- #1
Ado
- 26
- 4
Hi!
1. Homework Statement
From the website http://www1.uprh.edu/rbaretti/MomentumspaceIntegration8feb2010.htm
we can see the Fourier transform of the ground state hydrogenic wave function :
Φ(p) = ∫ ∫ ∫ exp(-i p r) (Z3/π )1/2 exp(-Zr) sin(θ) dθ dφ r² dr (1.1)
After intregation of the variable φ we have :
Φ(p) = 2π (Z3/π )1/2 ∫ ∫ { exp(-i pr cos(θ ) sin(θ) dθ } exp(-Zr) r² dr (1.2)
and next :
Φ(p) = 2π (Z3/π )1/2 ∫ { 2 sin(pr) /pr } exp(-Zr) r² dr (1.3)
= 4π (Z3/π )1/2 ∫ {sin(pr) /p} exp(-Zr) r dr
= 8 π1/2/ Z5/2 /( p² + Z² )²
And I want to understand these different steps..
I don't understand the relations used between (1.1), (1.2) and (1.3) :
(1.1) to (1.2) new cos(θ ) appears
(1.2) to (1.3) sin(pr)/pr appears
Can you explain me what happen between (1.1) and (1.3) ?
Tanks in advance !
1. Homework Statement
From the website http://www1.uprh.edu/rbaretti/MomentumspaceIntegration8feb2010.htm
we can see the Fourier transform of the ground state hydrogenic wave function :
Φ(p) = ∫ ∫ ∫ exp(-i p r) (Z3/π )1/2 exp(-Zr) sin(θ) dθ dφ r² dr (1.1)
After intregation of the variable φ we have :
Φ(p) = 2π (Z3/π )1/2 ∫ ∫ { exp(-i pr cos(θ ) sin(θ) dθ } exp(-Zr) r² dr (1.2)
and next :
Φ(p) = 2π (Z3/π )1/2 ∫ { 2 sin(pr) /pr } exp(-Zr) r² dr (1.3)
= 4π (Z3/π )1/2 ∫ {sin(pr) /p} exp(-Zr) r dr
= 8 π1/2/ Z5/2 /( p² + Z² )²
And I want to understand these different steps..
Homework Equations
I don't understand the relations used between (1.1), (1.2) and (1.3) :
(1.1) to (1.2) new cos(θ ) appears
(1.2) to (1.3) sin(pr)/pr appears
The Attempt at a Solution
Can you explain me what happen between (1.1) and (1.3) ?
Tanks in advance !