes a bit more intuitive sense to take about them.
If someone strums a string and let’s it vibrate then it’s very easy to define the frequency at which it’s vibrating, but pinpointing the exact point in time when it was played doesn’t make sense, it was played over a whole range of time. Going the other way if they pluck a string but then hold onto it, defining what single frequency it vibrates at doesn’t make sense, but the time range is much shorter.
This relationship between time and frequency (really energy) is also an uncertainty relation.
So when you take a fundamental particle and constrain it’s position, it’s momentum is really a wide range and not a single value. And if you pin down it’s momentum then it’s position is a wide range
I think the bottom line is that no partical ( physical mass) exists without its twin wave ( non physical ) massless energy. So one can be measured and the other cannot because it doesn’t seem to fully exist in this dimension. Plus our Measuring devices are still way to inadequate to capture objects on the quantum level. Best uneducated guess. 🙂
There are 6 postulates in quantum mechanics. These are the baseline principles upon which all of quantum mechanics is built on. The uncertainty principle is a logical conclusion that can be made from those postulates
1. The entire state of a system can be described with a wave function.
2. Any observable value has something called a “Hermitian operator” associated with it.
3. When you apply those “Hermitian operators” to a state, it gives some constant multiplied by that state.
4. A normalized wave function has an average value.
5. The Schrodinger equation describes time evolution.
6. The Pauli exclusion principle applies to fermions (protons, electrons, and some others)
The Heisenberg uncertainty principle is these applied, along with mathematics to reach the conclusions that position and momentum are mutually exclusive. It takes about 3 months of undergrad work to get to the point where you can do that “proof”
Step one to understanding quantum mechanics is accepting that all particles also behave as waves. This is probably the hardest step because it’s kind of wild.
What that means can be a bit hard to understand, but the crucial part here is that these waves (or wave packets) don’t have a specific location.
For instance, look at [this](https://upload.wikimedia.org/wikipedia/commons/thumb/1/10/Gaussian_wave_packet.svg/1280px-Gaussian_wave_packet.svg.png) wave packet. This is our electron. Where *exactly* is this electron? Is it the middle of the wave packet? The left side? The right side? It doesn’t have a specific location. We just know it’s vaguely in the area of the wave packet. It’s not so much that our knowledge is limited, but that the whole region is ‘electron’, yet the electron doesn’t actually occupy all of that space.
The energy of a particle is related to the frequency of its corresponding wave. Suppose you wanted to measure the frequency of a sound wave. You could do it by counting the number of peaks and troughs that hit your microphone in a given time. Say you measure for a second, and count 1000 peaks, so you report that the frequency is 1000 Hz. But there could have been an almost-peak at the end if the second that you didn’t count, so there’s a 1/1000 uncertainty in the measured frequency. If you count for 5 seconds instead, you can reduce that uncertainty by a factor of 5, but now there’s a cost — you can’t say exactly **when** that measurement represents, since it’s smeared over that 5 seconds.
Similarly, you can measure the energy of a particle, but the more precisely you try to measure it, the more you have to give up on knowing its exact position.
Let imagine we have two cars. We know the speed and initial position of one car and basically nothing about the other. We are going to assume that both drivers are going to keep their speed and direction constant and at some unknown point, they crash and we have their final positions after the crash. Now, what we would like to be able to do is figure out where the two cars crashed and how fast the other car was going. As it turns out, this is a really problem because the speed and position of the crash are interdependent. Where the crash happens would tell us what speed the other vehicle is moving at. Because the speeds and directions are consistent, we get a reasonably simple formula for describing the relationship between position and speed.
Now, lets imagine that both cars are swerving wildly across the road. This means that the vector describing their speed isn’t a constant and changes both direction and magnitude. This means even if we know the position of the crash, there is multiple possible sets of speeds & directions each car could be traveling which would produce the final positions after the crash, and the same of the crash position if we know only know the speed vectors. Knowing one of the position or speeds tells you basically nothing about the other.
Functionally, this is what is happening with particles and the uncertainty principle. We like to imagine particles moving in simple straight lines, but that isn’t really true. All particles vibrate a little tiny bit, but it’s enough to make things work like the cars swerving in the road. I also use the idea of speed as a simplification of momentum. It’s a general principle because this kind of problem whenever a property of a particle has a wave nature, and that happens for a bunch of them.
>What’s the underlying principle on why you can’t know the position and momentum of a particle at the same time?
I think you are approaching this as though this is a problem of knowledge about the particle. In other words, you are thinking that a quantum particle *has* both a position and momentum, we are just somehow limited in how to measure those two things simultaneously.
Think of looking down at a pool table and you see a cue stick hit the cue ball, and a sheet is pulled between you and the pool table the moment after it’s hit. After it bounces around long enough underneath this sheet on this ideal, frictionless table, your ability to predict where it is gets foggier and foggier until it could be anywhere under there. But this is a problem of knowledge; the cue ball itself “knows” where it is, it’s just that you don’t.
This is **not** a correct picture of the quantum model of a particle. The quantum model says that a particle *literally does not have* both of these properties to infinite resolution. It’s not that we can’t measure them, it’s that they are not simultaneously there.
So now picture that you’re looking down at the pool table and the cue stick hits the cue ball. At that moment, there is no sheet pulled. You can see everything at every moment. But, now the cue ball starts to spread out into a blob and dim. At any moment, you can stick a million pool cues down and have them all take a stroke, and when you do that, all of that energy will coalesce again into a solid cue ball in front of one of the cue sticks and go heading off in a new direction.
But which cue will be the one that hits the ball? Where on the table will that happen? We can only say with probability based on the last interaction. Again, this is not because we just lost track of it, but rather because the cue ball itself does not have a particular position and momentum. These properties of the ball’s relationship to the table literally disperse as things proceed from the last collapse of its wave function.
Now, keep in mind that I am talking about the quantum model here. This is *our model* of the universe. Is it what is literally happening in the real physical world? We think so *but it is a guess* and we also know that it is not a complete picture of what is actually happening, and we don’t know where the mismatches are. It is also possible that there’s something completely different going on that we haven’t fully grasped.
Because a particle is also a wave. The Uncertainty Principle is a statement about the fundamental nature of waves.
If you have a perfect sin wave stretched infinitely far across the universe, it has a perfectly defined energy/momentum. However, it doesn’t have a well defined position because we can’t pinpoint one single locations that’s where the wave is.
One the other extreme, if you scrunched the wave down to be infinitely compact, then we know exactly where it is, but because it’s all crammed into a single point, there’s no wavelength, so we can’t say what its energy/momentum is.
The Uncertainty Principle is saying just that, as well as setting the limits for the in-between cases.
[Here’s an illustration of what I was saying above](https://pbs.twimg.com/media/D7K4UYFUIAcwQpR.jpg) The top wave has no definable position, but we can describe its wavelength (and therefore its energy or momentum). The bottom wave packet we can point to where it is, but measuring the wavelength is very hard. The middle one we can point to a rough area of where the wave is, and we can sort of measure the wavelength, but neither is perfectly precise.
This requires a bit of knowledge about Fourier transforms in mathematics.
Say you have a signal that varies with time, e.g., a pulse of some sort like a short voltage pulse in electronics. The pulse can be decomposed as a sum of sinusoidal waves with varying frequencies, amplitudes and phases. You can compute the frequency components from the signal using a mathematical technique called the Fourier transform. The shorter and sharper the pulse in time (on the signal vs time graph), the more spread in frequency (i.e., signal vs frequency graph shows more spread). This is a mathematical consequence when using the Fourier transform to go from time to frequency. **There is a minimum value in the product of the ‘spread’ of the pulse in time and its ‘spread’ in frequency.**
In quantum mechanics, a particle has a probability wave (e.g., in position). Imagine a signal in position e.g. a pulse that gives the probability of finding a particle at a specific position. This pulse can be decomposed of a sum of sinusoids with different momentum. If you do the Fourier transform of a signal vs position, you get a signal vs momentum. As a consequence of the Fourier transform, there is a minimum in the product of the ‘spread’ of the pulse in position and the ‘spread’ in momentum. This ‘spread’ is basically an uncertainty in the position or momentum of a particle.
I had explained in a book this way once. You measure things by shooting some form of photon at it and then looking at what happened (xrays, visible light, etc). Normally, that doesnt affect the object you are measuring since a photon is so small. But, trying to measure a photon sized object with a photon changes affects the result purely by measuring it. So therefore if we shoot a photon at another electron, we know exactly where it was, but now know nothing about where it is going, or we can use tests to figure out where its going, but we dont know where it currently is. But we can find a nice middle ground were we reasonably know about where it is and where its going, which is the electron cloud.
Latest Answers