Equivalent expressions

1. Sep 18, 2011

intervoxel

In my research project I arrived at a particular case where the energy spectrum is given by the following transcendental equation

$\sqrt{-\frac{E}{E+V_D}}=\tan\frac{a\sqrt{\frac{2m}{\hbar^2}(E+V_D)}}{2}$

In the literature I found the equivalent expression below

$\cot a\sqrt{-\frac{2m}{\hbar^2}E}=\frac{2E+V_D}{2\sqrt{-E(E+V_D)}}$

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.

2. Sep 18, 2011

Staff Emeritus
Those are not equal.

3. Sep 18, 2011

intervoxel

You are right. There's an error now corrected:
$\sqrt{-\frac{E}{E+V_D}}=\tan\frac{a\sqrt{\frac{2m}{\hbar^2}(E+V_D)}}{2}$

$\cot a\sqrt{\frac{2m}{\hbar^2}(E+V_D)}=\frac{2E+V_D}{2 \sqrt{-E(E+V_D)}}$

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.

4. Sep 18, 2011

Staff Emeritus
Those aren't equal either.

5. Sep 18, 2011

olgranpappy

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.

6. Sep 19, 2011