Question

When solving for the allowed energies in certain materials (using the finite square well model) one comes across the following equation which must be solved for x:
√
1 + x2 = x tan(x)

When solving for the allowed energies in certain materials (using the finite square well model) one comes across the following equation which must be solved for x: V1 + x2 = x tan(x (in reality, the 1 should be replaced by a parameter zo that measures the width of the well, but we won't do that). Use Newton's method with initial guess 1 to generate a recursive sequence for the solution to this equation. Use a calculator to get the first estimate to three decimal places.

Answer 1

Jitar tanzent Un See solution & geven expression, J1+12 = n tann Situz-zetann og Allright, now, det fery = /1482 axtann Now, flan):=fltan -an tanki differentiating it with respect to sen f'c xn) = d [vieren - Intanen = f'(en) gelen ( tan xen + un sečen) 2tan Now, dan -( = f'(an) un Un Stan fchen using Mewton's formula, Anti XnT f'can] {iten rentanan? Jan 5) Xn+1 = an- Than (tank, Hn Seeen) =&nt = xy - Unitan tanantan Seehan (1+2 +2end it tenain an-Jitana (tanan-an see an] en seeran. Jitan - 1 Tant fansant anfitrints r51tan2 mfano Xne itsetan un ent= x² 17 th See an 4 xn-Jiten (tanana xn see hen) he, have our initial guess H=1 at n=o. Now I m=1, we have -1 H2 = (1) ²51112 See (1) 1 - 31+(Jan (1) -47 See (4) Ž 1,056 212 = 1.055604679 = 1.0556 1.179 635675 713 712 = 1.386 3232051 = 1.54 12 30665 1.570280525 1.57079610il at 7th iteration ns 16 017 n@=1.57096327 THE xo = 1.570 Scanned with CamScanner