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At the heart of quantum mechanics is the super awesome superposition. Pretty much all of the weirdness of quantum mechanics is due to superpositions. Ever heard of Schrödingers cat?

When we speak of objects, we generally classify them by their possible states. for example, a cat is either dead or alive, |\textrm{dead}\rangle or |textrm{alive}\rangle. A ball is rotating clockwise or anticlockwise, |\circlearrowright\rangle or |\circlearrowleft\rangle. And remember our discussion about the Stern Gerlach experiment? Atoms have spin up or spin down, |\uparrow\rangle or |\downarrow\rangle. Any object that you can think of is essentially described by its possible states, and how they change when things happen to the object.

When you go about your physics business, you may insert into your equations that a particle is spin up, then go ahead and compute, say, how it may interact with other particles. Well, interestingly, the mathematics of quantum mechanics does not care if you plug in, instead of just a spin up particle, something that looks like this: |\uparrow\rangle+|\downarrow\rangle, a superposition of two states!

Is this important? as it turns out, yes. First of all, when an object is in a superposition an we go to test if it is spin up or down, for example, it always comes up as either up or down. However, we can also go and retune our measuring apparatus, and detect this particle state. We must do a rotation and measure a long some new states, the plus state |+\rangle = \frac{1}{2}( |\uparrow\rangle+|\downarrow\rangle ) and minus state, |-\rangle = \frac{1}{2}( |\uparrow\rangle-|\downarrow\rangle ) . Note that adding these two newly defined states produces |\uparrow\rangle. Now if the particle was placed in the plus state, we would always detect it in the plus state.

Final question, so if we detect a particle in, say, the plus state, can we be sure it was in the plus state? Answer: Nope. Just like if we detect a particle in the spin up state, we cannot be sure it was in the spin up state, it could have been in spin up and down, in which case there is a fifty percent probability of detecting spin up or down. However, after the measurement we can be sure that if the particle is left alone, it is in the state that we detected it to be in!

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