WKB of modified Poschl-Teller potential

[dingo_d]

Homework Statement

I am given the modified Poschl-Teller potential:

$$V(x)=-\frac{U_0}{\cosh^2(\alpha x)}$$. I have to make WKB approximation of bound energy states and compare it to exact solution (analytical and numerical).

Homework Equations

Exact solution is given:

$$E_n=-\frac{\alpha^2\hbar^2}{2m}\left[\frac{1}{2}\sqrt{\frac{8m U_0}{\alpha^2\hbar^2}+1}-\left(n+\frac{1}{2}\right)\right]^2,\quad n\in\mathbb{Z^+}$$

WKB energy approximation is calculated from integral:

$$\int\limits_{x_1}^{x_2}\sqrt{2m(E-V(x))}dx=\left(n+\frac{1}{2}\right)\hbar\pi$$

$$x_1,\ x_2$$ are the turning points, places where $$V(x)=E$$.

The Attempt at a Solution

I found the solution in one article in Annals of Physics and it's:

$$E_n=-\frac{\alpha^2\hbar^2}{2m}\left[\sqrt{\frac{2m U_0}{\alpha^2\hbar^2}}-\left(n+\frac{1}{2}\right)\right]^2,\quad n\in\mathbb{Z^+}$$

And when I do the numerics via Python and Mathematica I get nice results (with Planck cons=m=alpha=1, and U_0=8) for ground state (n=0):

$$E_0^{exact/num}=-6.2344,\quad E_0^{WKB}=-6.125$$. So that really is the result. But how do I solve that nasty integral?!

It's:

$$\int\limits_{x_1}^{x_2}\sqrt{2m\left(E+\frac{U_0}{\cosh^2(\alpha x)}\right)}dx$$

Plus the turning points are something like arccosh which is a multivalued function.
Do I solve this by complex integration? And if so, how? Because usually the limits of complex integrals are usually something like infinities...

I tried substitution, $$z=\cosh^2(\aplha x)$$ but I had no luck. Plus the limits of the integral ends up the same when I substitute (probably because of the multivaluedness) :\

Any ideas?

 

[mark726]

Thank you for your post. The modified Poschl-Teller potential is a fascinating problem to work on. The WKB approximation can provide a good approximation for the bound energy states, especially for the ground state.

To solve the integral, you can try the following steps:

1. Use the substitution you mentioned, z=cosh^2(alpha x), and rewrite the integral as:

\int\limits_{z_1}^{z_2} \frac{dz}{\sqrt{z(z-z_1)(z-z_2)}}

where z_1 and z_2 are the roots of the polynomial in the square root.

2. Use the Euler substitution, z=t^2, to transform the integral into a standard form:

\int\limits_{t_1}^{t_2} \frac{2t}{\sqrt{(t^2-t_1)(t^2-t_2)}} dt

where t_1 and t_2 are the roots of the polynomial in the square root.

3. Use partial fraction decomposition to simplify the integrand.

4. Use the residue theorem to evaluate the integral.

Alternatively, you can also try using the method of steepest descent to evaluate the integral. This involves deforming the integration contour in the complex plane to a contour that passes through the saddle points of the integrand.

I hope this helps. Good luck with your calculations!