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PineApple2
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Hi. First of all, this question seems a bit obscure to me (like there are data missing) However, I quote it as it appeared in the source, so probably there aren't. Second, it is a question from a previous exam, so it is not graded or anything, and there is no problem helping me solving it...
After this preliminary, here is the question:
Two spinless particles are colliding elastically.
(a) Write a formal expression for the total scattering cross section as a sum of partial waves.
(b) We are given that the cross section has resonance at all angles except for when the particles collide at 90o. Far from resonance, the differential cross section is isotropic. If we write the scattering problem as a sum of partial waves, what angular momentum states correspond to the given conditions?
(c) Assume scattering with low energy such that the only contribution to the cross section comes from the two lowest channels. Rewrite the solution to section (a) given the condition you derived at section (b).
(d) What is the corresponding phase shift for a resonance?
(e) Given that when the two particles collide at 90o in resonance, the phase shift is 45o. The momentum at resonance is [itex]k_R[/itex]. Use the results of sections (a)-(d) to calculate the total cross section in resonance.
Ok. My attempt goes like this.
(a) The known expression for the cross section in terms of partial waves is:
[tex]
\sigma_{tot} = \frac{4\pi}{k^2}\sum_{\ell=0}^{\infty}(2\ell+1)sin^2\delta_{\ell}(k)
[/tex]
(no derivation here. I didn't take into account anything related to 2 particles...)
(b) In order to meet the given condition that [itex]\frac{d\sigma}{d\Omega}[/itex] diverges for every angle except for 90o. This must be the case with [itex]f_{\ell}[/itex] as well (since [itex]\frac{d\sigma}{d\Omega}=|f_{\ell}|^2[/itex]). So we need to assemble such a state that would make [itex]f_{\ell}[/itex] diverge for all angles except 90o. From here I am not sure how to continue...
(c) depends on section (b)
(d) The phase shift is 90o as is always the case with a resonance. Am I wrong?
(e) If one particle is scattered at an angle [itex]\theta[/itex], the other one is scattered at [itex]\frac{\pi}{2}-\theta[/itex]. So according to
[itex]\frac{d\sigma}{d\Omega}=|f(\theta)+f(\pi/2-\theta)|^2[/itex] (which is true for identical particles, a fact that we are not given) we need an angular momentum state that vanishes for each pair of angles [itex]\theta[/itex] and [itex]\frac{\pi}{2}-\theta[/itex]. Such a state could maybe consist of [itex]Y_2^0[/itex] (in which [itex]\cos^2(\pi/2-\theta)=\sin^2(\theta)[/itex] and so it cancels the angular dependence. Then adding up an appropriate factor of [itex]Y_0^0[/itex] would cancel out the scattering amplitude.
These are all sort of guesses. I am not sure about this question at all. I will be very happy for remarks on this and/or other suggestions.
Thanks!
[tex]
[/tex]
After this preliminary, here is the question:
Homework Statement
Two spinless particles are colliding elastically.
(a) Write a formal expression for the total scattering cross section as a sum of partial waves.
(b) We are given that the cross section has resonance at all angles except for when the particles collide at 90o. Far from resonance, the differential cross section is isotropic. If we write the scattering problem as a sum of partial waves, what angular momentum states correspond to the given conditions?
(c) Assume scattering with low energy such that the only contribution to the cross section comes from the two lowest channels. Rewrite the solution to section (a) given the condition you derived at section (b).
(d) What is the corresponding phase shift for a resonance?
(e) Given that when the two particles collide at 90o in resonance, the phase shift is 45o. The momentum at resonance is [itex]k_R[/itex]. Use the results of sections (a)-(d) to calculate the total cross section in resonance.
Homework Equations
The Attempt at a Solution
Ok. My attempt goes like this.
(a) The known expression for the cross section in terms of partial waves is:
[tex]
\sigma_{tot} = \frac{4\pi}{k^2}\sum_{\ell=0}^{\infty}(2\ell+1)sin^2\delta_{\ell}(k)
[/tex]
(no derivation here. I didn't take into account anything related to 2 particles...)
(b) In order to meet the given condition that [itex]\frac{d\sigma}{d\Omega}[/itex] diverges for every angle except for 90o. This must be the case with [itex]f_{\ell}[/itex] as well (since [itex]\frac{d\sigma}{d\Omega}=|f_{\ell}|^2[/itex]). So we need to assemble such a state that would make [itex]f_{\ell}[/itex] diverge for all angles except 90o. From here I am not sure how to continue...
(c) depends on section (b)
(d) The phase shift is 90o as is always the case with a resonance. Am I wrong?
(e) If one particle is scattered at an angle [itex]\theta[/itex], the other one is scattered at [itex]\frac{\pi}{2}-\theta[/itex]. So according to
[itex]\frac{d\sigma}{d\Omega}=|f(\theta)+f(\pi/2-\theta)|^2[/itex] (which is true for identical particles, a fact that we are not given) we need an angular momentum state that vanishes for each pair of angles [itex]\theta[/itex] and [itex]\frac{\pi}{2}-\theta[/itex]. Such a state could maybe consist of [itex]Y_2^0[/itex] (in which [itex]\cos^2(\pi/2-\theta)=\sin^2(\theta)[/itex] and so it cancels the angular dependence. Then adding up an appropriate factor of [itex]Y_0^0[/itex] would cancel out the scattering amplitude.
These are all sort of guesses. I am not sure about this question at all. I will be very happy for remarks on this and/or other suggestions.
Thanks!
[tex]
[/tex]
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