The elimination in Section 5.2 leads to the quadratic M_1M_2 , ^4-2K(M_1+M_2) ,…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Condensed Matter, Materials & Soft Matter · Foundations of mathematical language

Condensed Matter Physics, Ch Two atoms in the cell: acoustic and optical branches

Problem

The elimination in Section 5.2 leads to the quadratic M1M2 ω4−2K(M1+M2) ω2+4K2sin⁡2(qa/2)=0M_1M_2\,\omega^4-2K(M_1+M_2)\,\omega^2+4K^2\sin^2(qa/2)=0 for ω2\omega^2. Without using the quadratic formula, use the standard results for a quadratic Ax2+Bx+C=0Ax^2+Bx+C=0, that its two roots satisfy x++x−=−B/Ax_++x_-=-B/A and x+x−=C/Ax_+x_-=C/A, to write down ω+2+ω−2\omega_+^2+\omega_-^2 and ω+2 ω−2\omega_+^2\,\omega_-^2 directly in terms of KK, M1M_1, M2M_2, qq and aa. Then confirm that both expressions agree with the key result ω±2\omega_\pm^2 already found.

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