7.1.2.2 Riccati and Prüfer: Dirichlet BCs

The boundary conditions given by 7.5 translate to

$$z'(0) + \imath k = \frac{2 \imath k}{e^z(0)} ,$$ (7.6)

$$z'(L) = \imath \sqrt{k^2 - 2 m^{*} V_L / \hbar^2} .$$ (7.7)

If we set $z' = u$ we get (with constant mass in the electrodes)

$$u'(x) + {u(x)}^2 = 2 (V(x) - E) \frac{m^*}{\hbar^2}$$ (7.8)

with the Dirichlet boundary condition

$$u(L) = \sqrt{k^2 - 2 m^{*} V_L / \hbar^2} .$$ (7.9)

Solving Equation 7.8 fixes $z'$. The value of $z$ is fixed up to an additive constant $c$, which is then determined by the second boundary condition. For the Prüfer Equation 6.39 we get the Dirichlet boundary condition

$$\alpha(L) = \arctan(\sqrt{k^2 - 2 m^{*} V_L / \hbar^2})$$ (7.10)

We stress that with inflow boundary conditions the wave function $\psi$ cannot have nodes, hence in this case the Riccati equation has an everywhere defined solution. The main advantage with respect to the Schrödinger equation is that it is a (nonlinear) first order equation with a Dirichlet boundary condition.

We have implemented both the Prüfer and the Riccati equation as an alternative to the Schrödinger equation. The results are reviewed in the next section.