3.5.8.2 So­lu­tion pipeg-b

Ques­tion:

At what ra­tio of $\ell_y$$\raisebox{.5pt}{$/$}$​$\ell_x$ does the en­ergy $E_{121}$ be­come higher than the en­ergy $E_{311}$?

An­swer:

Us­ing the given ex­pres­sion for $E_{n_xn_yn_z}$,

$$E_{n_xn_yn_z} = \frac{n_x^2\hbar^2\pi^2}{2m\ell_x^2} + \frac... ...ar^2\pi^2}{2m\ell_y^2} + \frac{n_z^2\hbar^2\pi^2}{2m\ell_z^2},$$

$E_{121}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $E_{311}$ when

$$\frac{\hbar^2\pi^2}{2m\ell_x^2} + \frac{4\hbar^2\pi^2}{2m\el... ...ac{\hbar^2\pi^2}{2m\ell_y^2} + \frac{\hbar^2\pi^2}{2m\ell_z^2}$$

Can­cel­ing the terms that both sides have in com­mon:

$$\frac{3\hbar^2\pi^2}{2m\ell_y^2} = \frac{8\hbar^2\pi^2}{2m\ell_x^2}$$

and can­cel­ing the com­mon fac­tors and re­ar­rang­ing:

$$\frac{\ell_y^2}{\ell_x^2} = \frac{3}{8}.$$

So when $\ell_y$$\raisebox{.5pt}{$/$}$​$\ell_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{3/8}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0.61 or more, the third low­est en­ergy state is given by $E_{121}$ rather than $E_{311}$. Ob­vi­ously, it will look more like a box than a pipe then, with the $y$-​di­men­sion 61% of the $x$-​di­men­sion.