Physics:Quantum Perturbation theory - Revision history

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	== Introduction ==		== Introduction ==
	Perturbation theory is an important tool in [[Physics:Quantum\|quantum physics]] because exact solutions of the Schrödinger equation are known only for a limited number of idealized systems, such as the [[Physics:~~Hydrogen atom~~\|hydrogen atom]], the [[Physics:Quantum harmonic oscillator\|quantum harmonic oscillator]], and the particle in a box. By starting from such exactly solvable models, one can construct approximate solutions for more realistic systems.		Perturbation theory is an important tool in [[Physics:Quantum\|quantum physics]] because exact solutions of the Schrödinger equation are known only for a limited number of idealized systems, such as the [[Physics:Quantum atoms/hydrogen\|hydrogen atom]], the [[Physics:Quantum harmonic oscillator\|quantum harmonic oscillator]], and the particle in a box. By starting from such exactly solvable models, one can construct approximate solutions for more realistic systems.

	A standard example is the addition of an external electric field to the [[Physics:~~Hydrogen atom~~\|hydrogen atom]]. This produces small shifts in the spectral lines of hydrogen, known as the Stark effect. Although the resulting series expansions are generally not exact, they are often highly accurate when the expansion parameter is small. In [[Physics:Quantum electrodynamics\|quantum electrodynamics]] (QED), perturbative calculations of the electron's magnetic moment agree with experiment to extremely high precision.<ref>{{cite journal\|title=Tenth-order QED lepton anomalous magnetic moment: Eighth-order vertices containing a second-order vacuum polarization \| last1=Aoyama \| first1=Tatsumi \|last2=Hayakawa \|first2=Masashi \|last3=Kinoshita \|first3=Toichiro \|last4=Nio \|first4=Makiko \|s2cid=119279420 \|journal=Physical Review D \|volume=85 \|issue=3 \|pages=033007 \|year=2012 \|doi=10.1103/PhysRevD.85.033007\|arxiv=1110.2826\|bibcode=2012PhRvD..85c3007A}}</ref>		A standard example is the addition of an external electric field to the [[Physics:Quantum atoms/hydrogen\|hydrogen atom]]. This produces small shifts in the spectral lines of hydrogen, known as the Stark effect. Although the resulting series expansions are generally not exact, they are often highly accurate when the expansion parameter is small. In [[Physics:Quantum electrodynamics\|quantum electrodynamics]] (QED), perturbative calculations of the electron's magnetic moment agree with experiment to extremely high precision.<ref>{{cite journal\|title=Tenth-order QED lepton anomalous magnetic moment: Eighth-order vertices containing a second-order vacuum polarization \| last1=Aoyama \| first1=Tatsumi \|last2=Hayakawa \|first2=Masashi \|last3=Kinoshita \|first3=Toichiro \|last4=Nio \|first4=Makiko \|s2cid=119279420 \|journal=Physical Review D \|volume=85 \|issue=3 \|pages=033007 \|year=2012 \|doi=10.1103/PhysRevD.85.033007\|arxiv=1110.2826\|bibcode=2012PhRvD..85c3007A}}</ref>

	Perturbative expansions are usually asymptotic rather than convergent, so after sufficiently high order the approximation may worsen rather than improve. Nevertheless, perturbation theory remains one of the central methods of modern theoretical physics.<ref>{{cite journal\|doi=10.1002/qua.560210103\|title=Large orders and summability of eigenvalue perturbation theory: A mathematical overview\|year=1982\|last1=Simon\|first1=Barry\|journal=International Journal of Quantum Chemistry\|volume=21\|pages=3–25}}</ref>		Perturbative expansions are usually asymptotic rather than convergent, so after sufficiently high order the approximation may worsen rather than improve. Nevertheless, perturbation theory remains one of the central methods of modern theoretical physics.<ref>{{cite journal\|doi=10.1002/qua.560210103\|title=Large orders and summability of eigenvalue perturbation theory: A mathematical overview\|year=1982\|last1=Simon\|first1=Barry\|journal=International Journal of Quantum Chemistry\|volume=21\|pages=3–25}}</ref>

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	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

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	In [[Physics:Quantum mechanics\|quantum mechanics]], '''perturbation theory''' is a set of approximation schemes directly related to mathematical [[Physics:Perturbation theory\|perturbation]] for describing a complicated [[Physics:Quantum system\|quantum system]] in terms of a simpler one. The idea is to begin with a system whose mathematical solution is known, and then add a weak perturbing [[Physics:Hamiltonian (quantum mechanics)\|Hamiltonian]] representing a small disturbance. If the disturbance is not too large, the physical quantities of the perturbed system, such as its [[Physics:Energy level\|energy level]]s and [[Physics:Quantum state\|eigenstates]], can be written as corrections to those of the simpler system. In this way, perturbation theory makes it possible to study complicated unsolved systems by expanding around simpler solvable models.		In [[Physics:Quantum mechanics\|quantum mechanics]], '''perturbation theory''' is a set of approximation schemes directly related to mathematical [[Physics:Perturbation theory\|perturbation]] for describing a complicated [[Physics:Quantum system\|quantum system]] in terms of a simpler one. The idea is to begin with a system whose mathematical solution is known, and then add a weak perturbing [[Physics:Hamiltonian (quantum mechanics)\|Hamiltonian]] representing a small disturbance. If the disturbance is not too large, the physical quantities of the perturbed system, such as its [[Physics:Energy level\|energy level]]s and [[Physics:Quantum state\|eigenstates]], can be written as corrections to those of the simpler system. In this way, perturbation theory makes it possible to study complicated unsolved systems by expanding around simpler solvable models.
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			[[File:Quantum Perturbation Theory.png\|thumb\|280px\|Quantum Perturbation theory.]]
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	[[File:Quantum Perturbation Theory.png\|thumb\|400px\|Diagram illustrating quantum perturbation theory: a solvable Hamiltonian is modified by a weak perturbation, causing shifts in energy levels and mixing of eigenstates.]]		</div>

	== Introduction ==		== Introduction ==

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	{{Quantum article nav\|previous=Physics:Quantum Stationary states\|previous label=Stationary states\|next=Physics:Quantum Time-dependent perturbation theory\|next label=Time-dependent perturbation theory}}		{{Quantum article nav\|previous=Physics:Quantum Stationary states\|previous label=Stationary states\|next=Physics:Quantum Time-dependent perturbation theory\|next label=Time-dependent perturbation theory}}