The first two normalized wave functions for a harmonic oscillator are as follows: (a is a constant, -∞ ≤ x ≤ +∞0) Ground state: 4₁(x) = ()¹/4 e-ax²/2 = (14)4xe-ax3/2 (a) Verify that the above two wave functions are orthonormal. Standard integral: dx = [0x²e-ax² dx = ₂√²/² = (1²)¹/² 2a³/2 (b) What is the average displacement (x) of the harmonic oscillator in the ground state? First excited state: ₁(x) = (¹9³) ² Stoe-αx² е 1/2
The first two normalized wave functions for a harmonic oscillator are as follows: (a is a constant, -∞ ≤ x ≤ +∞0) Ground state: 4₁(x) = ()¹/4 e-ax²/2 = (14)4xe-ax3/2 (a) Verify that the above two wave functions are orthonormal. Standard integral: dx = [0x²e-ax² dx = ₂√²/² = (1²)¹/² 2a³/2 (b) What is the average displacement (x) of the harmonic oscillator in the ground state? First excited state: ₁(x) = (¹9³) ² Stoe-αx² е 1/2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
solve part a and b
![The first two normalized wave functions for a harmonic oscillator are as follows: (α is a constant, \(-\infty \leq x \leq +\infty\))
**Ground state:**
\[
\psi_0(x) = \left( \frac{\alpha}{\pi} \right)^{1/4} e^{-\alpha x^2/2}
\]
**First excited state:**
\[
\psi_1(x) = \left( \frac{4\alpha^3}{\pi} \right)^{1/4} x e^{-\alpha x^2/2}
\]
(a) Verify that the above two wave functions are orthonormal.
**Standard integral:**
\[
\int_{-\infty}^{+\infty} e^{-\alpha x^2} \, dx = \left( \frac{\pi}{\alpha} \right)^{1/2}
\]
\[
\int_{-\infty}^{+\infty} x^2 e^{-\alpha x^2} \, dx = \frac{\sqrt{\pi}}{2\alpha^{3/2}} = \left( \frac{\pi}{4\alpha^3} \right)^{1/2}
\]
(b) What is the average displacement \(\langle x \rangle\) of the harmonic oscillator in the ground state?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6957468-6fb1-4659-9083-987b06676d6d%2Fffc85b37-f3f1-4c3c-9814-dd4826e9d0c2%2Fhrr5d8r_processed.png&w=3840&q=75)
Transcribed Image Text:The first two normalized wave functions for a harmonic oscillator are as follows: (α is a constant, \(-\infty \leq x \leq +\infty\))
**Ground state:**
\[
\psi_0(x) = \left( \frac{\alpha}{\pi} \right)^{1/4} e^{-\alpha x^2/2}
\]
**First excited state:**
\[
\psi_1(x) = \left( \frac{4\alpha^3}{\pi} \right)^{1/4} x e^{-\alpha x^2/2}
\]
(a) Verify that the above two wave functions are orthonormal.
**Standard integral:**
\[
\int_{-\infty}^{+\infty} e^{-\alpha x^2} \, dx = \left( \frac{\pi}{\alpha} \right)^{1/2}
\]
\[
\int_{-\infty}^{+\infty} x^2 e^{-\alpha x^2} \, dx = \frac{\sqrt{\pi}}{2\alpha^{3/2}} = \left( \frac{\pi}{4\alpha^3} \right)^{1/2}
\]
(b) What is the average displacement \(\langle x \rangle\) of the harmonic oscillator in the ground state?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 30 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
