Chapter 10: Q35E (page 468)

By expanding an arbitrary $U (x)$ in a power series about a local minimum assumed to be at $x = a$ , prove that the effective spring constant is given by equation $(10 - 3)$ .

Short Answer

Expert verified

The effective spring constant is given by the expansion ${k = \frac{d^{2} U (x)}{d x^{2}}}_{a}$ .

Step by step solution

Given data

a power series expansion of $U (x)$ about local minimum at $x = a$ is given by the expansion ${k = \frac{d^{2} U (x)}{d x^{2}}}_{a}$

Concept of power series expansion

the expansion about $x = a$ is given by,

$f (a) + \frac{f' (a)}{1!} (x - a) + \frac{f' (a)}{2!} {(x - a)}^{2} + . . . .$

Determine the potential energy between two protons

We need to prove that a power series expansion of $U (x)$ about local minimum at $x = a$ is given by the expansion

${k = \frac{d^{2} U (x)}{d x^{2}}}_{a}$

Applying $(1)$ on $U (x)$ about $x = a$ , we get

$U (x - a) = U (a) + \frac{U' (a)}{1!} (x - a) + \frac{U' (a)}{2!} {(x - a)}^{2} + . . . . .$

The potential energy of diatomic molecule for small oscillation with an equilibrium separation and spring constant ' $k$ ' is given by

$U (x - a) \approx U_{0} + \frac{1}{2} k {(x - a)}^{2}$

Comparing the respective coefficients of (2) and (3), we get

$U_{0} = U (a) \frac{U' (a)}{2!} {(x - a)}^{2} = \frac{1}{2} k {(x - a)}^{2} k = U'' (a) {k = \frac{d^{2} U (x)}{d x^{2}}}_{a}$

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By expanding an arbitrary $U (x)$ in a power series about a local minimum assumed to be at $x = a$ , prove that the effective spring constant is given by equation $(10 - 3)$ .

Short Answer

Step by step solution

Given data

Concept of power series expansion

Determine the potential energy between two protons

One App. One Place for Learning.

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By expanding an arbitraryU(x) in a power series about a local minimum assumed to be atx=a , prove that the effective spring constant is given by equation(10-3) .

Short Answer

Step by step solution

Given data

Concept of power series expansion

Determine the potential energy between two protons

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physics of Motion

Electrostatics

Scientific Method Physics

Medical Physics

Turning Points in Physics

Electricity and Magnetism

Study anywhere. Anytime. Across all devices.

By expanding an arbitrary $U (x)$ in a power series about a local minimum assumed to be at $x = a$ , prove that the effective spring constant is given by equation $(10 - 3)$ .