4. Be sure to read the hint at the bottom. For the function 1 2² (2πσ2)1/4 40² f(x) = Δk2 = 2π exp answer the following. (a) Show that the normalization f(x) is such that f f(x)|² dx = 1. (b) The standard deviation of x is Ax, defined by Ax² = = x²|f(x)|²dx. Show that Ax = o. Extra hint: you can differentiate the main hint with respect to a or k. (c) Determine g(k) = f(x) e-ikx dx. (d) Show that f|g(k)|² dk = 2, consistent with Parseval's theorem. (e) The standard deviation of k is Ak, defined by 88 k² [g(k)|²³ dk. Find Ak. What is the product Ak Ax? Hint: you may assume the following definite integral: exp(-aa²) exp(±ika) da = a exp 4 4a
4. Be sure to read the hint at the bottom. For the function 1 2² (2πσ2)1/4 40² f(x) = Δk2 = 2π exp answer the following. (a) Show that the normalization f(x) is such that f f(x)|² dx = 1. (b) The standard deviation of x is Ax, defined by Ax² = = x²|f(x)|²dx. Show that Ax = o. Extra hint: you can differentiate the main hint with respect to a or k. (c) Determine g(k) = f(x) e-ikx dx. (d) Show that f|g(k)|² dk = 2, consistent with Parseval's theorem. (e) The standard deviation of k is Ak, defined by 88 k² [g(k)|²³ dk. Find Ak. What is the product Ak Ax? Hint: you may assume the following definite integral: exp(-aa²) exp(±ika) da = a exp 4 4a
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Question
Question 4.
![For the function
\[ f(x) = \frac{1}{(2\pi\sigma^2)^{1/4}} \exp \left( -\frac{x^2}{4\sigma^2} \right) \]
answer the following:
(a) Show that the normalization \( f(x) \) is such that \(\int_{-\infty}^{\infty}|f(x)|^2 dx = 1\).
(b) The standard deviation of \( x \), defined by
\[
\Delta x^2 = \int_{-\infty}^{\infty} x^2 |f(x)|^2 dx.
\]
Show that \(\Delta x = \sigma\). Extra hint: you can differentiate the main hint with respect to \(\alpha\) or \(k\).
(c) Determine \( g(k) = \frac{1}{2\pi}\int_{-\infty}^{\infty} f(x) e^{-ikx} dx \).
(d) Show that \(\int_{-\infty}^{\infty} |g(k)|^2 dk = \frac{1}{2\pi}\), consistent with Parseval's theorem.
(e) The standard deviation of \( k \) is \(\Delta k\), defined by
\[
\Delta k^2 = 2\pi \int_{-\infty}^{\infty} k^2 |g(k)|^2 dk.
\]
Find \(\Delta k\). What is the product \(\Delta k \Delta x\)?
Hint: you may assume the following definite integral:
\[
\int_{-\infty}^{\infty} \exp(-\alpha x^2) \exp(ikx) dx = \sqrt{\frac{\pi}{\alpha}} \exp\left(-\frac{k^2}{4\alpha}\right).
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b65ef36-cf51-4f81-80a6-74e205c9e9b1%2Fb0881363-fa8c-4767-9279-a3d70cd44667%2Fbyultwa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For the function
\[ f(x) = \frac{1}{(2\pi\sigma^2)^{1/4}} \exp \left( -\frac{x^2}{4\sigma^2} \right) \]
answer the following:
(a) Show that the normalization \( f(x) \) is such that \(\int_{-\infty}^{\infty}|f(x)|^2 dx = 1\).
(b) The standard deviation of \( x \), defined by
\[
\Delta x^2 = \int_{-\infty}^{\infty} x^2 |f(x)|^2 dx.
\]
Show that \(\Delta x = \sigma\). Extra hint: you can differentiate the main hint with respect to \(\alpha\) or \(k\).
(c) Determine \( g(k) = \frac{1}{2\pi}\int_{-\infty}^{\infty} f(x) e^{-ikx} dx \).
(d) Show that \(\int_{-\infty}^{\infty} |g(k)|^2 dk = \frac{1}{2\pi}\), consistent with Parseval's theorem.
(e) The standard deviation of \( k \) is \(\Delta k\), defined by
\[
\Delta k^2 = 2\pi \int_{-\infty}^{\infty} k^2 |g(k)|^2 dk.
\]
Find \(\Delta k\). What is the product \(\Delta k \Delta x\)?
Hint: you may assume the following definite integral:
\[
\int_{-\infty}^{\infty} \exp(-\alpha x^2) \exp(ikx) dx = \sqrt{\frac{\pi}{\alpha}} \exp\left(-\frac{k^2}{4\alpha}\right).
\]
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Step 1: Continuous probability distribution.
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