Solving a trascedental equation

[dingo_d]

Homework Statement

I have solved the problem of an infinite square well with a delta potential in the middle, what I got in the end is:

$$\cot(\frac{\sqrt{2 m E}}{\hbar}a)=-\frac{\Omega\hbar a \sqrt{2m E}}{2mE a}$$
, where a is the width of the well, m the mass of the particle in it, E the energy eigenvalue, and $$\Omega$$ some parameter that was with the delta potential.

Now I need to solve that equation and I have no idea how to do that :\

I'm not much of a numerical solver :\

Anywho the solution to Schrödinger eq. is:

$$u_n^+(x) =<br /> \begin{cases}<br /> A\sin(k_n^+(x+a)) & \text{if } -a\leq x < 0 \\<br /> -A\sin(k_n^+(x-a)) & \text{if } 0< x \leq a<br /> \end{cases}$$

And the $$k_n^+=\frac{n\pi}{2a}$$, where $$n=\pm 2,\pm 4,\ldots$$

 

[fzero]

You can use Newton's method:

http://en.wikipedia.org/wiki/Newton's_method

Just turn your equation into f(E) =0 and use Newton's method to find the roots. It usually helps to have a sketch of the function to determine good starting points.

 

[dingo_d]

Hmmm so I've set
$$f(ka)=\cot(ka)+\frac{\Omega}{ka}$$ and then with Mathematica I've made:
$$x_{n+1}=x_n+\frac{f(ka)}{f'(ka)}$$ and no matter what parameter I choose the solution doesn't converge :\

Now I dk if I'm doing sth wrong or not...

 

[fzero]

Hmmm so I've set
$$f(ka)=\cot(ka)+\frac{\Omega}{ka}$$ and then with Mathematica I've made:
$$x_{n+1}=x_n+\frac{f(ka)}{f'(ka)}$$ and no matter what parameter I choose the solution doesn't converge :\

Now I dk if I'm doing sth wrong or not...

There's a sign wrong in your iteration, it should be

$$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$

If you are using Mathematica, you can plot to get good starting points for Newton's method. You could also probably get a list of roots directly with Mathematica.

 

[dingo_d]

Oh, I've found it, I just hope it's right thing :D Thanks ^^