Transmission coefficient of an electron through a saddle-point potential in a magnetic field

We study the motion of a charged particle in two dimensions in a quadratic saddle-point potential ${\mathit{V}}_{\mathrm{SP}}$(x,y)=${\mathit{U}}_{\mathit{y}}$${\mathit{y}}^2$-${\mathit{U}}_{\mathit{x}}$${\mathit{x}}^2$+${\mathit{V}}_0$, in the presence of a perpendicular magnetic field. A simple, analytic expression is obtained for the transmission coefficient through the saddle point. We also calculate the transmission coefficient in the Wentzel-Kramers-Brillouin approximation, and find that this agrees well with the exact result, provided that the distance of the closest approach of the classical trajectory of the electron to the saddle point is not too small. Our analysis makes use of the fact that the Hamiltonian for this system can be expressed as a sum of two commuting Hamiltonians, one involving only the cyclotron coordinates, and the other involving only the guiding-center coordinates. The former has the form of a one-dimensional particle in a confining harmonic potential and describes the oscillations of the electron about the guiding-center position. The latter has the form of a one-dimensional particle in an inverted harmonic potential.