3.5.8.3 So­lu­tion pipeg-c

Ques­tion:

Shade the re­gions where the par­ti­cle is likely to be found in the $\psi_{322}$ en­ergy eigen­state.

An­swer:

The wave func­tion is

$$\psi_{322} = \sqrt{\frac{8}{\ell_x\ell_y\ell_z}} \sin\left(\... ...{2\pi}{\ell_y} y\right) \sin\left(\frac{2\pi}{\ell_z} z\right)$$

Now the trick is to re­al­ize that the wave func­tion is zero when any of the three sines is zero. Look­ing along the $z$-​di­rec­tion, you will see an ar­ray of 3 times 2 blobs, or 6 blobs:

Fig­ure 3.2: Eigen­state $\psi_{322}$.

The white hor­i­zon­tal cen­ter­line line along the pipe cor­re­sponds to $\sin(2{\pi}y/\ell_y)$ be­ing zero at $y$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac 12\ell_y$, and the two white ver­ti­cal white lines cor­re­spond to $\sin(3{\pi}x/\ell_x)$ be­ing zero at $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac 13\ell_x$ and $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac 23\ell_x$. The $\sin(2{\pi}z/\ell_z)$ fac­tor in the wave func­tion will split it fur­ther into six blobs front and 6 blobs rear, but that is not vis­i­ble when look­ing along the $z$-​di­rec­tion; the front blobs cover the rear ones. Seen from the top, you would again see an ar­ray of 3 times 2 blobs, the top blobs hid­ing the bot­tom ones.