Q22P The function聽j1(伪)=(伪cos伪鈭�... [FREE SOLUTION]

Chapter 7: Q22P (page 385)

The function $j_{1} (伪) = (伪 \cos 伪鈭� \sin 伪) / 伪$ is of interest in quantum mechanics. [It is called a spherical Bessel function; see Chapter 12, equation 17.4] Using problem 18, show that
${鈭�}_{0}^{鈭�} j_{1} (伪) \sin 伪 d 伪 = {\begin{cases} \begin{matrix} 蟺虫 / 2, & 鈭� 1 < x < 1 \end{matrix} \\ \begin{matrix} 0, & | x | > 1 \end{matrix} \end{cases}$

Short Answer

Expert verified

It has been verified that ${鈭�}_{0}^{鈭�} j_{1} (伪) \sin 伪 d 伪 = {\begin{matrix} 蟺 x / 2, & 鈭� 1 < x < 1 \\ 0, & | x | > 1 \end{matrix}$ using a spherical Bessel function and the result of problem 18.

Step by step solution

Given Information.

The given function $j_{1} (伪) = (伪 \cos 伪鈭� \sin 伪) / 伪$ is a spherical Bessel function.

Definition of fourier transform

The Fourier transform is a mathematical technique for expressing a function as the summation of sines and cosines functions.

Step 3: To show that ∫0∞j1(α)sinαdα={πx/2,−1<x<10,|x|>1

With reference to the equation 17.4, the Bessel function $j_{1} (x)$ is $j_{1} (x) = x (鈭� \frac{d}{d x}) \frac{\sin x}{x}$

$j_{1} (x) = \frac{\sin x 鈭� x \cos x}{x^{2}}$

With reference to the equation 12.18 , the result is $f (x) = \frac{2}{蟺} {鈭�}_{0}^{鈭�} \frac{\sin 伪鈭� 伪 \cos 伪}{伪^{2}} \sin (伪 x) d 伪$

$f (x) = \{\begin{cases} \begin{matrix} x, & x 鈭� [鈭� 1, 1] \end{matrix} \\ \begin{matrix} 0, & else \end{matrix} \end{cases}$

Thus, role="math" localid="1664277693592" $\begin{array}{l} f (x) = {鈭�}_{0}^{鈭�} j_{1} (伪) \sin (伪 x) d 伪 \\ f (x) = \{\begin{cases} \begin{matrix} \frac{蟺 x}{2} & x 鈭� [鈭� 1, 1] \end{matrix} \\ \begin{matrix} 0 & else \end{matrix} \end{cases} \end{array}$

Therefore, it has been verified that role="math" localid="1664277730512" ${鈭�}_{0}^{鈭�} j_{1} (伪) \sin 伪 d 伪 = \{\begin{cases} \begin{matrix} 蟺 x / 2, & 鈭� 1 < x < 1 \end{matrix} \\ \begin{matrix} 0, & | x | > 1 \end{matrix} \end{cases}$ using a spherical Bessel function and the result of problem 18.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Given Information.

Definition of fourier transform

Step 3: To show that ∫0∞j1(α)sinαdα={πx/2,−1<x<10,|x|>1

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Oscillations

Fluids

Quantum Physics

Rotational Dynamics

Nuclear Physics

Translational Dynamics

Study anywhere. Anytime. Across all devices.

91影视

The functionj1(伪)=(伪cos伪鈭�sin伪)/伪is of interest in quantum mechanics. [It is called a spherical Bessel function; see Chapter 12, equation 17.4] Using problem 18, show that鈭�0鈭�j1(伪)sin伪d伪={蟺虫/2,鈭�1<x<10,|x|>1

Short Answer

Step by step solution

Given Information.

Definition of fourier transform

Step 3: To show that ∫0∞j1(α)sinαdα={πx/2,−1<x<10,|x|>1

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Oscillations

Fluids

Quantum Physics

Rotational Dynamics

Nuclear Physics

Translational Dynamics

Study anywhere. Anytime. Across all devices.