quantum superposition and measurement
Cover a tossed coin with a palm and, until you open it, it is neither face nor back, both remaining as possibilities. The quantum goes one step further. Before a measurement the particle is one superposed state α|↑⟩ + β|↓⟩, the up state |↑⟩ and the down state |↓⟩ mixed by amplitudes. The first picture starts by turning the angle at which those two amplitudes mix.
The notation met here for the first time is the kets |↑⟩ and |↓⟩. They are the two branches, up and down, and the amplitudes in front set the size of each branch. Turn the mixing angle and the two bars α = cos(θ/2) and β = sin(θ/2) grow in opposite directions. One end is a pure |↑⟩, the other end is a pure |↓⟩, and the middle is one state in which the two are mixed evenly.
A superposed state breaks the moment it is measured. Press measure and the state that was mixed by amplitudes drops to one of |↑⟩ or |↓⟩, and that drop is called collapse. Which side comes out is a probability set by the ratio of |α|² and |β|², so tilting the angle to one side makes that result come more often. The same seed is used, so each press brings back the same randomness.
Where that probability comes from is what the Born rule says. The probability of the result |↑⟩ is the square of the amplitude |α|², and the probability of |↓⟩ is |β|². Draw the two bars and, however you turn the angle, their heights always add to 1. The particle has to land somewhere of the two. The point is that the probability is the square of the amplitude, not the amplitude itself.
A state that has collapsed once stays there. Drop it by a measurement to either |↑⟩ or |↓⟩ and measure again at once, and the same result comes out. Press again and again and it does not change. The first measurement fixed the state to that eigenstate. Reset and it returns to the superposed state, and only then can the result differ.
The act of observing itself changes the state. Turn observation on and a measurement happens at every moment, so the state is fixed as |↑⟩ or |↓⟩ and no longer waves. Turn observation off and the superposed state stays alive, the amplitudes remaining mixed. Not looking and looking differ in this way. A measurement does not stop at reading information. It changes the object.