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 Quantum Mechanics: The S...  
Schrodinger Equation -    • Schrodinger Equation Explained - Physics F...  
Nabla / Del -    • This Downward Pointing Triangle Means Grad...  
Continuity Equation -    • How Stuff Flows: Continuity Equation Expla...  

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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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