Understanding the HUP

Too many times I've seen this formula:

dx * dp > h_bar/2 (Heisenberg Uncertainty Principle)

accompanied by a description like this:

The better you know the position x, the larger the momentum p becomes, and
the better you know the momentum p, the larger the position x becomes.

Don't those of you spouting this realize that

1. this makes no sense! and
2. this is NOT what that formula states?

How can a POSITION as defined by a single coordinate in space ever become
large? It's a theoretical point. Points are by definition infinitesimally
small. Further, how can it be possible that the momentum of a stationary
elementary particle can become larger than the total momentum of our entire
galaxy? Since momentum requires motion, how can a stationary particle even
have an momentum?

Don't try to answer these questions. They only come up because of a
misunderstanding of what the HUP means. Here's the real breakdown...

dx <-> the error of the position value
Put another way, dx represents the range of possible values for the position
of the particle. It's not the actual position, but rather just a delta
corresponding to the experimental uncertainty in the position.

dp <-> the error of the momentum value
Much like dx, dp is just another range of values that might be the actual
momentum of the particle.

It's actually a matter of common sense why you cannot know x and p with near
absolute accuracy at the same time. Consider the formula for momentum:

p = m * v

Of course, there's a different formula for massless stuff, but this one
works well enough. Where

p <-> momentum
m <-> particle mass
v <-> particle velocity

we can replace v with dx/s where

dx <-> change in position
s <-> elapsed time

This leaves us with

p = m * dx / s

So for low values of dx, p will become less and less accurate. After all,
our experiment equipment is only just so accurate. Therefore dp will
increase as dx decreases. Likewise dp will decrease as dx increases. Similar
simple observations can be used to verify this interpretation of the HUP for
all of its variations.

Remember, the bottom line is that the HUP doesn't determine the values, or
for that matter even constrain them. The only thing the HUP does is give a
reasonable minimum MARGIN OF ERROR in measuring the values of certain
property pairs. Nothing more. Nothing less.

R.


.

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