The occurence of quantum events is interpreted entirely in terms of probability. We usually understand probability reasonably well where discrete events are involved. For instance,when a single die is thrown, we readily accept that there is a one-in-six chance of getting one of the possible numbers. There are six possible outcomes, each of which is equally likely (assuming a perfect cube for a die), and each one has a 1/6 probability of occuring.
Another way of defining probability is to throw the die many times, keeping track of the total number of throws (T), and the throws which produced a given number (say four) (F). As the total number of throws increased, the ratio F/T would approach 1/6.
What about multiple events? Take three dice. Throw the first one. The
probability of getting a 1 is 1/6. Throw the second one. The probability
of getting a 1 is also 1/6, and so on with the third. However, throw all
three together. The probability of getting a 1 AND ALSO getting another
1 AND ALSO getting a third 1 at the same time is 1/6 x 1/6 x 1/6 = 1/216.
This is fundamental about probability, where the outcome of multiple events
is characterized by the product of the probability of each individual event.
Sometimes you have to count carefully. With these three die, you can ask the chance of rolling a 7. Think carefully and you will find 15 distinct ways to get the number 7 (three 5-1-1, six 4-2-1, three 3-3-1, and three 3-2-2). The chance of getting a 7 is 15/216, or 5/72 or 0.06944 or 6.944%. By contrast the chance of getting a 3 (only 1 way, 1-1-1) is only 1/216 or 0.0046296 or 0.463%. By careful counting of the desired outcomes and the total attempts, we can easily determine probability in these discrete systems.
With these three dice, the numbers from 3 to 18 are possible to obtain. We can graph the probability against the number and obtain the following. Note how in a single throw of three dice, the probability of getting 10 is 0.125, or 12.5%, whereas the probability of getting the number 4 is 0.01389 or 1.389%. The probability of getting a 10 OR an 11, is 12.5 + 12.5 =25%. The probability of getting a 9 OR 10 OR 11 OR 12 is about 48.1%. To find the probability of a selection of altrnate outcomes, we add the probability of each event. If we throw the set of three dice twice the chance of getting a 10 the first time AND THEN getting an 11 the second time is 0.125 * 0.125 = 0.015625 or 1.56%. The probability of successive events occuring is the product of each individual event.
In Quantum Theory, we must deal with probabilities of a variable which
varies continuously, for instance, the x-coordinate. In such a case it
does not make any sense to ask, what is the probability of x = 0.250000...
occuring, since there are an infinite number of possible points on the
line and the probability of getting EXACTLY 0.25000... is infinitesimally
small. Instead, we speak about the probability of x lying within a small
interval on the axis, say between 0.25 and 0.30. When the interval is decreased
to an infinitesimal length, the interval between x and x + dx, the probability
as a function of position x is described by a function D(x) which is called
the probability density. Then the probability that the variable
will have a value between x and x + dx, is D(x)dx.
Imagine the probability density function to have some complicated shape,
as in the accompanying graph. To answer the question of the probability
of finding the value between A (say 1.5) and B (say 2.5), we add the probability
at each point between these two endpoints, exactly as we did in the discrete
case above. Except for a continuous variable, the summation is replaced
with an integral - we are looking for the area under the curve between
these two points. The purple region under the curve represents the probability
of finding the variable between the values 1.5 and 2.5. We have the relation
Probability is a real, non-negative number, so the function D(x) must
be real and positive everywhere. In quantum theory, the wavefunction can
be complex and negative, so that it cannot be a probability. However, the
SQUARE of the wavefunction is real-positive and the square of the wavefunction
is the probability density. You may note that for the curves drawn above,
I have scaled the probability so that if you add all of them up (or integrate
over the entire space), you get the number 1. This can be interpreted as
being the chance of getting ANY result out of all possible results is 1
- namely something must happen. With such scaled values, the probability
density is said to be NORMALIZED. A wavefunction, when squared and
integrated over all space gives the result 1, is also said to be normalized.
