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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 11: Q10P (page 416)

Find the scattering amplitude, in the Born approximation, for soft sphere scattering at arbitrary energy. Show that your formula reduces to Equation 11.82 in the low-energy limit.

Short Answer

Expert verified

For the scattering amplitude,

$r (胃) = - \frac{2 m V_{0}}{h^{2} k^{3}} \{\sin (k a) - k a \cos (k a)\} f (胃) = - \frac{2}{3} \frac{m V_{0} a^{3}}{h^{2}}$

Step by step solution

01

Given.

A soft sphere is defined by the potential

$V (r) = \{\begin{cases} V_{0} & r 鈮� a \\ 0 & r > a \end{cases}$ 鈥︹€� (1)

Where $V_{0} > 0$ is a constant.

For a spherical symmetry potential, the scattering amplitude:

role="math" localid="1658305775955" $f (胃) = - \frac{2 m}{h^{2} k} {鈭�}_{0}^{鈭�} r V (r) \sin (k r) d r$

02

To find the scattering amplitude.

For the potential above, we get:

$f (胃) = - \frac{2 m}{h^{2} k} V_{0} {鈭�}_{0}^{a} r \sin (k r) d r$

To do the integral, I used theintegral calculator, we get:

role="math" localid="1658305697027" $f (胃) = - \frac{2 m V_{0}}{h^{2} k^{3}} {[\frac{1}{k^{2}} \sin (k r) - \frac{r}{k} \cos (k r)]}_{0}^{a} f (胃) = - \frac{2 m V_{0}}{h^{2} k^{3}} \{\sin (k a) - k a \cos (k a)\}$

Where;

$k = 2 k \sin (\frac{胃}{2})$

In the low- energy limit $k a 鈮� 1$ , and hence,

$\sin (k a) 鈮� k a - \frac{1}{3!} {(k a)}^{3} \cos (k a) = 1 - \frac{1}{2} {(k a)}^{2}$

03

Therefore, the scattering amplitude.

Thus, the scattering amplitude is:

$f (胃) 鈮� \frac{2 m V_{0}}{h^{2} k^{3}} [k a - \frac{1}{6} {(k a)}^{3} - k a + \frac{1}{2} {(k a)}^{3}] f (胃) = - \frac{2}{3} \frac{m V_{0} a^{3}}{h^{2}}$

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Most popular questions from this chapter

Show that the ground state of hydrogen (Equation 4.80) satisfies the integral form of the Schr枚dinger equation, for the appropriateV and E(note that Eis negative, so $k = i k$ , where $k = \sqrt{- 2 m E / h}$ ).

Prove Equation 11.33, starting with Equation 11.32. Hint: Exploit the orthogonality of the Legendre polynomials to show that the coefficients with different values of l must separately vanish.

Nothing that show that the determinant of a Hermitian matrix is real, the determinant of a unitary matrix has modulus 1 (hence the name), and the determinant of an orthogonal matrix is or .

Find the scattering amplitude for low-energy soft-sphere scattering in the second Born approximation.

Construct the analogs to Equation 11.12 for one-dimensional and two-dimensional scattering.

See all solutions

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