# Applicability of wavefunction matching when Hamiltonian (not just potential) varies

A simple tunnelling calculation can be performed for a potential step by calculating the eigenfunctions for the Hamiltonian on either side of the step and matching the wavefunctions (and using boundary conditions). This approach can be used for scenarios where the Hamiltonian is of the form $H=H_0 + V$ where $V$ is a "constant" (different on either side of the step). However if I have a Hamiltonian of the form

$$H=\begin{pmatrix}A & B(k_x -i k_y)\\ B(k_x +i k_y)&A\end{pmatrix} + \begin{pmatrix}C(k_x^2+k_y^2) & D(k_x+i k_y)^2 \\ D(k_x-i k_y)^2 & C(k_x^2+k_y^2)\end{pmatrix}$$

where all upper case letters are constants, but take different values for $x<0$ and $x>0$ is this approach of wavefunction matching at $x=0$ valid or not, and why?

• The standard tunneling calculation you refer to is done in position representation. But the Hamiltonian you give is a two-level one in momentum representation. How did you obtain it? Does the "two levels" part represent a real spin or a two-level treatment of the tunneling problem? Can you link to the source that introduced this Hamiltonian?
– udrv
Jun 21 '16 at 2:21
• Ok if you consider $k_i = -i\hbar \partial_i$ and the upper case letters are proportional to the step function which is also in position representation. The two levels come from psuedospin, this is sort of like a bilayer graphene Hamiltonian
– Tom
Jun 21 '16 at 8:56
• In this case don't worry about matching wavefunctions at the origin, just about boundary conditions. Pseudospin approximates tunneling as transitions between two orthogonal subspaces, each with its own basis of spacewise wave functions (plane waves, oscillator wavefunctions, etc) which are already well-behaved (continuous and satisfying boundary conditions). The tunneling rate becomes transition rate between pseudo spin levels, possibly between asymptotic states in each pseudospin subspace.
– udrv
Jun 21 '16 at 13:39