91Ó°ÊÓ

Consider the wave packet defined by $$\psi(x) = \int ^\infty_0 B(k)cos kx dk$$ Let \(B(k) = e^{-a^2k^2}\). (a) The function \(B(k)\) has its maximum value at \(k\) = 0. Let \(k_h\) be the value of \(k\) at which \(B(k)\) has fallen to half its maximum value, and define the width of \(B(k)\) as \(w_k = k_h\) . In terms of \(\alpha\), what is \(w_k\) ? (b) Use integral tables to evaluate the integral that gives \(\psi(x)\). For what value of \(x\) is \(\psi(x)\) maximum? (c) Define the width of \(\psi(x)\) as \(w_x = x_h\) , where \(x_h\) is the positive value of \(x\) at which \(\psi(x)\) has fallen to half its maximum value. Calculate \(w_x\) in terms of \(\alpha\). (d) The momentum \(p\) is equal to \(hk/2\pi\), so the width of \(B\) in momentum is \(w_p = hw_k /2\pi\). Calculate the product \(w_p w_x\) and compare to the Heisenberg uncertainty principle.

Step by step solution

01

Determine Maximum and Half-Maximum of B(k)

The function \( B(k) = e^{-a^2k^2} \) is a Gaussian function with its maximum value at \( k = 0 \). To find \( k_h \), we solve for \( B(k_h) = \frac{1}{2}B(0) \). This gives: \[ e^{-a^2 k_h^2} = \frac{1}{2} \]Taking natural logarithms on both sides, we get:\[ -a^2 k_h^2 = \- \ln(2) \]Thus, solving for \( k_h \), we have:\[ k_h = \frac{\sqrt{\ln(2)}}{a} \] implies \( w_k = \frac{\sqrt{\ln(2)}}{a} \).

02

Evaluate the Integral for ψ(x)

To determine \( \psi(x) \), substitute \( B(k) = e^{-a^2k^2} \) into the integral:\[ \psi(x) = \int_0^\infty e^{-a^2 k^2} \cos(kx) \, dk \]Using integral tables or computational tools, this integral evaluates to:\[ \psi(x) = \frac{1}{2} \sqrt{\frac{\pi}{a^2}} \, e^{-\frac{x^2}{4a^2}} \]The maximum value occurs when the exponential term \( e^{-\frac{x^2}{4a^2}} \) is at its maximum, which is at \( x = 0 \).

03

Find Width of ψ(x) at Half-Maximum

The width \( w_x \) is found where \( \psi(x_h) = \frac{1}{2}\psi(0) \). Since \( \psi(x) = \frac{1}{2} \sqrt{\frac{\pi}{a^2}} \, e^{-\frac{x^2}{4a^2}} \):\[ \frac{1}{2} e^{-\frac{x_h^2}{4a^2}} = \frac{1}{4} \sqrt{\frac{\pi}{a^2}} \]Simplifying gives:\[ e^{-\frac{x_h^2}{4a^2}} = \frac{1}{2} \"\ln(2) \]Solving for \( x_h \), we have:\[ x_h = 2a\sqrt{\ln(2)} \]So \( w_x = 2a\sqrt{\ln(2)} \).

04

Calculate Momentum Width and Product wp wx

The width in momentum space is defined as \( w_p = \frac{h}{2\pi} w_k = \frac{h}{2\pi} \cdot \frac{\sqrt{\ln(2)}}{a} \).Thus, the product \( w_p w_x \) becomes:\[ w_p w_x = \left( \frac{h}{2\pi} \cdot \frac{\sqrt{\ln(2)}}{a} \right) \left( 2a\sqrt{\ln(2)} \right) = \frac{h}{2\pi} \cdot 2\ln(2) \]Finally, simplify to find:\[ w_p w_x = \frac{h \ln(2)}{\pi} \] This is compared to the Heisenberg uncertainty principle \( \Delta x \Delta p \geq \frac{\hbar}{2} \), where deviations show compliance based on constants.

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 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gaussian Function

The Gaussian function is a mathematical function of the form \( B(k) = e^{-a^2k^2} \). It is characterized by its bell-shaped curve, often likened to the shape of a bell, and is symmetric around its peak. Its maximum value is at the center, here at \( k = 0 \). The function decreases rapidly as the variable \( k \) moves away from the center.

• The Gaussian function is widely used in physics due to its properties of smoothness and localization.
• In the context of wave packets, it represents the distribution of wave vectors or momenta.
• This type of function naturally appears when considering the uncertainty and spread of properties, like position and momentum, as it exhibits behavior similar to probability distributions.

Overall, the Gaussian function provides a mathematical model for exploring the spread and intensity of wave packets.

Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle is a fundamental concept in quantum mechanics that limits our ability to simultaneously determine the exact position and momentum of a particle. It is formulated as \( \Delta x \Delta p \geq \frac{\hbar}{2} \), where \( \Delta x \) and \( \Delta p \) are the uncertainties in position and momentum, respectively, and \( \hbar \) is the reduced Planck's constant.

• The principle highlights the intrinsic limitations on precision due to the wave-like nature of particles.
• In terms of wave packets, this principle implies that a more localized particle (smaller \( \Delta x \)) will have a larger uncertainty in momentum (larger \( \Delta p \))).
• Understanding this principle is crucial for quantum mechanics, as it illustrates the probabilistic nature at the micro level.

This relationship is a key part of contemplating the widths \( w_x \) and \( w_p \) calculated for \( \psi(x) \) and \( B(k) \) respectively, as seen through their connection in the problem solution.

Momentum Space

Momentum space is an alternative framework to the more intuitive position space, where the momentum \( p = \frac{hk}{2\pi} \) of particles is considered instead of their position. In this space, the wave function is expressed in terms of momentum rather than spatial coordinates.

• By applying Fourier transforms, one can move between position and momentum spaces.
• Wave functions in momentum space provide insights into the distribution and likelihood of different momenta of a particle.
• This perspective is particularly useful in scenarios involving particle collisions, quantum tunneling, and other phenomena where momentum plays a crucial role.

In the provided exercise, the width in momentum space \( w_p \) relates to the distribution's spread in momentum, hinting at conjunction with the Heisenberg uncertainty principle.

Step by Step Solution

A step-by-step solution offers a methodical approach to solving problems, which is especially helpful in understanding complex physics concepts like wave packets.

• The process begins by identifying key features and functions, like \( B(k) \) in the original exercise, which is a Gaussian function.
• Proceeding with integral evaluation gives the wave packet \( \psi(x) \) and insights into its characteristics, such as maximum and widths in both position and momentum spaces.
• The related calculations provide a way to verify compliance with the Heisenberg uncertainty principle by comparing computed values.

This systematic approach turns intricate topics into understandable sequences, facilitating deeper comprehension of wave phenomena.

Integral Evaluation

Integral evaluation involves calculating the value of an integral, which is often essential in finding wave functions or other physical quantities in quantum mechanics. The integral \( \int \) itself represents area under a curve, often used to describe probabilities, densities, or, as in this case, wave packets.

• For the wave packet \( \psi(x) \), the integral \( \int_0^\infty e^{-a^2 k^2} \cos(kx) \, dk \) evaluates to \( \frac{1}{2} \sqrt{\frac{\pi}{a^2}} e^{-\frac{x^2}{4a^2}} \), demonstrating how different values integrate to depict the overall wave characteristics.
• Tables or computational tools are frequently employed for such integrals, given the complexity of real-world scenarios.
• As the result for \( \psi(x) \) shows, the integral determines not just the form but the precise behavior of wave functions across space.

Mastering the art of integral evaluation is crucial for predicting and understanding quantum behavior.