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## Tuesday 6 December 2022

### Physics Shorts: Macroscopic Uncertainty

Going to try something out. Some short physics offerings aimed at demystifying the alleged weirdness of things.

Here's why Heisenberg's Uncertainty Principle applies to macroscopic objects, but also why it appears not to.

Example:

Hesienberg’s Uncertainty Principle tells s that the uncertainty in position multiplied by the uncertainty in momentum must be greater than or equal to h-bar divided by two.

$\Delta x\Delta p\geq\dfrac{\hbar}{2}$

h-bar is the reduced Planck constant, and has a value of $1.05\times10^{-34}$ Joule-seconds. Numerically, it looks like this:

$0.0000000000000000000000000000000000105$

Let’s take an object - a bag of sugar, say - that weighs 500 grams. That’s a nice easy number to work with.

Let’s say we have a radar gun that can measure the speed to within .1 mph, or about .045 metres per second. That gives us 0.5 kilograms times .045 metres per second = 0.0225 kilogram metres per second. That’s our uncertainty.

Now we rearrange a little, and we get $\Delta x \geq \dfrac{\hbar}{2\Delta p}$. So, $2 \Delta p=0.045$, and we divide that long number above by that, which gives us a required uncertainty in position of

$\dfrac{0.0000000000000000000000000000000000105}{0.045}$

$\\left( =0.00000000000000000000000000000023333\\right)$ millimetres.

The result is, of course, a length many, many orders of magnitude less than the angular size of a standard half-kilo bag of sugar, let alone that of your corpulent self.