3.5.8.1 So­lu­tion pipeg-a

Ques­tion:

If the cross sec­tion di­men­sions $\ell_y$ and $\ell_z$ are one tenth the size of the pipe length, how much big­ger are the en­er­gies $E_{y1}$ and $E_{z1}$ com­pared to $E_{x1}$? So, by what per­cent­age is the one-di­men­sion­al ground state en­ergy $E_{x1}$ as an ap­prox­i­ma­tion to the three-di­men­sion­al one, $E_{111}$, then in er­ror?

An­swer:

The en­er­gies are

$$E_{x1}=\frac{\hbar^2\pi^2}{2m\ell_x^2}\quad E_{y1}=\frac{\hb... ...pi^2}{2m\ell_y^2}\quad E_{z1}=\frac{\hbar^2\pi^2}{2m\ell_z^2}.$$

If $\ell_y$ and $\ell_z$ are ten times smaller than $\ell_x$ then $E_{y1}$ and $E_{z1}$ are each 100 times larger than $E_{x1}$. So the one-di­men­sion­al ground state en­ergy $E_{x1}$ is smaller than the true ground state en­ergy $E_{111}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $E_{x1}+E_{y1}+E_{z1}$ by a fac­tor 201. Which means it is off by 20 000%.