Quantum Mechanics: Can the probability of finding a particle in the whole space be smaller or higher at certain times?
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A quantum system is described by a set of self-adjoint operators (A1…An,H) and a Hilbert space H. The mentioned operators represent the observables that you can experimentally measure and their eigenvalues the possible outcomes. Among them there is a special one, the Hamiltonian H, describing the time evolution of the system. A state is any element |ψ⟩∈H. What are the probabilities of being in a state (it could be a linear combination of several stationary states)? Assumptions provide that any state can be expanded onto a basis of eigenvectors of any of the observables, namely |ψ⟩=∑ici|ai⟩. This means that after having performed a measurement of the observable A your state can become any of its eigenvectors (namely any of its possible outcomes), collapsing into them with frequencies of |ci|2 if performing infinite measurements. What are the probabilities of being in a stationary state? It is not clear by stationary state. Any system changes in time, thus nothing is stationary. What can happen is that after a measurement of the energy your initial state becomes an eigenstate of the Hamiltonian (by definition of measurement); if so, and if the time evolution operator is diagonal onto the eigenstates of the Hamiltonian, then a subsequent measurement of the energy will give back the same state because the eigenvectors remain such.