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Starting from here, will it ultimately converge over time into an energy eigenstate corresponding to ##E_n## ... OR.. will it slosh around forever in a complicated way?

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In summary: So, in summary, the wave function will slosh around forever in a complicated way, as it is composed of a linear combination of energy eigenfunctions and can never converge to a single energy eigenstate.

- #1

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Starting from here, will it ultimately converge over time into an energy eigenstate corresponding to ##E_n## ... OR.. will it slosh around forever in a complicated way?

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$$\psi_j=\int_{-\infty}^{\infty} u_j^*(x) \psi(0,\vec{x}),$$

where ##u_j(x)## is the energy eigenfunction with eigenvalue ##E_j=(j+1/2)\omega##, ##j \in \{0,1,2,\ldots \}##. Then the wave function at any later time is given by

$$\psi(t,x)=\sum_{j=0}^{\infty} \exp(-\mathrm{i} E_j t) \psi_j u_j(x).$$

This immediately shows that you never converge to an energy eigenfunction but that for any time all components of the initial wave function stay involved. This must be so, because only the energy eigenfunctions represent stationary states, i.e., if initially you don't have the system prepared in an energy eigenfunction the state can never become an energy eigenstate later.

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Thanks !

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