Homework Help is Here – Start Your Trial Now!
Engineering
Computer Science
Problem 2.55 Find the ground state energy of the harmonic oscillator, to five significant digits, by the "wag-the-dog" method. That is, solve Equation 2.73 numerically, varying K until you get a wave function that goes to zero at large E. In Mathematica, appropriate input code would be Plot[ Evaluate[ u[x] /. NDSolve[ {w/[x] -(x² - K)*u[x] == 0, u[0] == 1, w[0] == 0}, u[x], {x, 0, b} ] ], {x, a, b}, PlotRange -> {c, d} (Here (a, b) is the horizontal range of the graph, and (c, d) is the vertical range-start with a = 0, b = 10, c = – 10, d = 10.) We know that the 2n + 1, so you might start with a “guess" of K = 0.9. Notice what the "tail" of the wave function does. Now try K = 1.1, and note correct solution is K = %3D that the tail flips over. Somewhere in between those values lies the correct solution. Zero in on it by bracketing K tighter and tighter. As you do so, you may want to adjust a, b, c, and d, to zero in on the cross-over point. d²y (2.73) where Kis the energy, in units of (1/2) hw:

Problem 2.55 Find the ground state energy of the harmonic oscillator, to five significant digits, by the "wag-the-dog" method. That is, solve Equation 2.73 numerically, varying K until you get a wave function that goes to zero at large E. In Mathematica, appropriate input code would be Plot[ Evaluate[ u[x] /. NDSolve[ {w/[x] -(x² - K)*u[x] == 0, u[0] == 1, w[0] == 0}, u[x], {x, 0, b} ] ], {x, a, b}, PlotRange -> {c, d} (Here (a, b) is the horizontal range of the graph, and (c, d) is the vertical range-start with a = 0, b = 10, c = – 10, d = 10.) We know that the 2n + 1, so you might start with a “guess" of K = 0.9. Notice what the "tail" of the wave function does. Now try K = 1.1, and note correct solution is K = %3D that the tail flips over. Somewhere in between those values lies the correct solution. Zero in on it by bracketing K tighter and tighter. As you do so, you may want to adjust a, b, c, and d, to zero in on the cross-over point. d²y (2.73) where Kis the energy, in units of (1/2) hw:

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
See similar textbooks
icon
Related questions
Question

Question related to Quantum Mechanics : Problem 2.55

Problem 2.55 Find the ground state energy of the harmonic oscillator, to five significant digits, by the "wag-the-dog" method. That is, solve Equation 2.73 numerically, varying K until you get a wave function that goes to zero at large E. In Mathematica, appropriate input code would be Plot[ Evaluate[ u[x] /. NDSolve[ {w/[x] -(x² - K)*u[x] == 0, u[0] == 1, w[0] == 0}, u[x], {x, 0, b} ] ], {x, a, b}, PlotRange -> {c, d} (Here (a, b) is the horizontal range of the graph, and (c, d) is the vertical range-start with a = 0, b = 10, c = – 10, d = 10.) We know that the 2n + 1, so you might start with a “guess" of K = 0.9. Notice what the "tail" of the wave function does. Now try K = 1.1, and note correct solution is K = %3D that the tail flips over. Somewhere in between those values lies the correct solution. Zero in on it by bracketing K tighter and tighter. As you do so, you may want to adjust a, b, c, and d, to zero in on the cross-over point.
Transcribed Image Text:Problem 2.55 Find the ground state energy of the harmonic oscillator, to five significant digits, by the "wag-the-dog" method. That is, solve Equation 2.73 numerically, varying K until you get a wave function that goes to zero at large E. In Mathematica, appropriate input code would be Plot[ Evaluate[ u[x] /. NDSolve[ {w/[x] -(x² - K)*u[x] == 0, u[0] == 1, w[0] == 0}, u[x], {x, 0, b} ] ], {x, a, b}, PlotRange -> {c, d} (Here (a, b) is the horizontal range of the graph, and (c, d) is the vertical range-start with a = 0, b = 10, c = – 10, d = 10.) We know that the 2n + 1, so you might start with a “guess" of K = 0.9. Notice what the "tail" of the wave function does. Now try K = 1.1, and note correct solution is K = %3D that the tail flips over. Somewhere in between those values lies the correct solution. Zero in on it by bracketing K tighter and tighter. As you do so, you may want to adjust a, b, c, and d, to zero in on the cross-over point.
d²y (2.73) where Kis the energy, in units of (1/2) hw:
Transcribed Image Text:d²y (2.73) where Kis the energy, in units of (1/2) hw:
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 6 steps with 6 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Fundamentals of Boolean Algebra and Digital Logics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Database System Concepts
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
Starting Out with Python (4th Edition)
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
C How to Program (8th Edition)
C How to Program (8th Edition)
Computer Science
ISBN:
9780133976892
Author:
Paul J. Deitel, Harvey Deitel
Publisher:
PEARSON
Database Systems: Design, Implementation, & Manag…
Database Systems: Design, Implementation, & Manag…
Computer Science
ISBN:
9781337627900
Author:
Carlos Coronel, Steven Morris
Publisher:
Cengage Learning
Programmable Logic Controllers
Programmable Logic Controllers
Computer Science
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education
SEE MORE TEXTBOOKS