Probability Wave Function Ψ(r,t): Time Independence

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The wave function Ψ(r,t) is inherently time-dependent, reflecting the dynamic nature of quantum systems. However, the probability density [Ψ(r,t)]² can appear time-independent under specific conditions, particularly when the wave function represents an energy eigenstate. In such cases, the system's energy remains constant, leading to a stable probability distribution. This phenomenon highlights the distinction between the wave function's evolution and the probability interpretation in quantum mechanics. Understanding these concepts is crucial for grasping the behavior of quantum systems.
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Wave function Ψ(r,t) is time dependent. But then why probability [Ψ(r,t)]2 is time independent
 
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It isn't.
 
Sometimes it is, when psi(x,t) is an energy eigenstate. (I hope I don't need to explain why.)
 
Demystifier said:
Sometimes it is, when psi(x,t) is an energy eigenstate. (I hope I don't need to explain why.)

Oh, I thought the question was intended in more general terms.
 

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