What actually "is" an orbital cloud?

When you see pictures of atoms, they are often more accurately described as fuzzy probabilistic clouds.

But, what actually is that cloud mathematically? Is it a ...... vector field? A Lie group? If I wanted to write down a formula that prescribed a "value" or a "probability" to each and every point in 3D space, how would one do that in a manner consistent with the shape you see in atoms?

I'd prefer not to delve into PDE right away, I'd prefer to start with a preliminary example first.

For instance, how would one describe a "fuzzy sphere" in 3D space whose influence decays over distance?

 

I cannot answer mathematically. But I can philosophically. Perhaps it will help...

An orbital cloud is a wave of potential energy. In this case the energy of the electron. When and if it interacts with something the wave collapses to a particle. Naturally the farther away you are. The probability of the electron existing as a particle at all...drops to none.

 

One could imagine a sphere of water...if you touch the water it becomes "ice" where you touched it. The same with measurement of the water. If you ask how deep does the water go. Then this would be your "sphere" of influence. Touching it being the collapse of the wave function.