Chapter 40: Q. 34 (page 1176)

A neutron is confined in a $10 fm$ -diameter nucleus. If the nucleus is modeled as a one-dimensional rigid box, what is the probability that a neutron in the ground state is less than $2.0 fm$ from the edge of the nucleus?

Short Answer

Expert verified

The probability is $9.72 % .$

Step by step solution

Given information

We know that ,

Diameter = $10 fm$

Neutron at positon = $2 fm$

We have to find the probability of the ground state.

Simplify

In general for the ground state for any particle the wave function is given by,

$φ (x) = \sqrt{\frac{2}{L}} \sin \frac{πx}{L}$

Then, the probability is defines as below with factor of 2 due to neutron is distributed symmetrical around the nucleus.

$P = 2 \int_{0}^{0.2 fm} {|φ (x)|}^{2} dx P = 2 \int_{0}^{0.2 fm} \frac{2}{L} \sin^{2} \frac{πx}{L} dx P = \frac{4}{L} \int_{0}^{0.2 fm} (1 - \cos \frac{2 πx}{L}) dx P = \frac{4}{L} {[x - \sin (\frac{2 πx}{L}) \frac{L}{2 π}]}_{0}^{2 fm} P = \frac{4}{10 fm} [2 fm - \sin (\frac{2 π \times 2 fm}{10 fm}) \frac{10 fm}{2 π}] P = 0.0972 P = 9.72 %$

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A neutron is confined in a $10 fm$ -diameter nucleus. If the nucleus is modeled as a one-dimensional rigid box, what is the probability that a neutron in the ground state is less than $2.0 fm$ from the edge of the nucleus?

Short Answer

Step by step solution

Given information

Simplify

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A neutron is confined in a 10fm-diameter nucleus. If the nucleus is modeled as a one-dimensional rigid box, what is the probability that a neutron in the ground state is less than 2.0fm from the edge of the nucleus?

Short Answer

Step by step solution

Given information

Simplify

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Energy Physics

Waves Physics

Magnetism

Modern Physics

Thermodynamics

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.

A neutron is confined in a $10 fm$ -diameter nucleus. If the nucleus is modeled as a one-dimensional rigid box, what is the probability that a neutron in the ground state is less than $2.0 fm$ from the edge of the nucleus?