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Does this problem have any sense?

Suppose an electron in an infinite well of length $0.5nm$. The state of the system is the superposition of the ground state and the first excited state. Find the time it takes the electron to go from one wall to the other.

Strictly speaking the electron isn't even moving and $\vert \Psi \vert ^2$ is zero at the walls so it doesn't even "touch" them.

It think the only solution would be a semiclassical interpretation.

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up vote 2 down vote accepted

Yes, I believe you have to think of it as if it were a semiclassical problem; you evaluate with QM the mean square velocity $\left< v^2 \right>$ of the particle, then calculate its square root; this should give you an estimate of the typical velocity of the particle. Once you have it, you divide the length of the well by it and find the time it takes the electron to go from one wall to the other as if it were a classical particle with velocity $\sqrt{\left< v^2 \right>}$

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Within the superposition of the ground and the first excited state, the wavefuncion oscillates between "hump at left" and "hump at right". Maybe you are asked to find the half-period of these oscillations?

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