Remeasurement of a quantum system

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After measuring a dynamical variable, the wavefunction collapses to the corresponding eigenfunction, and subsequent measurements yield the same results and probabilities for eigenvalues. This consistency occurs because the wavefunction remains in the eigenstate immediately after the measurement. Over time, the wavefunction begins to spread out due to the natural evolution of quantum states governed by the Schrödinger equation. The probability of observing a particular state later is determined by the overlap integral between the initial and final states. Understanding this process clarifies why eigenvalues remain unchanged upon repeated measurements.
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After making a measurement of a particular dynamical variable the wavefunction collapses into the corresponding eigenfunction. As I understand when the variable is then measured again the results and relative probabilities of eigenvalues are exactly the same as before. I don't understand why they are the same as before. What happens to the wavefunction?
 
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I think it slowly starts spreading out again.
 
ok. Why does it spread out though? Is there a reasoning behind why the eigenvalues will be the same when another measurement is made. There was a question in a book that kind of asked for an answer to this but I can't work out why?
 
Information. Start out with a state described by an eigenfunction then apply the propogator to it which describes the time evolution. At a later time the probability of observing some state is proportional to the overlap integral between the initial and final state...to put it simply
 

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