Analytical solution for bound state energies of infinite well

Question

I am trying to find bound state energies assuming infinite potential. I have been told it can be done by analytically solving Right Hand Side and Left Hand Side of an equation such as: $$\sqrt{E}\tan^{\frac{1}{2}}(\frac{2ma^2E}{4\bar{h}}) = \sqrt{V_0-E}$$ If solved properly, it should give one curve (RHS), crossed by several LHS curves. Intersection points are the answers I am looking for. Each intersection corresponds to one n. I am wondering if it can be done by Matlab or Mathematica? Sorry if it is too basic :) Thanks

Answer 1

If you mean you're trying to solve for $E$ $$ E^{1/2} \tan\left(\dfrac{2ma^2 E}{\hbar}\right)^{1/2} = (V_0 - E)^{1/2}$$ there are no "analytical", i.e. closed form, solutions. It can be solved numerically, given numerical values for the constants. You might also put this in non-dimensional form: if $x = 2 m a^2 E/\hbar$ and $v = 2 m a^2 V_0/\hbar$, the equation becomes $$ x \tan(x) = v - x$$