Chapters 11 12 Stationary Perturbation Theory And Its
Chapters 11, 12 Stationary Perturbation Theory And Its Applications
The eigenproblem 11. A. 1 • Let us assume that we have a system with an (unperturbed) Hamiltonian H 0, 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:
The eigenproblem 11. A. 1 • Next, a small perturbation of the system is introduced so that the new Hamiltonian is: • It is assumed that the perturbation is timeindependent: 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
The eigenproblem 11. A. 2 • 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
The eigenproblem 11. A. 2 • For the 0 th order: • For the 1 st order: • For the 2 nd order: • For the qth order: • One should now equate the coefficients of different powers of λ on both sides
The eigenproblem • For the 0 th order: • For the 1 st order: • For the 2 nd order: • For the qth order: • We will study the first three equations 11. A. 2
The eigenproblem 11. A. 2 • We chose the perturbed eigenstate to be normalized in a way that: • Then: • Thus:
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. B. 1 Perturbation of a non-degenerate level • Let us consider a non-degenerate eigenvalue of the unperturbed Hamiltonian • In this case: • Recall:
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. B. 1 Perturbation of a non-degenerate level • Let us consider a non-degenerate eigenvalue of the unperturbed Hamiltonian • In this case: • Recall:
11. B. 2 Perturbation of a non-degenerate level • Let us consider a non-degenerate eigenvalue of the unperturbed Hamiltonian • In this case: • Recall:
11. B. 2 Perturbation of a non-degenerate level • On the other hand: • Therefore:
11. B. 2 Perturbation of a non-degenerate level • Synopsizing:
Perturbation of a degenerate state 11. C • 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
Perturbation of a degenerate state • From • We obtain • There is a total of gn such relations 11. C
Perturbation of a degenerate state 11. C • 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 0 th degree one should diagonalize the matrix of the perturbation inside eigensubspace associated with
Perturbation of a degenerate state 11. C • 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
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 H 0 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.
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
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
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
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: 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: 12. B. 1
Spin-orbit coupling 12. B. 1 • 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:
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 L 2, S 2, and J 2 • The eigenvalues of are 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 L 2, S 2, and J 2 • The eigenvalues of are 12. C. 2
Spin-orbit coupling 12. C. 2 • 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 H 0 • In particular, we will calculate the 1 st order correction to the eigenvaules
Spin-orbit coupling • Recall: • In this case • Thus: • It is convenient to calculate this correction using representation 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 12. C. 2
- Slides: 30
