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.
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,
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! )
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
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'sgolf ball simulation, the particle could be ANYWHERE under that entire circular
( It could be at a lot of the other places too, its just not as
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
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 findingit 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"?