I don\'t really expect an answer to this because it seems likeeverytime i post a

Question

I don't really expect an answer to this because it seems likeeverytime i post a difficult question no one responds. but anyway,if you give me ANY sort of help ill rate you lifesaver. Here wego A particle of mass m and energy E is incident from r =infinity and moves toward the parabolic barrier                                              V(r) = V0 - 1/2 m2(r -R0)2, where R0 is the radial distance at which thepotential attains its maximum value V0. Show that theprobability of transmission T of the particle through this barrieris                                               T = exp [ 2(E-V0]/h-bar Use the barrier penetration formula T = exp (-2I), I = 2m/h-bar [V(r)-E] dr Also note that (a2-r2) dr= 1/2 [r(a2-r2) +a2sin-1 (r/abs a) where abs a means absolute value. note that in the I =integral, V(r) and E are in the numerator, to avoid confusion. I'mnot sure if this problem just involves evaluating a difficultintegral or if the harmonic oscillator is somehow involved. I don't really expect an answer to this because it seems likeeverytime i post a difficult question no one responds. but anyway,if you give me ANY sort of help ill rate you lifesaver. Here wego A particle of mass m and energy E is incident from r =infinity and moves toward the parabolic barrier                                              V(r) = V0 - 1/2 m2(r -R0)2, where R0 is the radial distance at which thepotential attains its maximum value V0. Show that theprobability of transmission T of the particle through this barrieris                                               T = exp [ 2(E-V0]/h-bar Use the barrier penetration formula T = exp (-2I), I = 2m/h-bar [V(r)-E] dr Also note that (a2-r2) dr= 1/2 [r(a2-r2) +a2sin-1 (r/abs a) where abs a means absolute value. note that in the I =integral, V(r) and E are in the numerator, to avoid confusion. I'mnot sure if this problem just involves evaluating a difficultintegral or if the harmonic oscillator is somehow involved.

Explanation / Answer

You have to evaluate the integral I directly. I = (int from r1 to r2)[(2m(V-E)/h^2)dr]           (note: h means h-bar and you have a typo in your formula) r1 and r2 are the points where V(r)-E is zero. first, change variables to x = r-Ro, A= Vo - E, =(1/2)m^2, a^2 = A/, dx = dr, I = (m/h) (int from x1 to x2)[(a^2-x^2)dx] where x1 and x2 are points satisfying a^2-x^2= 0 so x1 = -a, x2 = a Now you can evaluate the integral. only the sin-1term will contribute; the other term vanishes at the boundaries.andyou get the right result. hope this helps.

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