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Natural Science Forum / Physics / Research / August 2007WKB approximation in cylindrical coordinates
In the context of the solution of Schrodinger equation, is there any modified form of the WKB approximation which can be applied for the Schrodinger equation in cylindrical coordinates? More specifically, if the potential depends only on the radial distance i.e V=V(r), then one arrives at the radial equation after separation of variables; how can this equation be solved using the WKB method? Thanks. > In the context of the solution of Schrodinger equation, is there any > modified form of the WKB approximation which can be applied for the > Schrodinger equation in cylindrical coordinates? More specifically, if > the potential depends only on the radial distance i.e V=V(r), then one > arrives at the radial equation after separation of variables; how can > this equation be solved using the WKB method? Yes. If you use separation of variables, to write the wave function in terms of spherical coordinates psi(x) = R(r) Y_lm(theta,phi), where Y_lm are the spherical harmonics, then you'll get a radial Schroedinger equation for the radial wave function R(r). If you are in 2D, then you can replace spherical coordinates with polar coordinates and replace Y_lm by appropriate angular functions. Also, if you are in 3D and are dealing with cylindrical symmetry, you can replace Y_lm by appropriate functions of phi and z. Use the substitution S(r) = r*R(r) and rewrite the radial Schroedinger equation in terms of it. The kinetic term of this new equation will be identical to the kinetic term of 1D linear motion. The potential term will be a modified version of V(r). In this form the radial Schroedinger equation is amenable to treatment by the WKB method. More details in Sections 32 and 49 of Landau's book on Non-relativistic Quantum Mechanics (Course on Theoretical Physics, vol.3). Hope this helps. Igor > Use the substitution S(r) = r*R(r) and rewrite the radial Schroedinger > equation in terms of it. The kinetic term of this new equation will be > identical to the kinetic term of 1D linear motion. The potential term > will be a modified version of V(r). In this form the radial Schroedinger > equation is amenable to treatment by the WKB method. Eh, I kept thinking of a spherical coordinates as I wrote this. If you use r to represent the distance from the z-axis, as in cylindrical or polar coordinates, then the correct substitution is S(r) = sqrt(r)*R(r). Igor |
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