Demonstrate, for a particle scattering from a finite-range, spherically symmetric potential V(r), which is weak enough so that the Born approximation for symmetric potentials is valid, that the total cross section, at very low energies, is a linear function of the energy σ(E) = σ0(1+αE), where σ0 is related to the volume integral of the potential... Please see attached for full question.

The potential is of short range, so only r within some small region can contribute to the integral. When the energy goes to zero, the k vectors go to zero to, so the term (k-k') dot r in the exponential can be regarded as a small parameter which we are allowed to expand in. It is convenient to put d = k - k'. We will need the magnitude of d, so let's calculate this first.

In elastic collisions |k| = |k'|. It thus follows that:

d^2 = (k - k')^2 = k^2 + k'^2 - 2 k dot k' = ...

Solution Summary

A detailed solution is given. The Born approximation for symmetric potentials is valid, that the cross section is determined.

If np ≥ 5 and nq ≥ 5 estimate P (fewer than 2) with n= 13 and p= 0.4 by using the normal distribution as an approximation to the binomial distribution if np < 5 or nq <5 then state that the normal approximation is not suitable.
Select the correct choice below and if necessary fill in the answer box to complete our choic

Please also give Matlab solutions to the following problem.
Find the Pade approximation R_(5,4) (x) for f(x) = sin x. Plot the following graphs:
i) Comparison of R_(5,4) (x) versus f(x) = sin x, and
ii) the error E(x) = |R_(5,4) (x) - sin x|,
each on interval [-PI, PI].

Please provide formulas and logical conclusions for each. Include step by step calculations for each.
8. If the probability of having a boy is .47, find the following
a. For families with 10 children
i. The mean for the number of boys in a family
ii. The standard deviation for the number of boys
iii. The probability of h

(See attached file for full problem description with proper equations and exponents)
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1. Suppose that N(h) is an approximation to M for every h > 0 and that
M = N(h)+K1h2+K2h4+K3h6+.....
For some value K1, K2, K3... Use the values N(h), N(h/3), and N(h/9) to produce an O(h6) approximation to M.
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1. If n=100 and p= 0.02 in a binomial experiment, does this satisfy the rule for a normal approximation? why or why not?
2. See attached file for the graphs. What is the z-score for the standard normal distribution for graph B?

Using Legendre polynomials of degree 1, 2, and 3, find the least-squares approximation
for the function e^(-x) on [2, 4].
I am confused as to how to do this on the interval [2,4] and not [-1,1]
I have included the notes on this section, including an example (see the attached file).

A 56-question multiple choice test has 4 possible answers for each question and a student chooses the answers to each problem at random. A student selects 16 correct answers. Find the probability of the result using the normal curve approximation to the binomial distribution.

Suppose that 10% of all steel shafts produced by a certain process are nonconforming but can be re-worked (rather than having to be scrapped). Consider a random sample of 200 shafts, and let X denote the number among these that are nonconforming, what is the (approximate) probability that X is:
a. At most 30
b. Less than 30