How to Derive u_k in Kitaev's 1D p-Wave Superconductivity Model?

[DeathbyGreen]

Hello! So I'm really stuck in a personal quest to derive Kitaev's 1D p wave superconductivity model, and I'm stuck on the seemingly simplest part.

1. Homework Statement

In the Bogluibov transformation, we get two coefficients from the equations$|v_{k}|^{2}+ |u_{k}|^{2}= 1$

$v_{k}=(\frac{E_{Bulk}-\epsilon_{k}}{\Delta_{k}})\mu_{k}$

Where $E_{Bulk} = \sqrt{\epsilon_{k}^{2} + |\Delta_{k}|^{2}}$

The Attempt at a Solution

I cannot derive the correct expression for $u_{k}$

$u_{k} = \frac{\Delta_{k}}{|\Delta_{k}|}\frac{\sqrt{E_{Bulk}+\epsilon_{k}}}{\sqrt{2E_{Bulk}}}$I know it's just simple algebra, but I've been working on it for hours without any progress and I can't find any sources online that show the derivation :O

 

[Nate810]

2. Homework Equations |v_{k}|^{2}+ |u_{k}|^{2}= 1 v_{k}=(\frac{E_{Bulk}-\epsilon_{k}}{\Delta_{k}})\mu_{k} 3. The Attempt at a SolutionThe first step is to square both sides of the equation: |v_{k}|^{4}+ |u_{k}|^{4}= 1 Then, you can use the definition of v_{k} to substitute it in and rearrange it: |u_{k}|^{4} = 1 - (\frac{E_{Bulk}-\epsilon_{k}}{\Delta_{k}})^{4}\mu_{k}^{4}Next, you can take the square root of both sides: |u_{k}|^{2} = \sqrt{1 - (\frac{E_{Bulk}-\epsilon_{k}}{\Delta_{k}})^{4}\mu_{k}^{4}} Now you can use the definition of E_{Bulk} to substitute it in: |u_{k}|^{2} = \sqrt{1 - (\frac{\sqrt{\epsilon_{k}^{2} + |\Delta_{k}|^{2}}-\epsilon_{k}}{\Delta_{k}})^{4}\mu_{k}^{4}} Finally, you can rearrange it to get the desired expression for u_{k}: u_{k} = \frac{\Delta_{k}}{|\Delta_{k}|}\frac{\sqrt{E_{Bulk}+\epsilon_{k}}}{\sqrt{2E_{Bulk}}}