dxa"+m+Pexp(-2ax²). %3D 1. Explain in your own words what the terminology for each of the quantities in above expression is. 2. When k=n+m+p is an odd number, what can you say about the integral?
dxa"+m+Pexp(-2ax²). %3D 1. Explain in your own words what the terminology for each of the quantities in above expression is. 2. When k=n+m+p is an odd number, what can you say about the integral?
Related questions
Question
2. Please answer the question completely and accurately with full detailed steps.
![If we have trial functions, which depend on n, of the form \( \phi_n \equiv x^n \exp(-ax) \). We are interested in being able to perform integrals (for any values of n, m, and p) of the form:
\[
\langle \phi_n | x^p | \phi_m \rangle = \int_{-\infty}^{+\infty} dx \phi_n(x) x^p \phi_m(x) = \int_{-\infty}^{+\infty} dx x^{n+m+p} \exp(-2ax^2).
\]
1. Explain in your own words what the terminology for each of the quantities in the above expression is.
2. When \( k = n+m+p \) is an odd number, what can you say about the integral?
To perform these integrations we introduce the following integral and then employ integration by parts to develop a recursion relation:
\[
I_k(2\alpha) \equiv \int_{-\infty}^{+\infty} dx x^{2k} \exp(-2\alpha x^2)
\]
1. If \( u = x^{2k-1} \) what is \( du \)?
2. If \( dv = x \exp(-2\alpha x^2) dx \), what is \( v \)?
3. What is the value of \( uv \) at \( x = \infty \)? What is the value of \( uv \) at \( x = -\infty \)?
4. What is the value of \( I_{k-1}(2\alpha) \) in terms of \( I_k(2\alpha) \)?
---
**Table I: Complete the table above by filling in the value of \( I_k(2\alpha) \)**
\[
\begin{array}{c|c}
k & (m,p,n) \\
\hline
0 & \langle \phi_0 | x^0 | \phi_0 \rangle \\
1 & \langle \phi_1 | x^0 | \phi_1 \rangle = \langle \phi_0 | x^2 | \phi_0 \rangle = \langle \phi_0 | x^1 | \phi_1 \rangle \\
2 & \langle \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb460c0c-d029-4e90-a450-1d82490780a1%2F0398f7b4-016c-4b38-a8b3-3233a82230c9%2F19zyha7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:If we have trial functions, which depend on n, of the form \( \phi_n \equiv x^n \exp(-ax) \). We are interested in being able to perform integrals (for any values of n, m, and p) of the form:
\[
\langle \phi_n | x^p | \phi_m \rangle = \int_{-\infty}^{+\infty} dx \phi_n(x) x^p \phi_m(x) = \int_{-\infty}^{+\infty} dx x^{n+m+p} \exp(-2ax^2).
\]
1. Explain in your own words what the terminology for each of the quantities in the above expression is.
2. When \( k = n+m+p \) is an odd number, what can you say about the integral?
To perform these integrations we introduce the following integral and then employ integration by parts to develop a recursion relation:
\[
I_k(2\alpha) \equiv \int_{-\infty}^{+\infty} dx x^{2k} \exp(-2\alpha x^2)
\]
1. If \( u = x^{2k-1} \) what is \( du \)?
2. If \( dv = x \exp(-2\alpha x^2) dx \), what is \( v \)?
3. What is the value of \( uv \) at \( x = \infty \)? What is the value of \( uv \) at \( x = -\infty \)?
4. What is the value of \( I_{k-1}(2\alpha) \) in terms of \( I_k(2\alpha) \)?
---
**Table I: Complete the table above by filling in the value of \( I_k(2\alpha) \)**
\[
\begin{array}{c|c}
k & (m,p,n) \\
\hline
0 & \langle \phi_0 | x^0 | \phi_0 \rangle \\
1 & \langle \phi_1 | x^0 | \phi_1 \rangle = \langle \phi_0 | x^2 | \phi_0 \rangle = \langle \phi_0 | x^1 | \phi_1 \rangle \\
2 & \langle \
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
