It has been a long time since I've worked on periodic boundary conditions involving the Bloch theorem (quantum mechanics) so I am hoping to get some help on my next step.
Suppose I have a lattice with period $a$ with potential $V(x) = v_{0}\sum \delta (x-na)$. At any boundary the potential is infinite) and any interval between two adjacent boundary the potential is zero.
- General solution to the wavefunctions
On the interval $x \in (0, +a):$
on this interval $v(x) = 0$ by design.
Using the Time - independent Schrodinger equation: $-\frac{h^{2}}{2m}\frac{d^{2}}{dx^{2}} \psi_{I}(x) = E\psi_{I}(x)$
Two complex roots: the general solution is $\psi_{I}(x) = Ae^{i\tilde{k}x} + Be^{-i\tilde{k}x}$
On the interval $x \in (-a, 0):$
on this interval $v(x) = 0$ by design and since this interval is just a repeat of the interval $I$, the Bloch wave can be invoked.
The Bloch wave has a form $\psi(x) = e^{ikx}\phi(x)$ where $\phi(x)$ is periodic.
$\psi_{I}(x - (-a)) = \psi_{I}(x + a) = e^{ik(x+a)}\phi(x+a) = ... e^{ika}\psi_{II}(x)$.
The general solution for $\psi_{II}(x):$ $\psi_{II}(x) = e^{-ika}\psi_{I}(x+a) = e^{-ika}[Ae^{i\tilde{k}(x+a)} + Be^{-i\tilde{k}(x+a)}]$
- Applying the boundary conditions
The boundary conditions are
2.1 continuity of wavefunction: $\psi_{I}(0) = \psi_{II}(0)$
2.2 continuity of the first spatial derivative of the wavefunction: $$\psi_{I}^{'}(0) = \psi_{II}^{"}(0)$$
2.1 yields $A-Ae^{-ika}e^{i\tilde{k}a} + B - Be^{-ika}e^{-i\tilde{k}a} = 0$
2.2 requires $\psi_{I}^{'}(0) - \psi_{II}^{"}(0) = \frac{2mv_{0}}{\bar{h}}\psi(0)$ to give
$A(1 - e^{i(k-\tilde{k}a)-\frac{2mv_{0}}{\bar{h}^{2}i\tilde{k}}}) + B(-1 + e^{i(k-\tilde{k})a} - \frac{2mv_{0}}{\bar{h}^{2}i \tilde{k}}) = 0$
The matrix representation for 2.1 and 2.2 can be expressed as
\begin{bmatrix} (1 - e^{-ika}e^{i\tilde{k}a}), (1 - e^{-ika}e^{-i\tilde{k}a}) \\ (1 - e^{i(k-\tilde{k})a} - \frac{2mv_{0}}{\bar{h}^{2}i\tilde{k}}), (-1 + e^{i(k-\tilde{k})a }-\frac{2mv_{0}}{\bar{h}^{2}i\tilde{k}}) \end{bmatrix}
\begin{bmatrix} A \\ B \end{bmatrix}
=
\begin{bmatrix} 0 \\ 0 \end{bmatrix}
In proceeding with solving the determinant, what are we looking for?