Q55E To determine the classical expec... [FREE SOLUTION]

Chapter 5: Q55E (page 190)

To determine the classical expectation value of the position of a particle in a box is $\frac{L}{2}$ , the expectation value of the square of the position of a particle in a box isrole="math" localid="1658324625272" $\frac{L^{2}}{3}$ , and the uncertainty in the position of a particle in a box is $\frac{L}{\sqrt{12}}$ .

Short Answer

Expert verified

The uncertainty in the position of a particle in a box of length $L$ is $螖 x = \sqrt{\frac{鈭�}{x^{2} 鈭� {\overset{炉}{x}}^{2}}} = \sqrt{\frac{L^{2}}{3} 鈭� \frac{L^{2}}{4}} = \frac{L}{\sqrt{12}}$ .

Step by step solution

Step 1:

Given information: The classical probability per unit length of finding a particle in a box of length $L$ is $\frac{1}{L}$ along the entire length of the box i.e. $\frac{dP}{dx} = \frac{1}{L}$ .

Step 2:

The classical expectation value of the position of a particle in a box of length $L$ is

$\overset{炉}{x} = {鈭�}_{0}^{L} x dP = {鈭�}_{0}^{L} x \frac{dP}{dx} dx = \frac{1}{L} {鈭�}_{0}^{L} x dx = \frac{1}{L} 脳 \frac{L^{2}}{2} = \frac{L}{2} .$

The expectation value of the square of the position of a particle in a box of length role="math" localid="1658325211080" $L$ is

$\overset{炉}{x^{2}} = {鈭�}_{0}^{L} x^{2} dP = {鈭�}_{0}^{L} x^{2} \frac{dP}{dx} dx = \frac{1}{L} {鈭�}_{0}^{L} x^{2} dx = \frac{1}{L} 脳 \frac{L^{3}}{3} = \frac{L^{2}}{3}$

The uncertainty in the position of a particle in a box of length $L$ is

$螖 x = \sqrt{\frac{鈭�}{x^{2} 鈭� {\overset{炉}{x}}^{2}}} = \sqrt{\frac{L^{2}}{3} 鈭� \frac{L^{2}}{4}} = \frac{L}{\sqrt{12}}$

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91影视

To determine the classical expectation value of the position of a particle in a box is L2 , the expectation value of the square of the position of a particle in a box isrole="math" localid="1658324625272" L23 , and the uncertainty in the position of a particle in a box isL12 .

Short Answer

Step by step solution

Step 1:

Step 2:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physics of Motion

Mechanics and Materials

Electromagnetism

Fields in Physics

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Circular Motion and Gravitation

Study anywhere. Anytime. Across all devices.