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If we change width of the infinite quantum well $L$ to $2L$ slowly enough, how it does change energy levels.

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I'm pretty certain that Introduction to Quantum Mechanics by David J. Griffiths covers exactly this, but I don't have a copy to hand to check. I did a quick Google but failed to find a copy. However I did find a page on Physics Forums that does exactly this calculation, apparently as a problem set in the Griffiths book. Have a look at and see if it helps.

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