• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Help normalizing a wave function

1. Homework Statement
I don't see how the author normalizes ##u(r)=Asin(kr)##. From Griffiths, Introduction to Quantum Mechanics, 2nd edition, page 141-142:

http://imgur.com/a/bo8v6

2. Homework Equations
##\int_0^{\infty} \int_0^{\pi} \int_0^{2\pi}|A|^2 \sin^2(\frac{n\pi r}{a})r^2 \sin \theta drd\theta d\phi=1##

3. The Attempt at a Solution
My integral was
##\int_0^{\infty} \int_0^{\pi} \int_0^{2\pi}|A|^2 \sin^2(\frac{n\pi r}{a})r^2 \sin \theta drd\theta d\phi=1##

Mathematica simplifies the integral (without the ##A## for simplicity) to
##=\int_0^{\infty}4\pi r^2 \sin^2(\frac{n\pi r}{a})dr##

but it stops there. I don't think this integral converges. Did I make a mistake somewhere?
 
954
116
As Griffiths himself makes clear - the radial part of the wavefunction is ##R(r) = u(r) /r##, so that is what you should be putting in the integral instead. When he says normalising ##u(r)##, what he really means is finding the coefficient ##A##
 

TSny

Homework Helper
Gold Member
11,902
2,554
What is the value of the wavefunction for r > a?

The normalization condition for the radial part of the wavefunction R(r) is given in Griffith's equation [4.31] (not shown in your picture).
 
What is the value of the wavefunction for r > a?

The normalization condition for the radial part of the wavefunction R(r) is given in Griffith's equation [4.31] (not shown in your picture).
The wavefunction is ##0## for ##r>a##.
 
As Griffiths himself makes clear - the radial part of the wavefunction is ##R(r) = u(r) /r##, so that is what you should be putting in the integral instead. When he says normalising ##u(r)##, what he really means is finding the coefficient ##A##
Ah I see. Ok then but normalizing according to that equation yields ##\int_0^{\infty} |A|^2 sin^2 (n*\pi *r/a)dr##, which does not converge.
 

TSny

Homework Helper
Gold Member
11,902
2,554
Did you take account of the fact that the wavefunction vanishes for r > a?
 
Did you take account of the fact that the wavefunction vanishes for r > a?
But I thought whenever you normalize, you integrate over all space?
 

TSny

Homework Helper
Gold Member
11,902
2,554
Yes, you integrate over all space. But you have to use the correct wavefunction for all of space.
 

TSny

Homework Helper
Gold Member
11,902
2,554
To put it another way, is there a need to integrate over regions of space where the wavefunction is zero?
 
Yes, you integrate over all space. But you have to use the correct wavefunction for all of space.
So you mean the integral should technically be ##\int_0^a |A|^2 sin^2 (n\pi r/a)dr + \int_a^{\infty} 0 dr## ??
 

TSny

Homework Helper
Gold Member
11,902
2,554

Want to reply to this thread?

"Help normalizing a wave function" You must log in or register to reply here.

Related Threads for: Help normalizing a wave function

Replies
1
Views
8K
Replies
1
Views
2K
Replies
2
Views
3K
Replies
3
Views
3K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top