##### 3.5.8.1 So­lu­tion pipeg-a

Ques­tion:

If the cross sec­tion di­men­sions and are one tenth the size of the pipe length, how much big­ger are the en­er­gies and com­pared to ? So, by what per­cent­age is the one-di­men­sion­al ground state en­ergy as an ap­prox­i­ma­tion to the three-di­men­sion­al one, , then in er­ror?

An­swer:

The en­er­gies are

If and are ten times smaller than then and are each 100 times larger than . So the one-di­men­sion­al ground state en­ergy is smaller than the true ground state en­ergy by a fac­tor 201. Which means it is off by 20 000%.