Understanding Degenerate Perturbation Theory in Quantum Mechanics

Exploring time-independent perturbation theory, specifically focusing on non-degenerate and degenerate spectra. The lecture covers approximation schemes, treatment of multi-electron atom term values, and the effects of spin-orbit interaction. Concepts include evaluating expectation values, wavefunctions, and matrix eigenstates in the context of degenerate perturbation theory.

• Quantum Mechanics
• Perturbation Theory
• Degenerate Spectrum
• Schrödinger Equation
• Multi-Electron Atoms

oak_lay Follow

Uploaded on Sep 11, 2024 | 0 Views

Download Presentation

Please find below an Image/Link to download the presentation.

The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.

Presentation Transcript

1. PHY 741 Quantum Mechanics 12-12:50 AM MWF Olin 103 Plan for Lecture 26: Chap. 17 in Shankar: Time-Independent Perturbation Theory 1. Case of non-degenerate spectrum 2. Treatment of degenerate spectrum 11/01/2017 PHY 741 Fall 2017 -- Lecture 26 1

2. 11/01/2017 PHY 741 Fall 2017 -- Lecture 26 2

3. 11/01/2017 PHY 741 Fall 2017 -- Lecture 26 3

4. Approximation schemes for solving the time- independent Schr dinger equation = H n E n n = + 0 1 H H H In general, we approach the problem using the complete basis set of H = 0 : 0 0 0 n 0 H n E n However, consider the case when .... b N E E E = 0 0 n 0 n a n 11/01/2017 PHY 741 Fall 2017 -- Lecture 26 4

5. Degenerate perturbation theory, considering the effects on the -fold degenerate states: N = = 0 a 0 b 0 N 0 n 0 n 0 n , ,.... where ... n n n E E E a b N N = = 1 i 0 j i j For 1,2,... , assume N i n C n = 1 j The first-order wavefunctions wil N l be the + 0 j 0 1 0 i eigenstates of the matrix N N n H H n 11/01/2017 PHY 741 Fall 2017 -- Lecture 26 5

6. Example of degenerate perturbation theory in the treatment of the term values of multi-electron atoms: 2 e 2 2 Ze r = + H r ( ) h 2 i r ( ) h i r r i 2 m , i i j i i i j single electron terms Evaluating expectation values: electron-electron interaction 2 H for 2 LM LM p 5 25 1 25 10 25 = 2 0 2 R R ( ) E P ( 2 ,2 ;2 ,2 p p ) ( 2 ,2 ;2 ,2 p p ) e p p p p = + 2 0 2 R R ( ) ( 2 ,2 ;2 ,2 p p ) ( 2 ,2 ;2 ,2 p p ) E D e p p p p = + 2 0 2 R R ( ) ( 2 ,2 ;2 ,2 p ) ( 2 , p 2 ; p 2 , 2 ) p E S e p p p p 11/01/2017 PHY 741 Fall 2017 -- Lecture 26 6

7. Example of degenerate perturbation theory in the treatment of the effects of spin-orbit interaction: = L J =S+L J S L S S ( ) H G r SO Note that: ; JM ls + + 2 2 2 L = JM ls 2 = S L ; ( ) ; ; H G r JM ls JM ls SO 2 ( ) 2 G r ( ) + + + = J=l+1/2: ( 1) ( 1) ( 1) j j s s l l 2 ( ) G r ( ) ( ) + + = ; ; l M ls H l M ls l 1 2 1 2 SO 2 2( ) 2 J=l-1/2: G r ( ) ( ) ( ) = l + ; ; 1 l M ls H l M l s 1 2 1 2 SO 11/01/2017 PHY 741 Fall 2017 -- Lecture 26 7

8. Example of degenerate perturbation theory for a H atom in the degenerate states = 200 , 21 1 , 210 , 211 nlm 2 1 2 e = 0 all having zero-order energies E 2 2 2 a 0 In this case, consider a perturbation caused by an electrostatic field directed along the -axis causing polarization of the electron cos eFr H = Matrix elements: 2 2 ' ' 3 m l l lm H l m eFa = F : z 1 1 0 0 '0 m '1 Details: 1 eF r = / r a 1 4 2 200 210 2 cos co s H r dr e d 0 4 0 0 3 eF 16 = a a 0 1 a 0 11/01/2017 PHY 741 Fall 2017 -- Lecture 26 8

9. Degenerate perturbation theory example for the Stark effect -- continued Matrix elements: 0 3 2 2 ' ' 0 0 200 210 21 1 211 3 0 0 0 0 0 0 0 0 eFa 200 210 21 1 211 0 0 0 0 eFa = 0 1 lm H l m + 0 1 Eigenvalues of 2 2 ' ' : l m lm H H 0 2 for 21 1 E ( ( ) ) + 0 2 3 for 200 210 E eFa 1 0 2 + 0 2 3 for 200 210 E eFa 1 0 2 11/01/2017 PHY 741 Fall 2017 -- Lecture 26 9

10. Degenerate perturbation theory example for the Stark effect -- continued 0 1 Eigenvalues of 2 lm H H + 2 ' l m ' : 0 2 for 21 1 E ( ) + 0 2 1 2 E = 3 for 200 210 E eFa 1 0 2 ( ) + 0 2 3 for 200 210 E eFa 1 0 2 1 2 E F 0 2 E 11/01/2017 PHY 741 Fall 2017 -- Lecture 26 10

11. Degenerate perturbation theory example for effects of a constant magnetic field B on an atom 2 e c m + p A 1 e A r B Vector potential =2 = + + B S ( ) H V r 2 mc 2 p = + 0 ( ) H V r 2 m B Detail: 1 2 Keeping only terms to linear order in : e H mc ( ) = + 1 L S B 2 2 1 2 r B + = p r B p L B 11/01/2017 PHY 741 Fall 2017 -- Lecture 26 11

12. Degenerate perturbation theory example for effects of a constant magnetic field B on an atom -- continued For atoms with total orbital momentum and total spin : ; ( 1) S LM SM L L LM SM LM SM S S LM SM = + S L S = + = 2 2 L ; ; ; L LM SM M LM SM S z S S = 2 2 ; ( 1) ; 1)(2 L + ; ; S LM SM 1) S + M LM SM S S z S S S These states have a degeneracy of (2 Degenerate perturbation theory matrix for first order: e B c ( ) = + M M 1 ; '; ' 2 LM SM H LM SM M M S S S S MM 2 m 3 P S Example: atomic term: values of 1 0 1 1 0 1 M = 1 1 3 Paschen- Back effect 1 ; '; ' /( / 2 ) LM SM H LM SM e B m c S S = M = = S 3 1 1 2 0 2 M M 11/01/2017 PHY 741 Fall 2017 -- Lecture 26 12

13. Degenerate perturbation theory example for effects of a constant magnetic field B on an atom including the effects of spin-orbit interaction 2 e c m + p A e = + + + S L B S ( ) ( ) H V r G r 2 mc 2 p = + 0 ( ) H V r 2 m B Keeping only terms to linear order in : e H G r mc ( ) = + + 1 S L L S B ( ) 2 2 ( 2 ) G r e mc ( ) ( ) + + 2 2 2 J L S J S B = 2 11/01/2017 PHY 741 Fall 2017 -- Lecture 26 13