1+1D Dirac eq., low energy 2 components of spinor roughly =?

[Spinnor]

I found a paper that derives the Dirac equation in 1 + 1 dimensional space-time. It is equation 8, here,

http://academic.reed.edu/physics/fa...Miscellaneous Essays/A. 2D Dirac Equation.pdf

and here,

https://www.physicsforums.com/attachments/upload_2015-1-7_15-23-20-png.77402/ [Broken]

I tried to solve this and if my results were correct (not much confidence) then it looks as if unlike solutions to the 3 + 1 dimensional Dirac equation in 1 + 1 dimensions for small energy both components are roughly equal? Please see my solution below. I am not concerned about normalization factors.

Looking at solutions of the 3+1d Dirac equation I thought I could just throw away 2 of the 4 components and have a solution to the 1+1d Dirac equation, but when I substituted the guess in the 1+1d equation above things did not work out?

Thanks for any help!