D.77 Im­pen­e­tra­ble spher­i­cal shell

To solve the prob­lem of par­ti­cles stuck in­side an im­pen­e­tra­ble shell of ra­dius $a$, re­fer to ad­den­dum {A.6}. Ac­cord­ing to that ad­den­dum, the so­lu­tions with­out un­ac­cept­able sin­gu­lar­i­ties at the cen­ter are of the form

$$\psi_{Elm}(r,\theta,\phi) \propto j_l(p_{rm{c}}r/\hbar) Y_l^m(\theta,\phi) \qquad p_{rm{c}} \equiv \sqrt{2m(E-V)}$$ (D.53)

where the $j_l$ are the spher­i­cal Bessel func­tions of the first kind, the $Y_l^m$ the spher­i­cal har­mon­ics, and $p_{rm{c}}$ is the clas­si­cal mo­men­tum of a par­ti­cle with en­ergy $E$. $V_0$ is the con­stant po­ten­tial in­side the shell, which can be taken to be zero with­out fun­da­men­tally chang­ing the so­lu­tion.

Be­cause the wave func­tion must be zero at the shell $r$ $\vphantom0\raisebox{1.5pt}{$=$}$ $a$, $p_{rm{c}}a$$\raisebox{.5pt}{$/$}$​$\hbar$ must be one of the zero-cross­ings of the spher­i­cal Bessel func­tions. There­fore the al­low­able en­ergy lev­els are

$$E_{\bar{n}l} = \frac{hbar^2}{2ma^2} \beta_{\bar nl}^2 + V_0$$ (D.54)

where $\beta_{\bar{n}l}$ is the $\bar{n}$-th zero-cross­ing of spher­i­cal Bessel func­tion $j_l$ (not count­ing the ori­gin). Those cross­ings can be found tab­u­lated in for ex­am­ple [1], (un­der the guise of the Bessel func­tions of half-in­te­ger or­der.)

In terms of the count $n$ of the en­ergy lev­els of the har­monic os­cil­la­tor, $\bar{n}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 cor­re­sponds to en­ergy level $n$ $\vphantom0\raisebox{1.5pt}{$=$}$ $l+1$, and each next value of $\bar{n}$ in­creases the en­ergy lev­els by two, so

$$n = l - 1 + 2 \bar n$$