7 Questions
What is the general form of the total Hamiltonian in time-independent perturbation theory?
H = H0 + V
What is the physical significance of the first-order energy correction ΔE_n^1 in perturbation theory?
The expectation value of the perturbation V in the unperturbed state |n^0>
What is the formula for the first-order correction to the wave function |n^1> in perturbation theory?
|n^1> = Σ_(m≠n) [(V_nm) / (E_n^0 - E_m^0)] |m^0>
What is the assumption behind non-degenerate perturbation theory?
The unperturbed states are non-degenerate (i.e., distinct energies)
What is the condition for the perturbation V to be considered small in perturbation theory?
The perturbation V must be small compared to the energy level spacing of the unperturbed system
How can higher-order corrections be calculated in perturbation theory?
Using similar formulas to calculate corrections of any order
What is the purpose of time-independent perturbation theory?
To calculate the effects of a small perturbation on the energy levels and wave functions of a system
Study Notes
Hamiltonian Perturbation
Time-Independent Perturbation Theory
Perturbed Hamiltonian
- The total Hamiltonian (H) is split into an unperturbed part (H0) and a perturbation (V):
- H = H0 + V
- The unperturbed Hamiltonian (H0) has known eigenvalues (E_n^0) and eigenfunctions (|n^0)
First-Order Perturbation Theory
First-Order Energy Correction
- The first-order correction to the energy (ΔE_n^1) is given by:
- ΔE_n^1 =
- This represents the expectation value of the perturbation (V) in the unperturbed state |n^0>
First-Order Wave Function Correction
- The first-order correction to the wave function (|n^1>) is given by:
- |n^1> = Σ_(m≠n) [(V_nm) / (E_n^0 - E_m^0)] |m^0>
- This represents the correction to the unperturbed wave function due to the perturbation (V)
Higher-Order Perturbation Theory
- Higher-order corrections can be calculated using similar formulas
- The process can be extended to calculate corrections of any order
Limitations and Validity
- Non-degenerate perturbation theory assumes that the unperturbed states are non-degenerate (i.e., distinct energies)
- The perturbation (V) must be small compared to the energy level spacing of the unperturbed system
Hamiltonian Perturbation
Time-Independent Perturbation Theory
- The total Hamiltonian (H) is split into two parts: the unperturbed part (H0) and a perturbation (V)
- The unperturbed Hamiltonian (H0) has known eigenvalues (E_n^0) and eigenfunctions (|n^0)
First-Order Perturbation Theory
First-Order Energy Correction
- The first-order correction to the energy (ΔE_n^1) is the expectation value of the perturbation (V) in the unperturbed state |n^0>
- ΔE_n^1 represents the energy shift due to the perturbation
First-Order Wave Function Correction
- The first-order correction to the wave function (|n^1>) is a sum of corrections from other unperturbed states |m^0> (m ≠ n)
- The correction is proportional to the matrix element (V_nm) and inversely proportional to the energy difference (E_n^0 - E_m^0)
Higher-Order Perturbation Theory
- Higher-order corrections can be calculated using similar formulas
- The process can be extended to calculate corrections of any order
Limitations and Validity
- Non-degenerate perturbation theory assumes non-degenerate unperturbed states (i.e., distinct energies)
- The perturbation (V) must be small compared to the energy level spacing of the unperturbed system
Learn about time-independent perturbation theory, including the perturbed Hamiltonian and first-order energy correction. Understand the concepts of unperturbed Hamiltonian and perturbation.
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