The phase space thing isn't so hard to understand. Just a fancy name for a familiar thing.
Replace the x and y labels on your axis with any 2 other things. Tada. You made a phase space!
Here, we're replacing x with x ... uhh... it's fine. Don't worry about it.
And we're replacing y with p. P for momentum, obv. sheesh.
So instead of an xy graph, we have an xp graph. Phase space!
Now, the uncertainty relation they're talking about without saying it is that old, familiar
delta_x * delta_p >= ℏ
In our graph, the delta_x is a range of values in the x-axis, and likewise, the delta_p is a range of values in the p-axis.
Drawing these out, we see that they make a rectangle area on our phase space.
The thing you posted consistently misnamed this area as a volume.
Since the uncertainty relation is greater than a non-0 value, that means we can't shrink the rectangle down to a point, which would allow us to know the position and momentum to arbitrary precision at the same time.
If it were >= 0, then that's just the classical world where it *can* go to 0. I.e. the area can always be bigger than the minimum.
We can always know very little about either position or momentum. But as we try to know both better and better, we find that at a certain point... reducing the uncertainty in one direction (x or p) always increases it in the other direction. Such that there is a minimum area that rectangle can have.
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