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Equivalent expressions

  1. Sep 18, 2011 #1
    In my research project I arrived at a particular case where the energy spectrum is given by the following transcendental equation

    [itex]
    \sqrt{-\frac{E}{E+V_D}}=\tan\frac{a\sqrt{\frac{2m}{\hbar^2}(E+V_D)}}{2}
    [/itex]

    In the literature I found the equivalent expression below

    [itex]
    \cot a\sqrt{-\frac{2m}{\hbar^2}E}=\frac{2E+V_D}{2\sqrt{-E(E+V_D)}}
    [/itex]

    From the first expression I should get to the second one in order to show the consistency of the theory. But, no matter I have tried, I couldn't find out a solution for this problem.

    Can anyone help me, please?
     
  2. jcsd
  3. Sep 18, 2011 #2

    Vanadium 50

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    Those are not equal.
     
  4. Sep 18, 2011 #3
    You are right. There's an error now corrected:
    [itex]
    \sqrt{-\frac{E}{E+V_D}}=\tan\frac{a\sqrt{\frac{2m}{\hbar^2}(E+V_D)}}{2}
    [/itex]

    [itex]
    \cot a\sqrt{\frac{2m}{\hbar^2}(E+V_D)}=\frac{2E+V_D}{2 \sqrt{-E(E+V_D)}}
    [/itex]

    The eigenvalue E=-20.54769241 (for parameters V_D=50, a=1, m=1, hbar=1), for example, satisfies both equations.

    Still I'm not able to arrive at the other formula.

    I tried tan(x/2)=csc(x)-cot(x), etc.
     
  5. Sep 18, 2011 #4

    Vanadium 50

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    Those aren't equal either.
     
  6. Sep 18, 2011 #5

    olgranpappy

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    how about using:

    2cot(x)=((1/tan(x/2)) - tan(x/2))

    then plug in for tan(x/2), where x=a\sqrt(2m(E+V)/hbar^2), using the LHS of your first expression... and I guess be careful about signs and square roots.
     
  7. Sep 19, 2011 #6
    Thank you olgranpappy and vanadium50 for your valuable help.
     
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