Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

Because a nucleon is confined to a nucleus, we can take the uncertainty in its position to be approximately the nuclear radius r. Use the uncertainty $∆ p$ principle to determine the uncertainty in the linear momentum of the nucleon. Using the approximation $p \approx ∆ p$ and the fact that the nucleon is non-relativistic, calculate the kinetic energy of the nucleon in a nucleus with A = 100.

The kinetic energy of the nucleon in a nucleus with A = 100 is 30 Me V.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given data

Uncertainty in position is approximately the nuclear radius, $∆ x = r$
Approximation of momemtum, $p \approx ∆ p$
The nucleon is non-relativistic.
Mass or nucleon number, A =100

Step 2: Understanding the concept of uncertainty principle

The uncertainty principle states that the position and the velocity of an object cannot be measured simultaneously. Again, we are provided that the nucleon is non-relativistic, thus the energy can be found using the certainty condition of momentum and position considering the confinement of the nucleon with the nucleus.

Formulae:

The uncertainty relation of momentum-position, $∆ p = \frac{h}{∆ x} . . . . . . (1)$

The kinetic energy of a body in motion, $K = \frac{p^{2}}{2 m} . . . . . . . . . . . (2)$

The radius of an atom in a nucleus, $r = r_{0} A^{1 / 3} . . . . . . (3)$ $where r_{0} = 1.2 fm$

Step 3: Calculation of the kinetic energy of the nucleon

Substituting the value of value of momentum from equation (1) in equation (2), the kinetic energy of the nucleon can be calculated using the given data as follows:

$(for p \approx ∆ p, ∆ x = r, hc = 1240 MeV . fm, {mc}^{2} = 931.5 MeV)$

$K = \frac{h^{2}}{2 m r^{2}} = \frac{{(h c)}^{2}}{(2 m c^{2}) {(r_{0} A^{1 / 3})}^{2}} (∵ From equation (3), r = r_{0} A^{1 / 3}) = \frac{{(1240 MeV . fm)}^{2}}{(2 (931.5 MeV)) {((1.2 fm) (100^{1 / 3}))}^{2}} \approx 30 MeV$

Hence, the value of energy is 30 MeV.

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: Given data

Step 2: Understanding the concept of uncertainty principle

Step 3: Calculation of the kinetic energy of the nucleon

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