Quantum Theory at Its Minimum 2 : Superposition of Basis
The crazy situation of our quantum arrow is summarized as the followings.
(1) Alice can place the quantum arrow (spin 1/2) in any angle he chooses.
(2) Bob can decide an arbitrary direction of his choice, and the arrow would be found either in that direction or in the oposing direction.
(3) The probability of Bob finding the arrow in one direction and its opposite depends on both the Alice's choice of the direction of placement and Bob's choice of the direction of the observation.
The fact that Bob can find an arrow in both "up" and "down" with finite probabilities must mean that the arrow prepared by Alice in some sideway direction can be interpreted as some sort of mixture of the "up"-state and "down" state. In quantum mechanics, this mixture is expressed as superposition.
Since Alice can prepare an arrow in any direction, and Bob can always observe it as the the mixture of "up" and "down" states, it must mean that by the superposition of "up" and "down" states can produce aebitrary state which represents the quantum arrow in any angle. In this sense, it is possible to call "up" and "down" states as the basis states. Same argument should hold all for any direction and its opposite, for example, "right" and "left", and this must mean that any two states with opposite direction defines a basis states. This argument leads us to the proosition
(4) The superposition of two quantum state is again a legitimate quantum states.
(5) All states of spin 1/2 can be expressed as the superposition of a state representing an orientation of and anoher state representing the opposite orientation. The choice of the roientation defines a basis set, and there are infitenite amount of different choices.
The fact that the spin direction comes out totally random when measured along an axis perpendicular to the axis of spin preparation can be thought of as a manifestation of the celebrated Heisemberg's uncertainty principle applied to thespin direction.
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