Equivalent Expressions: Solve Energy Spectrum Transcendental Equation

[intervoxel]

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

<br /> \sqrt{-\frac{E}{E+V_D}}=\tan\frac{a\sqrt{\frac{2m}{\hbar^2}(E+V_D)}}{2}<br />

In the literature I found the equivalent expression below

<br /> \cot a\sqrt{-\frac{2m}{\hbar^2}E}=\frac{2E+V_D}{2\sqrt{-E(E+V_D)}}<br />

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?

 

[Vanadium 50]

Those are not equal.

 

[intervoxel]

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

<br /> \cot a\sqrt{\frac{2m}{\hbar^2}(E+V_D)}=\frac{2E+V_D}{2 \sqrt{-E(E+V_D)}}<br />

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.

 

[Vanadium 50]

Those aren't equal either.

 

[olgranpappy]

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.

 

[intervoxel]

Thank you olgranpappy and vanadium50 for your valuable help.