Here's a Weird One: Probability Can FLOW Like a Liquid (in Quantum Mechanics)

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In quantum mechanics, probability can flow through space and time, in exactly the same way as a fluid does!
It's worth recalling that in quantum mechanics, a system is described by its wave function - the mathematical function that contains all the information we can know about the system. And when we take the square modulus of the wave function, this can be used as a sort of probability density.
In other words, if we find the area under the wave function squared graph, between two points in space, then we calculate the probability of finding (e.g.) a particle between those two points in space. This can be extended to three dimensions, so the square modulus of the wave function can be integrated over a particular volume of space to give the likelihood of the particle being found in that volume.
The wave function of any system changes over time according to the Schrodinger equation. This can mean the wave function can simply "move" through space, hence the probability of finding our particle at different points in space can change, and the probability "flows" through space. Or a more complicated version is when the shape of the wave function changes. Either way, this results in a change in probability over time, which can be described as a probability flow.
The thing is though, the "flow" of probability through space is not some abstract concept - we can actually calculate this, and it turns out that the continuity equation describes this flow. The continuity equation is otherwise used to describe the flow of real fluids (i.e. liquids like water and juice, as well as gases), so it's almost surprising that probability in quantum mechanics follows the same equation.
The continuity equation looks at the density of a flowing quantity (whether that's a fluid or probability), and more specifically studies the rate of change of that density. In addition to this, it also looks at the divergence of the fluid or probability current. we take a brief look at this in the video, but for a more complete explanation check out the videos linked below.
In a nutshell, the equation considers an object flowing into a region of space, and equates this to the amount stored in the region + the amount leaving the region. This makes intuitive sense, but is only true if we consider the object flowing in to be a conserved quantity. For example, a real fluid must have a conserved amount of mass, meaning it cannot be created or destroyed in the region of space we are considering. Similarly, total probability is conserved anyway since the sum of all possibilities must always be 100%.
Videos linked in the cards for this video:
Wave Functions - • Wave Functions in Quan...
Schrodinger Equation - • Schrodinger Equation E...
Nabla / Del - • This Downward Pointing...
Continuity Equation - • How Stuff Flows: Conti...
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Timestamps:
0:00 - Probability Can Flow (Believe It Or Not)
0:33 - Wave Functions and Probability
1:59 - The Schrodinger Equation and Probability Flow
3:07 - Sponsor Message - Click the Link Below to Calculate Your Carbon Footprint!
4:52 - The Continuity Equation for Probability Flow
6:35 - The Continuity Equation for Fluids
7:04 - Interpreting the Continuity Equation for a Region of Space
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Опубликовано:

1 авг 2024

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Комментарии : 82

@ParthGChannel 2 года назад

Big thanks to Wren for sponsoring this video! Offset your carbon footprint on Wren: www.wren.co/start/parthg The first 100 people who sign up will have 10 extra trees planted in their name! And as always, let me know what other topics to cover in future videos :)

@lorigulfnoldor2162 2 года назад

Is there really no intuitive reason why "fluid behavior" should be a case? What about waves? Fluids make waves because of some wave-generating property, probability makes waves because of some wave-generating property?

@pratikiran 2 года назад

I am big fan of your videos, Could you give an introduction to Bell inequalities.

@balasubr2252 2 года назад

I have been theorizing that, there might be societal spacetimes for natural languages and if it is true would like to calculate the probabilities of miscommunication using the natural languages between different societal members. Would you like to make a video for such a use case?

@joaogabriel6424 2 года назад

The fluid analogy is consistente because the "mass" of the probability is also constant (the integral of rho is always 1), so the continuity equation and the fluid analogy works.

@NoSTs123 2 года назад

Particels behave like waves and probability like water. Ok, got it.

@marius3023 2 года назад

Probability can flow EXCACTLY like a fluid (kind of) 💀💀💀💀 😂

@sjzara 2 года назад

As probability is an abstract concept, it can have no causal effects and so can’t be the thing that is flowing. This is called “confusing the map with the territory”. Whatever is flowing can be modelled as a probability amplitude, but we don’t know what is flowing: perhaps we never shall!

@alwaysdisputin9930 2 года назад

I think maybe the Pauli exclusion principle is because : If you try to cram 2 fermions (eg electrons) into the same space you get: ◠◡ + ◡◠ = ---------------------- zero probability so there's no chance of it happening? ↖ (Parth G's previous video said fermions have an 'antisymmetric wave function' i.e. a wave function that becomes negative when you swap your 2 electrons around) So isn't this an example of probability having causal effects? & in her video 'What's "inside" a black hole?' Dr Becky says if gravity's strong enough, electrons can be pushed together close enough for them to merge with protons thus forming neutrons & thus a neutron star is born. So I think this probability gives rise to a physically repulsive force in the Pauli exclusion principle but if you apply enough force you can overwhelm it.

@davidwright8432 2 года назад

There's no confusion,. That distinction was mentioned in the video. 'Probability flow' is a metaphor - an 'as if'. Distinct from an 'is.'.

@alwaysdisputin9930 2 года назад

@@davidwright8432 So when photons are fired 1 by 1 like tennis balls through the slits in the double slit experiment, what are the waves made of? They must be waves because they spread out from the slits like if you threw a stone into a pond. & then they exhibit the wave phemonenon of constructive & destructive interference to form a zebra pattern on the screen. But waves of what? Probability?

@lorigulfnoldor2162 2 года назад

@@alwaysdisputin9930 ....waves of quantum field?

@Rudol_Zeppili Год назад

The changes in probability do have a casual effect, they make it statistically more likely or less likely to interact with a particle in a region of space. You might say “well in any given interaction, the particle can be anywhere therefore there’s no effect” but that’s not true to begin with in a single interaction since the probability distribution effects how likely you’ll encounter a particle in a single interaction, and with multiple interactions the changes in probability distribution cause a noticeable change as now interactions happen mostly in one given area and much rarely in others so that changes how other particles interact with that particle, and thus has a significant effect. if probability distributions didn’t have effects we wouldn’t be able to observe them, which we absolutely can.

@hamsandwichlover3075 2 года назад

you explain things so clearly, love your videos :)

@dennisant6876 2 года назад

Path G, thanks a lot for this interesting video! I would love to see how is this continuity equation for probability derived .

@PTGaonkar 2 года назад

You deserve much more views man love your videos specially your amazing simple explanation of much more complicated topics and it also shows your understanding in physics. As Albert Einstein said "if you can't explain it to a 6 years old then you don't understand yourself...

@mariocesarsousa 2 года назад

Thanks a lot. That's really interesting.

@System.Error. 9 месяцев назад

This is also well explained in the new Quantum Mechanics book by Jackson, which shows that using the probability current makes the Schrodinger equation like a continuity equation in fluid mechanics

@lepidoptera9337 3 месяца назад

If Jackson writes that, then he has lost his mind completely. There are no probabilities in quantum mechanics until we apply the Born rule.

@Stephanpar23 2 года назад

In addition, there is an intuitive deduction that can be made about probability fluid in terms of wave-particle duality. Combining this with the Time-Dependent Schrodinger model, we can see that as particles propagate through space, they all share the same kinetic and potential energy states in terms of interaction.

@davidwright8432 Год назад

Great content, as usual - because crisp and clear! - but when you say 'this video' and point to ... something, that 'something' doesn't appear on the screen. Maybe say, 'See playlist before comments'. But mainly, thanks for such a wonderful series of videos.

@matheuso.7204 2 года назад

That's a great video >

@phy_dude 2 года назад

Top notch 💚

@SaeedNeamati 2 года назад

you're a gift for minds. thank you for your valuable contents.

@matthiasfreiburghaus4202 2 года назад

I remember when we had to show this in our quantum mechanics course as an exercise. It made me so happy :)

@schmetterling4477 10 месяцев назад

You were happy about a definition? Wow... just imagine what will happen once you meet the right woman. Your head will explode. ;-)

@buccaneerrex 2 года назад

Warning: Lay physics ramble below: I have always thought that in a sense, what 'banged' in the big bang was the possibility for change to happen. That phase change propagates outward from the origin, and this wavefront of possibility is the present moment of time. After all, if the universe is infinite in all directions and infinitely dense (or at least at a much higher energy density per unit space) the only direction in which things can change is not likely to be one of the three spatial dimensions we're familiar with. I thought about the idea of a vacuum decay event propagating outward at c, and have seen speculation that the initial expansion event and formation of matter and energy in the universe was from a similar vacuum phase change. But since the big bang happened everywhere, it can't propagate 'out'. Unless you've got an extra dimension handy. So the idea that probability can 'flow' seems intuitive in that sense. 'energy', the probability evolution of particles and radiation, are different descriptions for the same phenomenon of the universe's substance undergoing that phase change. The big bang is still banging, and we're not what's left, we're the wave as it ripples 'out' from the center. With the 'center' in this case being the origin of time.

@alphalunamare 2 года назад

So does the probability remaining in the volume correspond to the measurement?

@axelperezmachado3500 2 года назад

so how far does the analogy extends? can wave functions have viscosity?

@yahiazakaria3953 2 года назад

Thanks. Please best book quantum mechanics?

@m.c.4674 2 года назад

If you want to know why the formula describe a physical fluid . I suggest you should find out when , what and why these formula were originally created . This all goes back to a mechanical description of gravity and light.

@Stephanpar23 2 года назад

So I am guessing that the continuity equation can be rewritten also as the Time-Dependent Schrodinger equation with the derivative of rho with respects to time being our kinetic energy value while the divergence times the fluid current would be our potential energy value given the total energy value of our expression as the wave function propagates through time and space.

@fotnite_ 2 года назад

As a stats student, it's interesting to see the different ways in which this kind of thing is talked about

@JohnDlugosz 2 года назад

So what would turbulence look like in probability density?

@johnholly7520 2 года назад

Awesome

@davidsweeney111 2 года назад

What’s the furthest distance an electron of say lithium could be found from the nucleus? How do we calculate that?

@bullpup1337 2 года назад

There is no furthest distance. The probability just goes down a lot if you go far away. I think the calculation will involve solving the Schrödinger equation. But it would be easier for hydrogen with only one electron and proton.

@ydnas5309 2 года назад

Hello. Have you ever consider to major in biophysics?

@xjuhox 2 года назад

Well, one future topic might be the hydrodynamical formulation of QM. Besides the continuity equation, the quantum potential takes its form. The strange quantum potential illustrates how squeezing the wave function loads additional energy into the system.

@schmetterling4477 10 месяцев назад

Unitarity is nothing special. It's simply the consequence of QM being an ensemble theory, which is a human made definition. Nature knows nothing about that.

@sitaramar13 2 года назад

have a doubt sir. In wave propagation water or sound , how individual particle vibrations are transmitted from particle to particle in the direction of wave propagation? There are gaps between particles in solids, liquids and gases. Is this not against principle of locality

@schmetterling4477 10 месяцев назад

The individual interactions in gases and liquids happen through energy and momentum exchange by collision, but please do not mistake what you are seeing here for a classical liquid. The meaning is completely different.

@geoffrygifari3377 2 года назад

hmmm so analogous to conservation of mass, i guess because the particle has to be found *somewhere* gives rise to the continuity equation...

@schmetterling4477 10 месяцев назад

No, not really. What is conserved here is the number of systems in the quantum mechanical ensemble. That "the particle has to be found somewhere" is not even good physics. Take the case of a decaying nucleus. The probability to find the original nucleus somewhere decreases exponentially in time. The physical description is still unitary because the probability of finding either the nucleus or its decay products is still the same.

@joaqbadillo4673 2 года назад

Nice video! Although I wouldn't say it flows *exactly* like a fluid since the continuity equation is not the only equation used to describe fluids. I feel clickbaited :( jk I really like your content and it was an interesting perspective :D

@iu6iu6 2 года назад

@salec7592 2 года назад

Conservation of total probability (equals 1) is axiomatic, given in the definition of probability. And, whenever there's conservation, there's continuity equation, right? So, is this strictly quantum physics related then?

@feiyuqiu7912 2 года назад

That is my idea ten years ago

@mitchellbriggs7856 2 года назад

What about into the fourth dimension?

@schmetterling4477 10 месяцев назад

What about it? You can build ensembles in any number of dimensions you like.

@pedronobre3898 2 года назад

How come squaring the wave function gives us the probability of finding the particle in a given region?

@alwaysdisputin9930 2 года назад

This video by Dos shows the wave function has a complex component + a real component: 'A Simple Explanation of Quantum Wavefunctions' ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-sOI4DlWQ_1w.html Say you have a complex number e.g. (4 - i) & you square it: (4 - i)(4 - i) = 16 - 4i - 4i - i² = 16 - 8i - i² but i means √ - 1 therefore i² = -1 = 16 - 8i - -1 = 17 - 8i So that would give a probability of 17 - 8i % What the heck does that even mean? So instead of squaring they do "multiplying by its complex conjugate" like this: (4 - i)(4 + i) = 16 - 4i + 4i - i² = 17 Thus "multiplying by its complex conjugate" is a trick they use to get rid of the complex part of the wave function so it's an easy-to-understand wave of (real) probabilities

@pedronobre3898 2 года назад

@@alwaysdisputin9930 Oh, i thought that somehow squaring the wave function gave us the probability whereas the non-squared function didn't, but in reality they both do but "squaring it" up gets rid of the annoying imaginary part. Did i get it right? Also, thanks for taking your time to explain it to me.

@itsbs 2 года назад

Parth - do you really believe "probability density" and "probability current"? Why would you abstract it from Schrodinger's own design of electric charge density and current? The experiments to prove that the math predicts the outcome do not use "probability testers/meters? What equipment do we ACTUALLY use in experiments to determine the result?

@schmetterling4477 10 месяцев назад

We are always using the same equipment: energy sources and energy measuring devices. Did nobody ever explain physics to you?

@parthabrata2003 Год назад

My name is also parth. ,,, love from india

@ChaineYTXF 2 года назад

neat

@yitzakIr 2 года назад

Im gonna drink the quantum fluid

@physics-theworkingofeveryt6086 2 года назад

👍

@alwaysdisputin9930 2 года назад

Very interesting. TY. I must make flashcards of this continuity equation & get software to show me the flashcards every day so I don't forget it. So is the Pauli exclusion principle because : If you try to cram 2 fermions (eg electrons) into the same space you get: ◠◡ + ◡◠ = ---------------------- zero probability so there's no chance of it happening? ↖ (Your previous video said fermions have an antisymmetric wave function i.e. a wave function that becomes negative when you swap your 2 electrons around)

@yoloswag6242 2 года назад

After seeing this video, my layman brain tells me that we need about 10 more 'general relativity level' breakthroughs to even come close to understanding the nature of the universe. Am I right????

@Godakuri 2 года назад

Just in fluid mechanics alone, we are desperately in need of an "Einstein." But yes, you're right. We are far from understanding everything. Our latest understanding is summed up through Quantum Field Theory, which is my main end goal of self teaching

@rutam_.k 2 года назад

Why does his accent have little sprinkle of irish in it here and there lol thats new, no complaints tho

@schmetterling4477 10 месяцев назад

Why is that weird? That's how WE defined it. Dude... :-)

@CMDRunematti 2 года назад

I mean it's literally called wave function... Waves of the probability...

@schmetterling4477 10 месяцев назад

That's a common mistake you can find in many entry level QM textbooks. The wave function does NOT describe a probability. Only the Born rule does.

@GIACOMOANGELI 2 года назад

I can't believe someone this smart believes in the global worming BS

@davidwright8432 2 года назад

Yes, sponsors are essential. But having a disruption in the middle of a demanding narrative is a pain. It also makes the viewer (this one, anyhow) impatient with the sponsor. Not good - for the sponsor.

@bullpup1337 2 года назад

yeah… and you cannot pay your way out of it like with normal adverts and youtube premium

@alwaysdisputin9930 2 года назад

@@bullpup1337 have you ever seen the movie Big where Tom Hanks tells his secretary to obtain a video of a sport (some sort of strange US sport) with all the advertisements removed? You need a secretary like that, mate.

@esorse 2 года назад

An existential guarantee is assumed in a zero lower bounded, monotonically increasing temporal sequence, where no term is less than the previous one and given this, if a robot simultaneously throws four coins the same in every respect, implying an equal chance of either an H, or a T and subsequently, four coins are also consecutively thrown, with the respective outcome snapshots : H | H | T | T and H, H, T, T , implying that H | H | T | T has unequivocally occured in a world, characterized by a partitioned entity called an wholistic ontology, in contrast to H, H, T, T in four different temporally and hence space-timeally signatured worlds, then the probability of an H from presuming the world is described by either H | H | T | T, or H, H, T, T,, is two out of four throws equals 2/4 = 1/2 = 0.5 = 50 % in English language, rational number, decimal number and percentage form, where "=" means equals.

@bullpup1337 2 года назад

is this some kind of new random text generator your are using? It almost makes sense.