Non-degenerate perturbation theory

Definition

Non-degenerate perturbation theory is a method used in quantum mechanics to approximate the eigenstates and eigenvalues of a quantum system when a small perturbation is applied to a system with non-degenerate energy levels. This approach relies on the assumption that the unperturbed states do not share the same energy, allowing for simpler calculations of how these states and energies change due to the perturbation. This method is particularly useful for analyzing systems where the added potential or interaction does not lead to degeneracy among the energy levels.

5 Must Know Facts For Your Next Test

1. In non-degenerate perturbation theory, the first-order energy correction is given by the expectation value of the perturbing Hamiltonian in the unperturbed state.
2. The perturbed eigenstates can be expressed as linear combinations of the original eigenstates, with coefficients determined by the first-order corrections.
3. The method assumes that the perturbation is small compared to the differences between unperturbed energy levels.
4. Non-degenerate perturbation theory simplifies calculations since it only requires consideration of one state at a time, unlike degenerate cases which may involve more complex interactions.
5. Applications of non-degenerate perturbation theory include atomic and molecular systems, where external fields or interactions can be treated as small perturbations.

Review Questions

• How does non-degenerate perturbation theory differ from degenerate perturbation theory in terms of its approach and applicability?
  • Non-degenerate perturbation theory applies to systems where energy levels are distinct and does not involve any degeneracy. It simplifies calculations by treating each unperturbed state individually. In contrast, degenerate perturbation theory must account for multiple states sharing the same energy level, requiring a more complex analysis of how these states interact under perturbations.
• What role do eigenvalues and eigenstates play in non-degenerate perturbation theory when calculating corrections due to perturbations?
  • In non-degenerate perturbation theory, eigenvalues represent the original energy levels of a quantum system before any perturbation is applied. When a perturbation is introduced, corrections to these eigenvalues are calculated using expectation values of the perturbing Hamiltonian. The corresponding eigenstates also undergo changes and are expressed as linear combinations of the original states, with adjustments made according to first-order corrections.
• Evaluate how non-degenerate perturbation theory can be applied in real-world scenarios, particularly in atomic physics.
  • Non-degenerate perturbation theory is widely used in atomic physics to understand how atoms interact with external fields, such as electromagnetic radiation. For example, it helps predict shifts in atomic energy levels when an atom is exposed to light or electric fields. By applying this method, physicists can calculate transition probabilities and spectral lines in atomic spectra, allowing for better understanding and predictions in areas like laser technology and quantum optics.

Related terms

Perturbation Theory: A mathematical approach used to find an approximate solution to a problem that cannot be solved exactly, by starting from a known solution of a related problem.

Degenerate States: States of a quantum system that have the same energy level, leading to complications when applying perturbation theory.

Eigenvalues and Eigenstates: Eigenvalues are the possible measured values of an observable, while eigenstates are the corresponding states of the system associated with those values.