Semiclassical Bohr quantization with a magnetic potential

[zetafunction]

Homework Statement

given the Hamiltonian in one dimension $$H= \frac{(p-eA)^{2}}{2m}+ V(x)$$

use the Bohr-Sommerfeld quantization in one dimension to obtain n=n(E)

Homework Equations

Hamiltonian , quantization

The Attempt at a Solution

from the usual quantization algorithm in one dimension i get

$$\int_{0}^{a}dx (2m)^{1/2}(E-V(X))^{1/2}+ \int_{0}^{a}dxA(x) =nh$$

here 'a' is a turning point so V(a)=E , simply the contribution to n(E) by the magnetic potential is an integral of A(x)

 

[NateD]

over the range of 0 to a , so n(E)= (2m)^{1/2}\int_{0}^{a}dx (E-V(X))^{1/2} +\frac{e}{h}\int_{0}^{a}A(x) dx