Quantum mechanics1
- 2. 2 Perturbation Theory is an extremely important method of seeing how a Quantum System will be affected by a small change in the potential. ... Perturbation theory is one among them. Perturbation means small disturbance
- 3. 3 Degeneracy in quantum mechanics refers to the situation when more than one eigenstate corresponds to the same energy. Conversely, non-degeneracy occurs when each eigenstate corresponds to a unique energy the hydrogen atom: in the absence of any external field, and ignoring spin, an electron in the nth energy level can have orbital quantum numbers and magnetic quantum numbers
- 4. 4 In the following derivations, let it be assumed that all eigenenergies and eigenfunctions are normalized. To find the 1st - order energy correction due to some perturbing potential, begin with the unperturbed eigenvalue problem ………..(1) If some perturbing Hamiltonian is added to the unperturbed Hamiltonian, the total Hamiltonian becomes where it is required that this perturbing Hamiltonian be small compared to the unperturbed Hamiltonian. The perturbed eigenvalue problem then becomes ………..(2) ………..(3)
- 5. 5 Where, ………..(7) …..(5) …..(4) We can use some parameter to keep track of the higher and higher order corrections to the unperturbed value, such that ………..(6) Thus, as 0 Substituting the new expressions for n and En into the perturbed eigenvalue problem yields ....(10) ………..(9) ………..(8) Expanding and collecting like terms: ....(11)
- 6. 6 Now, since The previous equation is in the form, which means that Clearly we get the original unperturbed portion of the problem back: ………..(14) ………..(13) ………..(12) we get the corrections to the unperturbed problem due to the perturbing potential as well: Collecting like terms, Equation (17) tells us that 0 operates on n (1) suggesting that n (1) may be expressed as the superposition of the eigenstates corresponding to 0 ,namely ………..(18) ………..(17) ………..(16) ………..(15)
- 7. 7 Now, switching to Dirac notation.. Now the LHS of equation (20) can be separated into two terms and multiplying from the left by < j (0)|, equation (17) becomes The first of these two terms can be re-written and this bra-ket yields the unperturbed energy in the j-basis; therefore The only value in the series that is non-zero occurs when i = j : ………..(24) ………..(23) ………..(22) ………..(20) ………..(21) ………..(19)
- 8. 8 Now we re-write equation (20) in terms of the new LHS and RHS: The second term in the LHS can be re-written in a similar way: Therefore, the LHS of equation (20) becomes The RHS of equation (20) can be simplified in a similar way: where we use the shorthand notation or ………..(25) ………..(26) ………..(27) ………..(28) ………..(29) ………..(30)
- 9. 9 When n = j, equation (30) becomes the final expression for the 1st order energy correction due to the perturbation potential : This equation states that the 1st order correction to the energy corresponding to some non-degenerate energy level is simply the perturbing potential ' “averaged” over the unperturbed eigenstate of the system. Thus the new eigenenergy is, ………..(32) ………..(31)
- 10. Derivation of 1st-order Eigenstate Correction 10 Then for n /= j equation (30) becomes To find the 1st order eigenstate correction, remember equation (18), which says that may be expressed as the superposition of the eigenstates corresponding to 0 ;namely But j is simply one particular state out of all i. So in general ………..(36) ………..(35) ………..(34) ………..(33) or
- 11. 100% 11 Substituting this equation back into equation (33) yields Thus, the new eigenstate is ………..(38) ………..(37)
- 12. EXAMPLES: 12