Fisika Modern 08 schrodinger eqinhydrogenatom

1,064 views

Published on

Published in: Education
0 Comments
2 Likes
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
1,064
On SlideShare
0
From Embeds
0
Number of Embeds
25
Actions
Shares
0
Downloads
0
Comments
0
Likes
2
Embeds 0
No embeds

No notes for slide

Fisika Modern 08 schrodinger eqinhydrogenatom

  1. 1. Pertemuan 6-8 Fisika Modern Application Schrodinger Equation In Hydrogen Atom Hadi Nasbey, M.Si <ul><li>Jurusan Fisika </li></ul><ul><li>Fakultas Matematika dan Ilmu Pengetahuan Alam </li></ul>07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  2. 2. Outline <ul><li>Application of the Schrödinger Equation to the Hydrogen Atom </li></ul><ul><li>The Schrödinger Equation in Spherical Coordinates </li></ul><ul><li>Solution of the Schrödinger Equation for Hydrogen </li></ul>07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  3. 3. <ul><li>The potential energy of the electron-proton system is electrostatic: </li></ul><ul><li>Use the three-dimensional time-independent Schrödinger Equation. </li></ul><ul><li>For Hydrogen-like atoms (He + or Li ++ ), replace e 2 with Ze 2 ( Z is the atomic number). </li></ul><ul><li>In all cases, for better accuracy, replace m with the reduced mass,  . </li></ul>Application of the Schrödinger Equation to the Hydrogen Atom 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  4. 4. <ul><li>The potential (central force) V ( r ) depends on the distance r between the proton and electron. </li></ul>Spherical Coordinates Transform to spherical polar coordinates because of the radial symmetry. 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  5. 5. The Schrödinger Equation in Spherical Coordinates Transformed into spherical coordinates, the Schrödinger equation becomes: 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  6. 6. Separable Solution <ul><li>The wave function   is a function of r ,  ,  . This is a potentially complicated function. </li></ul><ul><li>Assume instead that  is separable , that is, a product of three functions, each of one variable only: </li></ul>This would make life much simpler, and it turns out to work. 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  7. 7. <ul><li>Start with Schrodinger’s Equation: </li></ul>Solution of the Schrödinger Equation for Hydrogen Multiply both sides by  r 2 sin 2  / R f g : Substitute: 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  8. 8. r and  appear only on the left side and  appears only on the right side. The left side of the equation cannot change as  changes. The right side cannot change with either r or  . Each side needs to be equal to a constant for the equation to be true. Set the constant to be − m ℓ 2 Solution of the Schrödinger Equation for H azimuthal equation 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  9. 9. satisfies the azimuthal equation for any value of m ℓ . The solution must be single valued to be a valid solution for any  : Solution of the Schrödinger Equation for H m ℓ must be an integer (positive or negative) for this to be true. Specifically: So: 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  10. 10. <ul><li>Now set the left side equal to − m ℓ 2 : </li></ul>Solution of the Schrödinger Equation for H Now, the left side depends only on r , and the right side depends only on  . We can use the same trick again! Rearrange it and divide by sin 2 (  ) : 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | − m ℓ 2
  11. 11. <ul><li>Set each side equal to the constant ℓ(ℓ + 1) . </li></ul>Solution of the Schrödinger Equation for H Radial equation Angular equation We’ve separated the Schrödinger equation into three ordinary second-order differential equations, each containing only one variable. 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |
  12. 12. TERIMA KASIH 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

×