the point being that such elemental mathematics don't hold for e.g. spin (or magnetic spin quantum numbers).
Let's look at the following experiment:
You have a stream of particles of total spin 1/2 and have them fly in x-direction through a magnetic field along the z-axis. Turns out those particles (or rather their spin) has two distinct possibilities of arranging itself to the magnetic field (up or down). Which means that those particles will either fly upwards (z-axis) by a certain amount or downwards by the same amount. There's no telling ahead for each particle which way it's going to go.
Now you select those with spin up (say you just block those going down). If you do the same experiment again, you will find that they only go upwards. No problem yet...
But, say you were interested and sent that selected beam (remember only spin up) through another magnetic field in y-axis (perpendicular to the first experiment). You will find that they will again split in two groups one going into the positive, the other into the negative y-direction. Still no problem...
Now you select for either positive or negative y-direction (doesn't matter), and being the mean person that you are, send the beam through another magnetic field in z-direction.
The result is that you again get two distinct beams. The particles have lost their information pertaining to their magnetic alignment with respect to the z-axis.
so you see, such basic terms don't always hold true at the microscopic level...
regards,
(please don't forget to thank Profs. Stern and Gerlach for that nice experiment)
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