 Waves

# Waves

If a particle exists somewhere along a line (call it the x axis) then even though you might not have a clue as to "where" exactly it is, if you add up all the little probabilities of its being here or there etc, they simply have to add up to unity, 1.

That's just another way of saying "the particle has to be SOMEWHERE".

Also, although it may be more likely for the particle to be in some places than others,
and you might not even have a clue about the actual SHAPE of the probability curve
(that is; a line who's height represents the probability of finding the particle at that particular x),
even so, its a fair bet that the curve DECREASES in height as x goes out to "minus infinity" on the left and also as x goes out to "plus infinity" on the right.

Why?

because if it was the other way round, the sum of all the 'little' probabilities wouldn't be so little anymore! It would be infinite! And that's a lot bigger than 1.

Anyway, with assumptions such as these and such as "the value you get for a location measurement is the value you get for a location measurement" (in other words "x=x"), it is possible to derive the fact that the probability of finding the particle at this or that place, actually oscillates!

That is, if a particle has a well defined momentum in the x_direction, then there is a kind of a periodicity in the probability of the particle being found at various locations along the x axis.
This probability is the "square" of the "probability amplitude" at x, and this probability amplitude can be negative.

This is the "wave" nature of matter.

It DOESN'T mean that "a point particle is a wave" or any other such drivel.
A mathematical point cannot be a wave, and a wave, which is a distributed concept, cannot be a mathematical point.
( and any 'physicist' who tells you any different is incompetent! )

Anyway...

Imagine a wave, like a ripple on a pond.
The square of the amplitude of the ripple-crest gives you the probability of finding the particle at that radius.

But there is no suggestion that the particle is 'moving' like a surfer on the ripple, or anything even remotely like a straight line trajectory, but rather, as in the previous section's golf ball simulation, the particle could be ANYWHERE under that entire circular crest.

( It could be at a lot of the other places too, its just not as likely)

Anyway, it is this probability amplitude that gets 'defracted' going through the famous double slits
( actually particles are sick and tired of tourists asking them to go through the double slits 'one more time' just for a selfie, so give it a break ).

The question of "which slit" the particle "goes through" is a red herring because the "point particle" NEVER moves in ANYTHING LIKE a classical trajectory ANYWAY!
Only the maxima and minima of "its most likely position" move like that,
and THEY interfere and diffract etc.
(think of the point particle as 'teleporting' here and there bizillions of times every femptosecond, yet always conforming to the overall 'wave shape') If you put your finger on it / detect it on a screen, it is just your average eigenpeanut!

** There is a difference between merely imagining a point particle as having a location, and actually finding it to be at a location.

When a particle is actually found at some location or other, then by that very fact, the 'wave' describing the probability of finding it,
obviously no longer exists.

The particle does not 'collect itself together' from all those locations upon being detected.
Rather the particle is detected where it is detected, because it happened to be just there (for no causalistic 'reason' but by pure probabilistic randomness) at that instant of its detection!

Now what's this 'quantumey' stuff about "infinite dimensions"?

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