Physics Tutorials

Rayleigh-Schrödinger perturbation theory is a method used in quantum mechanics to calculate the energy levels and wave functions of a system under the influence of a perturbation. The expressions and equations used in Rayleigh-Schrödinger perturbation theory are:

Expressions:

- Unperturbed Hamiltonian: H0
- Perturbation: V
- Total Hamiltonian: H = H0 + V
- Unperturbed energy levels: E_n^0
- Unperturbed wave functions: |n^0
- Perturbed energy levels: E_n
- Perturbed wave functions: |n

Equations:

1. Schrödinger equation: H|n= E_n|n
2. Unperturbed Schrödinger equation: H0|n^0= E_n^0|n^0
3. Perturbation equation: (H0 + V)|n= E_n|n
4. Rayleigh-Schrödinger perturbation theory equations:

a) First-order correction to energy:

E_n^1 = ∫|n^0|V|n^0

b) First-order correction to wave function:

|n^1= ∑_(m≠n) |m^0(V|n^0/E_n^0 - E_m^0)

c) Second-order correction to energy:

E_n^2= ∑_(m≠n) |∫|m^0|V|n^0|^2/(E_n^0 - E_m^0)

d) Second-order correction to wave function:

|n^2= ∑_(m≠n) |m^0(V|n^1/E_n^0 - E_m^0)

These equations are used to calculate the energy levels and wave functions of the perturbed system to first and second order in the perturbation.

Note: The Rayleigh-Schrödinger perturbation theory is a more rigorous and systematic approach than the simple perturbation theory, and it is widely used in quantum mechanics to study the behavior of systems under the influence of perturbations.

Degenerate perturbation theory is a extension of perturbation theory used when the unperturbed energy levels are degenerate (i.e., same energy level). The expressions used in degenerate perturbation theory are:

1. Unperturbed Hamiltonian: H0
2. Perturbation: V
3. Total Hamiltonian: H = H0 + V
4. Unperturbed energy levels: E_n^0 (degenerate)
5. Unperturbed wave functions: |n^0 (degenerate)
6. Perturbed energy levels: E_n
7. Perturbed wave functions: |n

Equations:

1. Schrödinger equation: H|n= E_n|n
2. Unperturbed Schrödinger equation: H0|n^0= E_n^0|n^0
3. Perturbation equation: (H0 + V)|n= E_n|n
4. Degenerate perturbation theory equation:

(V - E_n^1)|n^1= ∑_(m≠n) |m^0(V|n^0/E_n^0 - E_m^0)

where:

- E_n^1 is the first-order correction to the energy
- |n^1 is the first-order correction to the wave function

The degenerate perturbation theory expression is used to calculate the energy levels and wave functions of the perturbed system, taking into account the degeneracy of the unperturbed energy levels.

Note: The expression is similar to the non-degenerate perturbation theory, but with an additional sum over the degenerate states.