Hello I understand how to approach finite potential well. However i am disturbed by equation which describes number of states ##N## for a finite potential well (##d## is a width of a well and ##W_p## is potential):(adsbygoogle = window.adsbygoogle || []).push({});

$$

N \approx \dfrac{\sqrt{2m W_p}d}{\hbar \pi}

$$

I am sure it has something to do with one of the constants ##\mathcal L## or ##\mathcal K## defined this way:

\begin{align}

\mathcal L &\equiv \sqrt{\tfrac{2mW}{\hbar^2}} & \mathcal{K}&\equiv \sqrt{ \tfrac{ 2m(W_p-W) }{ \hbar^2 }}

\end{align}

and the transcendental equations for ODD and EVEN solutions:

\begin{align}

&\frac{\mathcal K}{\mathcal L} = \tan \left(\mathcal L \tfrac{d}{2}\right) &&-\frac{\mathcal L}{\mathcal K} = \tan \left(\mathcal L \tfrac{d}{2}\right)\\

&\scriptsize{\text{transc. eq. - EVEN}} &&\scriptsize{\text{transc. eq. - ODD}}

\end{align}

QUESTION:Could anyoe tell me where does 1st equation come from? I mean ##\tan(W)## repeats every ##\pi##, but if i insert ##\mathcal L## in transcendental equation i have ##\tan(\sqrt{W})##! On what intervals does the latter repeat itself? Does this has something to do with it? It sure looks like it... Please help me to synthisize all this in my head.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Energies and numbers of bound states in finite potential well

**Physics Forums | Science Articles, Homework Help, Discussion**