4.1.4.3 So­lu­tion harmd-c

Ques­tion:

Write down the ex­plicit ex­pres­sion for the eigen­state $\psi_{213}$ us­ing ta­ble 4.1, then ver­ify that it looks like fig­ure 4.2 when look­ing along the $z$-​axis, with the $x$-​axis hor­i­zon­tal and the $y$-​axis ver­ti­cal.

An­swer:

The generic ex­pres­sion for the eigen­func­tions is

$$\psi_{n_xn_yn_z}=h_{n_x}(x) h_{n_y}(y) h_{n_z}(z)$$

and sub­sti­tut­ing $n_x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 2, $n_y$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 and $n_z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 3, you get

$$\psi_{213} = h_2(x) h_1(y) h_3(z).$$

Now sub­sti­tute for those func­tions from ta­ble 4.1:

$$\psi_{213} = {\displaystyle\frac{[2(x/\ell)^2-1][2y/\ell][2(... ...ht)^{3/4}}} e^{-x^2/2\ell^2}e^{-y^2/2\ell^2}e^{-z^2/2\ell^2}$$

where the con­stant $\ell$ is as given in ta­ble 4.1.

The first poly­no­mial within square brack­ets in the ex­pres­sion above is zero at $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\ell$$\raisebox{.5pt}{$/$}$​$\sqrt 2$ and $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$$\ell$$\raisebox{.5pt}{$/$}$​$\sqrt 2$, pro­duc­ing the two ver­ti­cal white lines along which there is zero prob­a­bil­ity of find­ing the par­ti­cle. Sim­i­larly, the sec­ond poly­no­mial within square brack­ets is zero at $y$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, pro­duc­ing the hor­i­zon­tal white line. Hence look­ing along the $z$-​di­rec­tion, you see the dis­tri­b­u­tion:

Fig­ure 4.2: En­ergy eigen­func­tion $\psi_{213}$.

Seen from above, you would see four rows of three patches, as the third poly­no­mial be­tween brack­ets pro­duces zero prob­a­bil­ity of find­ing the par­ti­cle at $z$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-\sqrt{3/2}\ell$, $z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, and $z$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{3/2}\ell$, split­ting the dis­tri­b­u­tion into four in the $z$-​di­rec­tion.

This ex­am­ple il­lus­trates that there is one more set of patches in a given di­rec­tion each time the cor­re­spond­ing quan­tum num­ber in­creases by one unit.