If we change width of the infinite quantum well $L$ to $2L$ slowly enough, how it does change energy levels.
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 http://www.physicsforums.com/showthread.php?t=453180 and see if it helps.