N.5 Why bound­ary con­di­tions are tricky

You might well ask why you can­not have a wave func­tion that has a change in wave func­tion value at the ends of the pipe. In par­tic­u­lar, you might ask what is wrong with a wave func­tion that is a nonzero con­stant in­side the pipe and zero out­side it. Since the sec­ond de­riv­a­tive of a con­stant is zero, this (in­cor­rectly) ap­pears to sat­isfy the Hamil­ton­ian eigen­value prob­lem with an en­ergy eigen­value equal to zero.

The prob­lem is that this wave func­tion has “jump dis­con­ti­nu­ities” at the ends of the pipe where the wave func­tion jumps from the con­stant value to zero. (Graph­i­cally, the func­tion is bro­ken into sep­a­rate pieces at the ends.) Sup­pose you ap­prox­i­mate such a wave func­tion with a smooth one whose value merely drops down steeply rather than jumps down to zero. The steep fall-off pro­duces a first or­der de­riv­a­tive that is very large in the fall-off re­gions, and a sec­ond de­riv­a­tive that is much larger still. There­fore, in­clud­ing the fall-off re­gions, the av­er­age ki­netic en­ergy is not close to zero, as the con­stant part alone would sug­gest, but ac­tu­ally al­most in­fi­nitely large. And in the limit of a real jump, such eigen­func­tions pro­duce in­fi­nite en­ergy, so they are not phys­i­cally ac­cept­able.

The bot­tom line is that jump dis­con­ti­nu­ities in the wave func­tion are not ac­cept­able. How­ever, the cor­rect so­lu­tions will have jump dis­con­ti­nu­ities in the de­riv­a­tive of the wave func­tion, where it jumps from a nonzero value to zero at the pipe walls. Such dis­con­ti­nu­ities in the de­riv­a­tive cor­re­spond to kinks in the wave func­tion. These kinks are ac­cept­able; they nat­u­rally form when the walls are made more and more im­pen­e­tra­ble. Jumps are wrong, but kinks are fine. (Don't break the wave func­tion, but crease it all you like.)

For more com­pli­cated cases, it may be less triv­ial to fig­ure out what sin­gu­lar­i­ties are ac­cept­able or not. In gen­eral, you want to check the ex­pec­ta­tion value, as de­fined later, of the en­ergy of the al­most sin­gu­lar case, us­ing in­te­gra­tion by parts to re­move dif­fi­cult-to-es­ti­mate higher de­riv­a­tives, and then check that this en­ergy re­mains bounded in the limit to the fully sin­gu­lar case. That is math­e­mat­ics far be­yond what this book wants to cover, but in gen­eral you want to make sin­gu­lar­i­ties as mi­nor as pos­si­ble.