Uncertainty‏

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The way you usually talk about the uncertainty principle

is to discuss measurements of position and momentum.

But it goes deeper than that.

It is an intrinsic limitation on 

what position and momentum mean.

Ultimately, position and momentum are not numbers.

They are more complicated objects called operators,

which I won't try to describe except to say that 

they are perfectly precise mathematical constructions

-just more complicated than numbers.

The uncertainty principle arises from 

the difference between numbers and operators.

The quantity ∆x is not just 

the uncertainty of measurement;

it is the irreducible uncertainty

of the particle's position.

What the uncertainty principle captures

is not a lack of knowledge,

but a fundamental fuzziness

of the subatomic world (…)

The uncertainty 

in the momentum of an electron

is essentially the momentum itself,

because of this uncertainty in direction.

The overall picture is that the electron

is trapped by its attraction to the nucleus,

but quantum mechanics does not permit

the electron to rest in its trap.

Instead, it wanders ceaselessly in a way 

that quantum mechanics describes.

This insistent wandering 

is what gives the atom it size.

If the electron were permitted to sit still,

it would do so inside the nucleus,

because it is attracted to the nucleus.

Matter itself would then collapse

to the density of the nucleus,

which would be very uncomfortable!

So the quantum wanderings 

of the electrons inside the atoms

is a blessing.

Although the electron

in a hydrogen atom

has an uncertainty position

and an uncertain momentum,

it has a definite energy.

Actually, it has several possible energies.

The way physicists describe the situation

is to say that electron's energy is 'quantized'.

The Little Book of String Theory

Steven S. Gubser

Princeton University Press (2010)