# Majorana wavefunction

I'm trying to compute the wavefunction for a Majorana state in an nanowire/superconductor hybrid system, like arXiv: Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures.

I use the same ansatz for the wavefunction $\Psi\left(x\right) = Ae^{zx}$ and obtain the characteristic polynomial

$$z^{4} + 4\left(\mu + 1\right)z^{2} + 8\lambda\Delta z + 4\left(\mu^{2} + \Delta^{2} - V^{2}\right) = 0\text{,}$$

where $\mu$ the chemical potential, $V$ the Zeeman field, $\Delta$ the superconducting gap and I know that $u_{\sigma} = \lambda v_{\sigma}$ where $u_{\sigma}$ describes electron states and $v_{\sigma}$ describes hole states with $\lambda = \pm 1$.To solve the above equation I use the fundamental theorem of algebra:

$$z^{4} + 4\left(\mu + 1\right)z^{2} + 8\lambda\Delta z + 4\left(\mu^{2} + \Delta^{2} - V^{2}\right) = z^4 - \left(z_1 + z_2 + z_3 + z_4\right)z^3 + \left(z_1 z_2 + z_1 z_3 + z_1 z_4 + 2_2 z_3 + z_2 z_4 + z_3 z_4\right)z^2 - \left(z_1 z_2 z_3 + z_2 z_3 z_4 + z_1 z_3 z_4\right)z + z_1 z_2 z_3 z_4 = 0\text{,}$$

where $z_{i}$ are the roots. We see directly that

$$\sum_{i = 1}^{4} z_i = 0 \text{ and } \prod_{i = 1}^{4} z_i = 4\left(\mu^{2} + \Delta^{2} - V^{2}\right)\text{.}$$

Now we can study different cases: the most interested case is when $z_{1/2}$ are complex and $z_{3/4}$ real. In this case we obtain

$$z_{1/2} = a \pm ib$$

and

$$z_{3/4} = -a \pm \sqrt{a^{2} - \frac{4\left(\mu^{2} + \Delta^{2} - V^{2}\right)}{a^2 + b^2}}\text{.}$$

In the publication they write that they have 4 boundary condition and 1 condition from the normalization.Okay, two from the fact that the wavefunction must localized at ends, so that I can write

$$\Psi\left(0\right) = \Psi\left(L\right) = 0\text{,}$$

and the same fact for the derivative of the wavefunction

$$\Psi^\prime\left(0\right) = \Psi^\prime\left(L\right) = 0\text{.}$$

Edit:

In the paper above they study two cases. In the first $\left(\mu^{2} + \Delta^{2} - V^{2}\right) > 0$ they say that they have 4 boundary condition and 1 condition from normalization but only 4 coefficients. I can explain that they neglect the two complex wavefunctions. However, in the opposite case $\left(\mu^{2} + \Delta^{2} - V^{2}\right) < 0$ they write that they have 6 coefficients and 6 Condition to solve the equation.

But now I'm completly confused! Why now 6 coefficients and 6 conditions?

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Nobody has an idea? –  Lars Milz Oct 27 at 12:30