Perturbation Theory and Approximation Methods
Most quantum problems cannot be solved exactly, so approximation methods are essential; perturbation theory treats a system as a solvable one plus a small correction, while the variational and WKB methods bound or estimate energies and wavefunctions in other regimes.
Definition
Approximation methods in quantum mechanics are systematic techniques for estimating energies, states, and transition rates when the Schrodinger equation cannot be solved exactly, the principal ones being perturbation theory, the variational method, and the semiclassical WKB approximation.
Scope
The area covers time-independent perturbation theory for energy and state corrections including degenerate cases, time-dependent perturbation theory and Fermi's golden rule for transition rates, the variational principle that bounds ground-state energies from above, and the WKB semiclassical approximation for slowly varying potentials and tunneling.
Sub-topics
Core questions
- How are energy levels and states corrected when a small perturbation is added?
- How are transition rates between states computed under a time-dependent influence?
- How can the ground-state energy be bounded without solving the equation exactly?
- When does a semiclassical approximation give accurate results?
Key concepts
- perturbation expansion
- degenerate perturbation theory
- Fermi's golden rule
- variational principle
- trial wavefunction
- WKB approximation
Key theories
- Perturbation theory
- Expanding energies and states in powers of a small perturbation gives corrections order by order, with the leading energy shift equal to the expectation of the perturbation, and its time-dependent form yields Fermi's golden rule for transition rates between states.
- Variational and WKB methods
- The variational principle guarantees that the expectation of the Hamiltonian in any trial state is an upper bound on the ground-state energy, while the WKB approximation builds wavefunctions from a slowly varying local wavelength, accurate when the potential changes little over a wavelength.
Clinical relevance
These methods make quantum mechanics applicable to real systems: perturbation theory predicts the Stark and Zeeman splittings and atomic transition rates, the variational method gives accurate ground-state energies in quantum chemistry, and WKB explains tunneling rates and quantization conditions across atomic, nuclear, and solid-state physics.
History
Rayleigh and Schrodinger developed time-independent perturbation theory in the 1920s; Dirac formulated time-dependent perturbation theory and Fermi popularized the golden rule for transition rates, while the WKB method was introduced independently by Wentzel, Kramers, and Brillouin in 1926.
Key figures
- Erwin Schrodinger
- Paul Dirac
- Enrico Fermi
- Lord Rayleigh
Related topics
Seminal works
- sakurai2017
- landau1977
Frequently asked questions
- When does perturbation theory fail?
- It breaks down when the perturbation is not small compared with the energy spacing, when levels are nearly degenerate so denominators blow up, or when the series does not converge; in such cases variational, semiclassical, or numerical methods are needed instead.
- Why does the variational method always overestimate the ground-state energy?
- Any trial state is a mixture of the true eigenstates, and because all excited-state energies lie above the ground state, the expectation of the Hamiltonian is a weighted average that can never fall below the lowest eigenvalue.