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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.

[[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.]]

== Introduction ==
Perturbation theory is an important tool in [[Physics:Quantum|quantum physics]] because exact solutions of the [[Physics:Schrödinger equation|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 [[Physics:Particle in a box|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 [[Physics:Electric field|electric field]] to the [[Physics:Hydrogen atom|hydrogen atom]]. This produces small shifts in the [[Physics:Spectral line|spectral line]]s of hydrogen, known as the [[Physics:Stark effect|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 [[Physics:Magnetic moment|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>

== Approximate Hamiltonians ==
In perturbation theory one writes the Hamiltonian as
<math display="block">H = H_0 + \lambda V,</math>
where <math>H_0</math> is the exactly solvable unperturbed Hamiltonian, <math>V</math> is the perturbation, and <math>\lambda</math> is a dimensionless parameter that measures the strength of the perturbation. The parameter is often introduced formally and set equal to 1 at the end of the calculation.

The goal is to determine how the eigenvalues and eigenstates of <math>H_0</math> are modified by the perturbation. If the perturbation is weak, the energies and states can be expanded in powers of <math>\lambda</math>.

== Applying perturbation theory ==
Perturbation theory is applicable when the problem cannot be solved exactly but can be formulated as a small correction to an exactly solvable one. The method works best when the matrix elements of the perturbation are small compared with the relevant differences of unperturbed energy levels.

If the perturbation is too large, or if the system contains qualitatively new states not connected smoothly to the unperturbed states, perturbation theory may fail. This occurs, for example, in low-energy [[Physics:Quantum chromodynamics|quantum chromodynamics]], where the coupling constant becomes too large for a perturbative expansion to remain valid. It also fails for genuinely non-perturbative phenomena such as certain [[Physics:Bound state|bound state]]s, [[Physics:Soliton|soliton]]s, and collective effects like [[Physics:Superconductivity|superconductivity]], where other methods such as the [[Chemistry:Variational method (quantum mechanics)|variational method]], [[Physics:WKB approximation|WKB approximation]], or numerical approaches such as [[Density functional theory|density functional theory]] may be required.<ref>{{cite journal |last1=van Mourik |first1=T. |last2=Buhl |first2=M. |last3=Gaigeot |first3=M.-P. |title=Density functional theory across chemistry, physics and biology |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=10 February 2014 |volume=372 |issue=2011 |pages=20120488 |doi=10.1098/rsta.2012.0488 |pmc=3928866 |pmid=24516181 |bibcode=2014RSPTA.37220488V }}</ref>

== Time-independent perturbation theory ==
Time-independent perturbation theory deals with perturbations that do not depend explicitly on time. It was presented by [[Biography:Erwin Schrödinger|Erwin Schrödinger]] in 1926, building on earlier work of [[Biography:John William Strutt, 3rd Baron Rayleigh|Lord Rayleigh]], and is therefore often called '''Rayleigh–Schrödinger perturbation theory'''.<ref>{{cite journal|first= E. |last=Schrödinger|journal= Annalen der Physik|volume= 80|pages= 437–490 |doi=10.1002/andp.19263851302 |issue=13|year=1926|title=Quantisierung als Eigenwertproblem|trans-title=Quantization as an eigenvalue problem|language=German|bibcode=1926AnP...385..437S}}</ref><ref>{{cite book|first=J. W. S.|last=Rayleigh|title=Theory of Sound|edition=2nd |volume= I|pages=115–118|publisher= Macmillan|location= London |year=1894|isbn=978-1-152-06023-4}}</ref><ref>{{Cite journal|date=2018-07-01|title=Aspects of perturbation theory in quantum mechanics: The BenderWuMathematica® package|journal=Computer Physics Communications|language=en|volume=228|pages=273–289| doi=10.1016/j.cpc.2017.11.018 | issn=0010-4655 |last1=Sulejmanpasic |first1=Tin |last2=Ünsal |first2=Mithat |s2cid=46923647 |bibcode=2018CoPhC.228..273S |doi-access=free}}</ref>

Suppose the unperturbed Hamiltonian satisfies
<math display="block">H_0 \left |n^{(0)} \right \rang = E_n^{(0)} \left |n^{(0)} \right\rang.</math>

For a weak perturbation,
<math display="block">H = H_0 + \lambda V.</math>

The perturbed energies and states are expanded as
<math display="block">\begin{align}
E_n &= E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \cdots,\\[1ex]
|n\rang &= \left |n^{(0)} \right \rang + \lambda \left |n^{(1)} \right \rang + \lambda^2 \left |n^{(2)} \right \rang + \cdots.
\end{align}</math>

=== First-order corrections ===
For a non-degenerate level, the first-order shift in the energy is
<math display="block">E_n^{(1)} = \left \langle n^{(0)} \right | V \left |n^{(0)} \right \rang,</math>
which is the [[Physics:Expectation value (quantum mechanics)|expectation value]] of the perturbation in the unperturbed state.

The first-order correction to the eigenstate is
<math display="block">\left |n^{(1)} \right \rang = \sum_{k \ne n} \frac{\left \langle k^{(0)} \right |V\left |n^{(0)} \right \rang}{E_n^{(0)} - E_k^{(0)}} \left |k^{(0)} \right \rang.</math>

This expression shows that nearby energy levels contribute most strongly to the mixing of states.

=== Second-order and higher-order corrections ===
To second order, the energy becomes
<math display="block">E_n(\lambda) = E_n^{(0)} + \lambda \left \langle n^{(0)} \right |V\left |n^{(0)} \right \rang + \lambda^2\sum_{k \ne n} \frac{\left |\left \langle k^{(0)} \right |V\left |n^{(0)} \right \rang \right |^2} {E_n^{(0)} - E_k^{(0)}} + O(\lambda^3).</math>

The third-order energy correction can also be written explicitly as<ref>{{cite book| first1=L. D. | last1=Landau|first2= E. M.|last2= Lifschitz| title=Quantum Mechanics: Non-relativistic Theory|edition=3rd| isbn=978-0-08-019012-9|year=1977|publisher=Pergamon Press }}</ref>
<math display="block">E_n^{(3)} = \sum_{k \neq n} \sum_{m \neq n} \frac{\langle n^{(0)} | V | m^{(0)} \rangle \langle m^{(0)} | V | k^{(0)} \rangle \langle k^{(0)} | V | n^{(0)} \rangle}{\left(E_n^{(0)} - E_m^{(0)} \right) \left(E_n^{(0)} - E_k^{(0)} \right)} - \langle n^{(0)} | V | n^{(0)} \rangle \sum_{m \neq n} \frac{|\langle n^{(0)} | V | m^{(0)} \rangle|^2}{\left( E_n^{(0)} - E_m^{(0)} \right)^2}.</math>

Higher-order corrections can be developed systematically, though the expressions rapidly become cumbersome.

=== Effects of degeneracy ===
When two or more unperturbed states have the same energy, ordinary non-degenerate perturbation theory fails because denominators such as <math>E_n^{(0)}-E_k^{(0)}</math> vanish. In this case one must first diagonalize the perturbation within the degenerate subspace. This leads to '''degenerate perturbation theory''', in which the perturbation lifts the degeneracy and defines the correct zeroth-order basis for further expansion.

Near-degenerate states must also be treated with care, since even a small perturbation can produce substantial mixing and level splitting.

=== Generalization to multi-parameter case ===
Time-independent perturbation theory can be generalized to Hamiltonians depending on several small parameters <math>x^\mu</math>. In this formulation the Hamiltonian may be written
<math display="block">H(x^\mu)= H(0) + x^\mu F_\mu,</math>
where the <math>F_\mu</math> are generalized force operators. The derivatives of the energies and states with respect to these parameters can be computed systematically using the [[Physics:Hellmann–Feynman theorem|Hellmann–Feynman theorem]]s,
<math display="block">\partial_\mu E_n=\langle n|\partial_\mu H | n\rangle,</math>
<math display="block">\langle m|\partial_\mu n\rangle=\frac{\langle m|\partial_\mu H | n\rangle}{E_n-E_m}.</math>

This differential-geometric viewpoint is especially useful in modern treatments of parameter-dependent quantum systems and effective Hamiltonians.<ref>{{cite book|chapter-url=https://books.google.com/books?id=38m2QgAACAAJ|title=Symmetry and Strain-induced Effects in Semiconductors|isbn=978-0-470-07321-6|last1=Bir|first1=Gennadiĭ Levikovich | last2=Pikus|first2=Grigoriĭ Ezekielevich|year=1974|chapter=Chapter 15: Perturbation theory for the degenerate case | publisher=Wiley }}</ref><ref>{{cite journal|last=Soliverez|first= Carlos E. | url=https://www.academia.edu/attachments/11648957/download_file | via=Academia.Edu | title=General Theory of Effective Hamiltonians | journal = Physical Review A | volume=24 | issue= 1 | pages= 4–9 | year=1981 | doi=10.1103/PhysRevA.24.4 | bibcode = 1981PhRvA..24....4S }}</ref><ref>{{cite journal | vauthors = Hogervorst M, Meineri M, Penedones J, Salehi Vaziri K | title = Hamiltonian truncation in Anti-de Sitter spacetime | journal = Journal of High Energy Physics | date = 2021 | volume = 2021 | issue = 8 | page = 63 | doi = 10.1007/JHEP08(2021)063 | arxiv = 2104.10689 | bibcode = 2021JHEP...08..063H | s2cid = 233346724 }}</ref>

== Time-dependent perturbation theory ==
Time-dependent perturbation theory, developed by [[Biography:Paul Dirac|Paul Dirac]], studies a time-dependent perturbation <math>V(t)</math> added to a time-independent Hamiltonian <math>H_0</math>.<ref>{{Citation |last=Dick |first=Rainer |title=Time-Dependent Perturbations in Quantum Mechanics |date=2020 |url=https://doi.org/10.1007/978-3-030-57870-1_13 |work=Advanced Quantum Mechanics: Materials and Photons |pages=265–310 |editor-last=Dick |editor-first=Rainer |access-date=2023-10-24 |series=Graduate Texts in Physics |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-030-57870-1_13 |isbn=978-3-030-57870-1}}</ref><ref>[[Biography:Albert Messiah|Albert Messiah]] (1966). ''Quantum Mechanics'', North Holland, John Wiley & Sons. {{ISBN|0486409244}}; J. J. Sakurai (1994). ''Modern Quantum Mechanics'' (Addison-Wesley) {{ISBN|9780201539295}}.</ref>

The Hamiltonian is
<math display="block">H = H_0 + V(t).</math>

If the unperturbed system has eigenstates <math>|n\rang</math>, the general state may be written as
<math display="block"> |\psi(t)\rang = \sum_n c_n(t) e^{- i E_n t / \hbar} |n\rang.</math>

The coefficients <math>c_n(t)</math> satisfy coupled differential equations,
<math display="block"> \frac{dc_n}{dt} = \frac{-i}{\hbar} \sum_k \lang n|V(t)|k\rang \,c_k(t)\, e^{-i(E_k - E_n)t/\hbar}.</math>

This framework is used to calculate transition amplitudes and transition probabilities between states, and leads to important results such as [[Physics:Fermi's golden rule|Fermi's golden rule]] and the [[Physics:Dyson series|Dyson series]]. It is particularly useful in areas such as [[Physics:Laser|laser]] physics, atomic transitions, particle decay, and line broadening.

== Strong perturbation theory ==
A complementary expansion exists for very large perturbations. In this case one may exchange the roles of the unperturbed Hamiltonian and the perturbation, leading to a '''dual Dyson series'''. This strong-coupling expansion is related to the adiabatic approximation and, in suitable limits, to the Wigner–Kirkwood series and semiclassical methods.<ref name="fra1">{{cite journal| doi=10.1103/PhysRevA.58.3439| first=M. |last=Frasca| s2cid=2699775 |title= Duality in Perturbation Theory and the Quantum Adiabatic Approximation|journal=Physical Review A|volume= 58|pages=3439–3442 |year=1998|arxiv = hep-th/9801069 |bibcode = 1998PhRvA..58.3439F| issue=5 }}</ref><ref name="most">{{cite journal| doi= 10.1103/PhysRevA.55.1653| first= A. | last=Mostafazadeh | s2cid= 17059815 |title=Quantum adiabatic approximation and the geometric phase |journal=Physical Review A|volume=55|pages=1653–1664 |year=1997|arxiv = hep-th/9606053 |bibcode = 1997PhRvA..55.1653M| issue= 3 }}</ref><ref name="fra2">{{cite journal| first=Marco |last=Frasca | s2cid=19783654 | title=A strongly perturbed quantum system is a semiclassical system|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences| doi=10.1098/rspa.2007.1879| volume=463|year=2007|arxiv = hep-th/0603182 |bibcode = 2007RSPSA.463.2195F| issue=2085 |pages=2195–2200}}</ref>

== Examples ==
=== Quartic oscillator ===
For a quantum harmonic oscillator with quartic perturbation,
<math display="block">H = -\frac{\hbar^2}{2 m} \frac{\partial^2}{\partial x^2}+\frac{m \omega^2 x^2}{2}+\lambda x^4,</math>
the first-order correction to the ground-state energy is
<math display="block">E_0^{(1)}=\frac{3}{4}\frac{\hbar^2 \lambda}{m^2 \omega^2}.</math>

=== Quantum pendulum ===
For the quantum pendulum with perturbation
<math display="block">V=-\cos \phi,</math>
the first-order correction vanishes,
<math display="block">E_n^{(1)}=0,</math>
while the second-order correction is
<math display="block">E_n^{(2)}=\frac{ m a^2}{\hbar^2 }\frac{1}{4 n^2-1}.</math>

=== Potential energy as a perturbation ===
For weak spatial potentials acting on free-particle states, perturbation theory yields approximate scattered wave functions in one, two, and three dimensions. These results are widely used in scattering theory.<ref>Lifshitz, E. M., & LD and Sykes Landau (JB). (1965). Quantum Mechanics; Non-relativistic Theory. Pergamon Press.</ref>

== Applications ==
Perturbation theory has many applications in quantum physics, including:
* [[Physics:Rabi cycle|Rabi cycle]]
* [[Physics:Fermi's golden rule|Fermi's golden rule]]
* [[Physics:Muon spin spectroscopy|Muon spin spectroscopy]]
* [[Physics:Perturbed angular correlation|Perturbed angular correlation]]

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
{{reflist|3}}

== External links ==
* {{cite web|title=L1.1 General problem. Non-degenerate perturbation theory|date=14 February 2019|publisher=MIT OpenCourseWare | website=YouTube | url=https://www.youtube.com/watch?v=_OZXEb8FxZQ |archive-url=https://ghostarchive.org/varchive/youtube/20211212/_OZXEb8FxZQ| archive-date=2021-12-12 |url-status=live}} (lecture by [[Biography:Barton Zwiebach|Barton Zwiebach]])

{{Author|Harold Foppele}}

{{Sourceattribution|Perturbation theory (quantum mechanics)|1}}

Physics:Quantum Perturbation theory - Revision history

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