Perturbation Theory and Its Applications in Quantum Mechanics

1. Chapters 11, 12 Stationary Perturbation Theory And Its Applications

2. 11.A.1 The eigenproblem • Let us assume that we have a system with an (unperturbed) Hamiltonian H0, the eigenvalue problem for which is solved and the spectrum is discrete: • The spectrum can be degenerate; index i distinguishes different states corresponding to an energy value with the same integer index p • The eigenstates form a complete orthonormal basis:

3. 11.A.1 The eigenproblem • Next, a small perturbation of the system is introduced so that the new Hamiltonian is: • It is assumed that the perturbation is time-independent: stationary perturbation • Usually it is convenient to introduce a small dimensionless parameter λ and rewrite the perturbed Hamiltonian as: • The perturbation may or may not remove a degeneracy of a specific energy level

4. 11.A.2 The eigenproblem • Thus, the modified eigenproblem is: • We assume that the following expansions in powers of λ can be made: • Substituting these expansion into the eigenproblem: • One should now equate the coefficients of different powers of λ on both sides

5. 11.A.2 The eigenproblem • For the 0th order: • For the 1st order: • For the 2nd order: • For the qth order: • One should now equate the coefficients of different powers of λ on both sides

6. 11.A.2 The eigenproblem • For the 0th order: • For the 1st order: • For the 2nd order: • For the qth order: • We will study the first three equations

7. 11.A.2 The eigenproblem • We chose the perturbed eigenstate to be normalized in a way that: • Then: • Thus:

8. 11.B.1 Perturbation of a non-degenerate level • Let us consider a non-degenerate eigenvalue of the unperturbed Hamiltonian • In this case: • Recall:

9. 11.B.1 Perturbation of a non-degenerate level • Let us consider a non-degenerate eigenvalue of the unperturbed Hamiltonian • In this case: • Recall:

10. 11.B.1 Perturbation of a non-degenerate level • Let us consider a non-degenerate eigenvalue of the unperturbed Hamiltonian • In this case: • Recall:

11. 11.B.1 Perturbation of a non-degenerate level • Let us consider a non-degenerate eigenvalue of the unperturbed Hamiltonian • In this case: • Recall:

12. 11.B.2 Perturbation of a non-degenerate level • Let us consider a non-degenerate eigenvalue of the unperturbed Hamiltonian • In this case: • Recall:

13. 11.B.2 Perturbation of a non-degenerate level • On the other hand: • Therefore:

14. 11.B.2 Perturbation of a non-degenerate level • Synopsizing:

15. 11.C Perturbation of a degenerate state • Let us now consider a degenerate eigenvalue of the unperturbed Hamiltonian: • The perturbation can (or not) split this level into distinct sublevels • If the number of different sublevels is smaller than the degeneracy of the level, some of the sublevels will remain degenerate • We will use the same approach to calculate the corrections for both the eigenvalues and the eigenvectors

16. 11.C Perturbation of a degenerate state • From • We obtain • There is a total of gn such relations

17. 11.C Perturbation of a degenerate state • This expression is nothing else but a matrix representation the eigenproblem • The representation is obtained in the subspace spanned by vectors • Thus to calculate the first-order corrections to the eigenvalues and the eigenstates in the 0th degree one should diagonalize the matrix of the perturbation inside eigensubspace associated with

18. 11.C Perturbation of a degenerate state • Such diagonalization procedure may be repeated for the eigensubspaces of all the degenerate energy values • This problem is much simpler mathematically than the initial problem of diagonalizing the Hamiltonian in the entire state space • It is advantageous to find a basis in which diagonalization calculations are as simple as possible

19. 12.A Application of the stationary perturbation theory for a hydrogen atom • We completely solved the problem for the following Hamiltonian describing hydrogen atom: • Modern spectroscopy however reveals effects that cannot be explained in terms of this Hamiltonian • The reason behind these discrepancies is in the fact that the H0 Hamiltonian is only an approximation • It does not take into account many factors such as relativistic nature of motion, magnetic effects associated with spin, etc.

20. 12.A Application of the stationary perturbation theory for a hydrogen atom • As a result, the so called fine structure appears in emission/absorption spectra of hydrogen and other elements • Its origins: • a) Relativistic motion • b) Spin-orbit coupling • c) Quantization of electric field

21. 12.A Application of the stationary perturbation theory for a hydrogen atom • Further inspection of high-resolution spectra reveals additional weaker splitting, the hyperfine structure, originating from the interaction between dipole moments of the electron and the proton

22. 12.A Application of the stationary perturbation theory for a hydrogen atom • All these corrections to the original Hamiltonian are described by different mathematical expressions which can be treated in terms of the stationary perturbation theory • Let us consider, as an example, the fine structure corrections produced by the spin-orbit coupling • They originate from the interaction between the electron’s spin and the magnetic field arising due to the proton-electron relative motion

23. 12.B.1 Spin-orbit coupling • The electron is moving in the electrostatic field created by a proton • From the electron’s point of view, the proton is moving, thus creating a magnetic field:

24. 12.B.1 Spin-orbit coupling • The electron is moving in the electrostatic field created by a proton • From the electron’s point of view, the proton is moving, thus creating a magnetic field: • Because of its spin, the electron possesses an intrinsic magnetic moment: • On the other hand, the electrostatic field is:

25. 12.B.1 Spin-orbit coupling • The interaction energy between a magnetic field and a magnetic moment: • This expression is derived for an inertial frame of reference; account of the non-inertial nature yields:

26. 12.C.2 Spin-orbit coupling • This term can be treated as a perturbation of the Hamiltonian • Recall: • Although the perturbed Hamiltonian no longer commutes with the orbital angular momentum and the spin, it commutes with L2, S2, and J2 • The eigenvalues of are

27. 12.C.2 Spin-orbit coupling • This term can be treated as a perturbation of the Hamiltonian • Recall: • Although the perturbed Hamiltonian no longer commutes with the orbital angular momentum and the spin, it commutes with L2, S2, and J2 • The eigenvalues of are

28. 12.C.2 Spin-orbit coupling • The scales of energies associated with the spin-orbit coupling are about two order of magnitude smaller then the energies associated with the discreet atomic levels • Therefore the stationary perturbation theory can be used to calculate corrections due to the spin-orbit coupling • As an illustration, let us use the Hamiltonian of hydrogen atom as H0 • In particular, we will calculate the 1st order correction to the eigenvaules

29. 12.C.2 Spin-orbit coupling • Recall: • In this case • Thus: • It is convenient to calculate this correction using representation

30. 12.C.2 Spin-orbit coupling • The radial part can be calculated separately: • After this one needs to diagonalize this matrix: • Obviously it is prudent to change the basis: • All these values are well known and tabulated