> When the position probability distribution is made narrower, the momentum probability distribution becomes wider ... Note that this is not a limitation of measurement
I see this a lot, but what's the reason we know it's not the case?
Quantum physicist here. This is the way I like to think about it. A quantum system corresponds to some vector (i.e. the state of the system) and there are an infinitude of possible bases to write down the vector. Position and momentum are two bases of the same vector space that don’t share any element (not only that, they are particularly incompatible in the sense that you need ALL the element of one basis to decompose a single element of the other), so if you have a very localized system (e.g. that roughly corresponds to only one basis element in the position basis) you’ll need a lot of elements of the momentum basis to describe the same vector, and vice versa. The key to the puzzle is that there’s no escaping the uncertainty principle because position and momentum are two incompatible points of view of the same vector space (and of the same vector if you think about the state).
Update: I think part of what’s confusing about this is that classically we can have pretty much any combination of position and momentum, but quantum mechanically it’s the opposite: once you know the state in the position basis, its momentum representation is also determined and vice versa.
Imagine you want to calculate what frequencies are present in a sound. At the width of the time window you are looking at goes to zero, you just have a single point. There's no frequency content of a single point! It's just a number. As you look at a longer time window, you can see what frequencies are present over the window, but you lose some sense of time "resolution" over which that is meaningful. This is fundamental and inherent in the Fourier transform.
Put differently, you can't have a signal that is simultaneously both "band-limited" and "time-limited". Other conjugate variables, like position and momentum, work the same way.
They are conjugate variables (i.e. Fourier transform duals) which arise in quantum mechanics simply due to the matter wave nature of all quantum objects
I think this is fundamentally a question of interpretation of quantum mechanics. We (society/physicists) do some set of experiments, and come up with a model to explain all of the resulting measurements. Quantum mechanics is a model that explains almost all of the phenomena that we observe. However, it is fundamentally only a description of our observations.
To answer the question directly, in quantum mechanics, a state (of some particle, or collection of particles, as in the blog post) is represented as some vector, and we represent "operators" (which can represent position, momentum, spin, etc...) as matrices acting on the state vectors. When we do a measurement, the state is projected onto an eigenvector of the operator. If we do multiple measurements we have to project multiple times, along different bases. This process is not necessarily commutative. If I measure position and then momentum I will get a fundamentally different result than if I measure momentum and then position.
If we make measurements of two observables that have the same basis, then the two matrices will commute, and there is no such limitation. However, with non-commuting observables this is a fundamental limitation, no matter how good your measurement is, you will always be projecting the initial state in different ways depending on how you measure the different observables.
In quantum mechanics, the impossibility of localizing in both position and momentum simultaneously is a corollary of the more fundamental principle of wave-particle duality:
1. Everything has both particle and wave characteristics.
2. Position comes from looking at the thing from the particle perspective, while momentum comes from looking at the thing from the wave perspective.
3. It's mathematically impossible for a thing to be a "perfect" point particle and also a "perfect" wave (i.e. a sine function).
> the uncertainty principle is inherent in the properties of all wave-like systems... it arises in quantum mechanics simply due to the matter wave nature of all quantum objects. Thus, the uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology.
Because it is part of the theory of quantum mechanics itself. It is not just an empirical fact, it is a prediction of the theory itself.
Similar behavior shows up in Fourier Transforms [1]. When an audio sample is converted into frequency-space, it shows similar "narrowing/widening" relationship with the original audio sample. This again isn't a limitation of measurement, its just the relationship between the two bases
Are you asking why we say that the momentum is physically spread through a distribution instead of a precise value that we just happen to not know?
The hypothesis that it has a precise value is called "hidden variable theory" if you want to search for it. We know that several variants of it are false because of how particles interfere with each other and how they can become correlated. The interference math is quite simple (the fact that particles have destructive interference means they aren't just a lot of stuff with unknown properties), but the math on correlation is complicated.
There is still an hypothesis called "superdeterminism" that say that all the information about the entire past and future of particles always existed with infinite precision. That one we can't rule out.
It has to do with the wave-particle duality of matter. As you more finely attempt to measure the momentum (a more fundamental measurement compared to kinetic energy) and its location, you run into the wave-like nature of matter. It becomes more of a distribution rather than a point measurement. I won’t attempt to show the mathematics behind this, I’ll leave that for a physicist or mathematician.
In classical mechanics momentum is the derivative of position with respect to time. In quantum mechanics momentum is the gradient, i.e. the combination of the partial derivatives with respect to the three directions of space, of the wave function [1], i.e. the probability amplitude of the particle as a function of position.
In consequence, classically one can prepare a particle with any position and any momentum independently, i.e. one fixes the position and the time derivative of the position. Quantum mechanically fixing the position distribution also fixes the momentum distribution as the later is the derivative of the former with respect to space and therefore can not be specified independently. Similarly to the classical case one can also fix the time derivative of the wave function but that corresponds to the energy, not the momentum.
You can't specify the time derivative of the wavefunction independently, because the Schrodinger equation is first-order (unlike classical dynamical laws which are second-order).
It has been a while since I studied solid state physics but yes, exactly correct.
The fun effects of semiconductors rely on the electronic properties of certain solids (or their electrons, equivalently their absence ie ‘holes’) which take place at a small scale, hence the use of quantum mechanics.
Some of these effects can manifest at larger scales, and can be modelled classically too.
While every mosfet transistor have mode could considered genuine quantum (tunnel effect), it is not used in computers, it used in microwave generators/detectors.
What used in computers, are just switch mode (electric field creating/moving/changing conducting zone), which is mostly classic physics and created as fruit of garage engineering, without understanding of quantum mechanics.
And yes, this is very abstract border, as I know, 100% quantum effects considered when working with single atom or single electron (something <= 1nm), but for example, 100nm transistors, usually considered as classic physics.
BTW it is very interest, what will show image sensor with ~1nm sized cells. Now their cells are much larger than wavelength, but already created superlenses, could focus on nm-sized things.
[+] [-] sdflhasjd|3 years ago|reply
I see this a lot, but what's the reason we know it's not the case?
[+] [-] ziofill|3 years ago|reply
Update: I think part of what’s confusing about this is that classically we can have pretty much any combination of position and momentum, but quantum mechanically it’s the opposite: once you know the state in the position basis, its momentum representation is also determined and vice versa.
[+] [-] frisco|3 years ago|reply
Put differently, you can't have a signal that is simultaneously both "band-limited" and "time-limited". Other conjugate variables, like position and momentum, work the same way.
[+] [-] platz|3 years ago|reply
https://en.wikipedia.org/wiki/Uncertainty_principle#cite_ref...
https://en.wikipedia.org/wiki/Uncertainty_principle#Introduc...
https://en.wikipedia.org/wiki/Conjugate_variables
[+] [-] aviplane|3 years ago|reply
To answer the question directly, in quantum mechanics, a state (of some particle, or collection of particles, as in the blog post) is represented as some vector, and we represent "operators" (which can represent position, momentum, spin, etc...) as matrices acting on the state vectors. When we do a measurement, the state is projected onto an eigenvector of the operator. If we do multiple measurements we have to project multiple times, along different bases. This process is not necessarily commutative. If I measure position and then momentum I will get a fundamentally different result than if I measure momentum and then position.
If we make measurements of two observables that have the same basis, then the two matrices will commute, and there is no such limitation. However, with non-commuting observables this is a fundamental limitation, no matter how good your measurement is, you will always be projecting the initial state in different ways depending on how you measure the different observables.
[+] [-] civilized|3 years ago|reply
1. Everything has both particle and wave characteristics.
2. Position comes from looking at the thing from the particle perspective, while momentum comes from looking at the thing from the wave perspective.
3. It's mathematically impossible for a thing to be a "perfect" point particle and also a "perfect" wave (i.e. a sine function).
Wikipedia covers this well: https://en.wikipedia.org/wiki/Uncertainty_principle
> the uncertainty principle is inherent in the properties of all wave-like systems... it arises in quantum mechanics simply due to the matter wave nature of all quantum objects. Thus, the uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology.
[+] [-] azeemba|3 years ago|reply
Similar behavior shows up in Fourier Transforms [1]. When an audio sample is converted into frequency-space, it shows similar "narrowing/widening" relationship with the original audio sample. This again isn't a limitation of measurement, its just the relationship between the two bases
[1]: https://en.wikipedia.org/wiki/Fourier_transform
[+] [-] marcosdumay|3 years ago|reply
The hypothesis that it has a precise value is called "hidden variable theory" if you want to search for it. We know that several variants of it are false because of how particles interfere with each other and how they can become correlated. The interference math is quite simple (the fact that particles have destructive interference means they aren't just a lot of stuff with unknown properties), but the math on correlation is complicated.
There is still an hypothesis called "superdeterminism" that say that all the information about the entire past and future of particles always existed with infinite precision. That one we can't rule out.
[+] [-] l33t233372|3 years ago|reply
[+] [-] illini1|3 years ago|reply
[+] [-] shonenknifefan1|3 years ago|reply
https://en.wikipedia.org/wiki/Uncertainty_principle?wprov=sf...
[+] [-] chadcmulligan|3 years ago|reply
[+] [-] russdill|3 years ago|reply
[+] [-] whatshisface|3 years ago|reply
[+] [-] danbruc|3 years ago|reply
In consequence, classically one can prepare a particle with any position and any momentum independently, i.e. one fixes the position and the time derivative of the position. Quantum mechanically fixing the position distribution also fixes the momentum distribution as the later is the derivative of the former with respect to space and therefore can not be specified independently. Similarly to the classical case one can also fix the time derivative of the wave function but that corresponds to the energy, not the momentum.
[1] In position space.
[+] [-] snarkconjecture|3 years ago|reply
[+] [-] tromp|3 years ago|reply
Does that mean that quantum effects play an essential role in the operation of a transistor?
[+] [-] mikrl|3 years ago|reply
The fun effects of semiconductors rely on the electronic properties of certain solids (or their electrons, equivalently their absence ie ‘holes’) which take place at a small scale, hence the use of quantum mechanics.
Some of these effects can manifest at larger scales, and can be modelled classically too.
[+] [-] simne|3 years ago|reply
While every mosfet transistor have mode could considered genuine quantum (tunnel effect), it is not used in computers, it used in microwave generators/detectors.
What used in computers, are just switch mode (electric field creating/moving/changing conducting zone), which is mostly classic physics and created as fruit of garage engineering, without understanding of quantum mechanics.
And yes, this is very abstract border, as I know, 100% quantum effects considered when working with single atom or single electron (something <= 1nm), but for example, 100nm transistors, usually considered as classic physics.
BTW it is very interest, what will show image sensor with ~1nm sized cells. Now their cells are much larger than wavelength, but already created superlenses, could focus on nm-sized things.
[+] [-] justaj|3 years ago|reply
[+] [-] azeemba|3 years ago|reply
Do you mind sharing what browser/platform you are using?
[+] [-] simne|3 years ago|reply
[+] [-] mahathu|3 years ago|reply
[+] [-] TDiblik|3 years ago|reply