QM-22 WKB Approximation

Using WKB approximation find the energy levels of a particle in a gravitational potential of the earth. You may assume the acceleration due to gravity to remain constant and that the surface of the earth is a rigid boundary. 

Find the energy levels of a particle in well   $$ V(x) = A |x|  \qquad A > 0$$     using WKB approximation. How does your answer compare with the large $n$  limit            $$ E_n = (n-1/4)^{\tfrac{2}{3}} \Big( \frac{3\pi}{2\surd 2}\Big)^{\tfrac{2}{3}} \Big( \frac{A^2\hbar^2}{m} \Big)^{\tfrac{1}{3}} $$  of the exact answer.      [Mavromatis]

      Show that the energy levels of a particle in potential well  $$ V(x) = A |x|^p  \qquad A>0 $$  using the WKB quantization rule are given by

                  $$ E_n = (n+\tfrac{1}{2})^{2p/(p+2)}\left( \frac{2\pi  \hbar|A|^{1/p}} {\sqrt{2m}}\, I_p \right) $$

      where $I(p)$ is given by   $$  I(p)=2 \int_0^1 \sqrt{1-|x|^p} dx  $$

      [Mavromatis]

 Find the energy levels of a particle in a potential well   $$v(x)= A|x|^{1/2}$$ using the WKB approximation.

                $$ E_n= n^{2/5}\Big(\frac{15 \hbar A^2}{8\sqrt{2m}} \Big)^{\frac{2}{5}}$$