Problem 19.1 Peskin: Find Harmonic Oscillator Solution

[JCCJ]

Hi,
I'm trying to do this problem (19.1 from Peskin) that apparently should be quite straightforward but when I plug the anzat given at c) into the equation I don't get an harmonic oscillator as the book indicates. Could please anyone tell me what is wrong?
Thanks

 

[Vanadium 50]

This is a first - not only does the student not type the answer, he doesn't even type the question!

 

[JCCJ]

I think the question/problem is quite clear: The problem 19.1 from the peskin QFT book. I give as a fact that anyone in the high energy physics section has acces to this book to refer to the problem.
Anyway the problem is the following:
given the equation $$-i \sigma \cdot D \psi_R=0,$$ where $$A^{\mu}=(0,0,B x^1, A)$$ with $$B$$ a constant and $$A$$ depends only adiabatically in time.
Now when I put the anzat given by the book $$\psi_R=\left(\begin{array}{c}\phi_1(x^1)\\ \phi_2(x^1)\end{array}\right)e^{i(k_2x^2+k_3 x^3)}$$ and eliminate one of the $$\phi$$ I'm supossed to get an harmonic oscillator eqn. How? I get an eqn of the form $$\phi_i ''(x^1)+\omega^2(x^1) \phi_i(x^1)=0$$ with an explicit dependence of $$\omega$$ in $$x$$.

 

[Vanadium 50]

Not everyone is within arm's reach of Peskin. I don't even own a copy.

 

[mfb]

You get a harmonic oscillator at every $x^1$, where the frequency depends on that coordinate.

By the way: with $-tags, you don't get new lines for every symbol.$