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Energies and numbers of bound states in finite potential well

  1. Apr 5, 2013 #1
    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):
    $$
    N \approx \dfrac{\sqrt{2m W_p}d}{\hbar \pi}
    $$
    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.
     
  2. jcsd
  3. Apr 5, 2013 #2
    It's a strange question, I'm not sure to have understood it.
    tan(x) is periodic with period = π; it means, for example, that tan(W) = 0 for: W = kπ where k is an integer; if it's tan(√(W)) = 0 it means that √(W) = kπ → W = k2π2.
     
  4. Apr 6, 2013 #3
    This did help. Thank you.
     
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