Question

3. (a) Calculate the scattering amplitude $f_k( heta)$ and the differential scattering cross section $dsigma/dOmega$ in the (first) Born approximation for the potential $V(r) = V_0 e^{-alpha r^2}$. (b) Use the knowledge of $f_k( heta)$ derived in part (a) to show that in the first Born approximation the s-wave ($l=0$) phase shift should be given by $sin delta_0 = -frac{mu V_0}{2hbar^2 k} sqrt{frac{pi}{alpha}} (1 - e^{-k^2/alpha})$, (Hint: Assume that the phase shift $delta_0$ is small. Use the identity: $f_k( heta) = frac{1}{k} sum_{l=0}^{infty} (2l + 1) P_l(cos heta) e^{i delta_l} sin delta_l$, and the orthogonality integral for the Legendre polynomials.)

          3. (a) Calculate the scattering amplitude $f_k(	heta)$ and the differential scattering cross section $dsigma/dOmega$ in the (first) Born approximation for the potential

$V(r) = V_0 e^{-alpha r^2}$.

(b) Use the knowledge of $f_k(	heta)$ derived in part (a) to show that in the first Born approximation the s-wave ($l=0$) phase shift should be given by

$sin delta_0 = -frac{mu V_0}{2hbar^2 k} sqrt{frac{pi}{alpha}} (1 - e^{-k^2/alpha})$,

(Hint: Assume that the phase shift $delta_0$ is small. Use the identity:

$f_k(	heta) = frac{1}{k} sum_{l=0}^{infty} (2l + 1) P_l(cos 	heta) e^{i delta_l} sin delta_l$,
and the orthogonality integral for the Legendre polynomials.)

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$3. (a) Calculate the scattering amplitude fk( heta) and the differential scattering cross section dsigma/dOmega in the (first) Born approximation for the potential V(r) = V0 e^-alpha r^2. (b) Use the knowledge of fk( heta) derived in part (a) to show that in the first Born approximation the s-wave (l=0) phase shift should be given by sin delta0 = -fracmu V02hbar^2 k sqrtfracpialpha (1 - e^-k^2/alpha), (Hint: Assume that the phase shift delta0 is small. Use the identity: fk( heta) = frac1k suml=0^infty (2l + 1) Pl(cos heta) e^i deltal sin deltal, and the orthogonality integral for the Legendre polynomials.)$

Added by Luis Miguel G.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sri K

08/31/2023

Step 1

It is particularly useful for weak potentials. The first Born approximation is the simplest form and is used when the potential is weak enough that the scattered wave can be treated as a small perturbation of the incident wave. Show more…

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Transcript

00:01 Here in the first part of the question, we have to calculate the scattered amplitude that is fk of theta for the differential scattering cross sectional area which is given as d sigma divided by the d ohm in the first bond approximation for the potential where vr is equals to v0 e raised to the power minus alpha of r raised to the power 2.

00:21 So this is given here.

00:23 So we can see that the scattering amplitude from here is given as f of x of theta that becomes equals to minus 2 mu which is divided by the x squared of q integral is from 0 to infinity of r multiplied by the v of r multiplied by the sin of q of r which is multiplied by the dr.

00:42 So that from here is equals to 2 mu which is divided by the 2 square of q multiplied by the curly and divided by the curly q integral.

00:49 That is from 0 to infinity of v of r multiplied by the cos of q of r that is multiplied by the dr.

00:55 Simplifying the term from here, this term from here will be equals to mu of v0 which is divided by h square of q multiplied by the pi.

01:03 That is divided by alpha.

01:05 Multiply by the curly divided by the curly q of e raised to the power minus q square which is divided by q r plus e raised to the power minus its 4r.

01:14 E raised to the power minus q square, which is divided by 4r.

01:18 So this from here is equals to mu v0 which is divided by h square of q that is multiplied by the pi divided by alpha e raised to the power minus q square which is divided by 4 of alpha multiplied by the minus of q 2q which is divided by 4 alpha...

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