Perturbation theory

Perturbation theory is a method for the approximate solution of problems in theoretical physics , applicable when a small parameter is present in the problem, and if the parameter is neglected, the problem has an exact solution.

Physical quantities calculated by perturbation theory have the form of a series

A=A^{(0)}+\varepsilon A^{(1)}+\varepsilon ^{2}A^{(2)}+...

{\ displaystyle A = A ^ {(0)} + \ varepsilon A ^ {(1)} + \ varepsilon ^ {2} A ^ {(2)} + ...}

A = A ^ {{((0)}} + \ varepsilon A ^ {{((1)}} + \ varepsilon ^ {2} A ^ {{(2)}} + ...

Where $A^{(0)}$ ${\ displaystyle A ^ {(0)}}$ $A ^ {{(0)}}$ - solving an unperturbed problem, $\varepsilon$ ${\ displaystyle \ varepsilon}$ $\ varepsilon$ Is a small parameter. Odds $A^{(n)}$ ${\ displaystyle A ^ {(n)}}$ $A ^ {{(n)}}$ are found by successive approximations, i.e. $A^{(n)}$ ${\ displaystyle A ^ {(n)}}$ $A ^ {{(n)}}$ expressed through $A^{(0)},...,A^{(n-1)}$ ${\ displaystyle A ^ {(0)}, ..., A ^ {(n-1)}}$ $A ^ {{((0)}}, ..., A ^ {{(n-1)}}$ . It is used in celestial mechanics , quantum mechanics , quantum field theory , etc.

Content

In Celestial Mechanics

Historically, the first discipline in which the perturbation theory was developed was celestial mechanics. The task of finding the motion of the planets of the solar system is the task $N$ ${\ displaystyle N}$ bodies, which, unlike the two-body problem , does not have an exact analytical solution. Its solution, however, is facilitated by the fact that, due to the small mass of the planets, the attraction of the planets to each other is much weaker than their attraction by the Sun. Neglecting the masses of planets, the task boils down to $N-1$ ${\ displaystyle N-1}$ independent tasks of two bodies that are solved exactly; each planet moves in the gravitational field of the Sun in an elliptical orbit according to Kepler’s laws . This is a solution to the unperturbed problem , or zero approximation . Forces from other planets lead to distortion, or disturbance of these elliptical orbits. To calculate the trajectory of the planet, taking into account the perturbation, the following method is used.

The position of the planet in space and its speed can be set using six quantities (according to the number of degrees of freedom ): the semimajor axis and the eccentricity of the orbit, the inclination of its orbit to the ecliptic plane, the longitude of the ascending node , the longitude of the perihelion and the moment it passes through the perihelion. These quantities (we denote them for simplicity $a_{i}$ ${\ displaystyle a_ {i}}$ ) compares favorably with Cartesian coordinates and velocity components in that for an unperturbed motion they are constant:

a_{i}(t)=a_{i}^{(0)}={\rm {const}},

{\ displaystyle a_ {i} (t) = a_ {i} ^ {(0)} = {\ rm {const}},}

therefore, the equations of motion of the planet written through them contain a small parameter on the right side:

{\frac {da_{i}}{dt}}=\varepsilon f_{i}(a_{1},a_{2},...a_{6},t)\qquad \qquad (*)

{\ displaystyle {\ frac {da_ {i}} {dt}} = \ varepsilon f_ {i} (a_ {1}, a_ {2}, ... a_ {6}, t) \ qquad \ qquad (* )}

In view of this, it is convenient to solve the equations of motion by the method of successive approximations. In a first approximation, we substitute the solutions of the unperturbed equation to the right side, and we find:

a_{i}(t)=a_{i}^{(0)}+\varepsilon a_{i}^{(1)}(t)=a_{i}^{(0)}+\varepsilon \int _{0}^{t}f_{i}(a_{i}^{(0)},\tau )d\tau .

{\ displaystyle a_ {i} (t) = a_ {i} ^ {(0)} + \ varepsilon a_ {i} ^ {(1)} (t) = a_ {i} ^ {(0)} + \ varepsilon \ int _ {0} ^ {t} f_ {i} (a_ {i} ^ {(0)}, \ tau) d \ tau.}

To find the second approximation, we substitute the found solution in the right-hand side (*) and solve the resulting equations, etc.

In quantum mechanics

Perturbation theory in quantum mechanics is used when the system Hamiltonian can be represented as

H=H^{(0)}+V

{\ displaystyle H = H ^ {(0)} + V}

Where $H^{(0)}$ ${\ displaystyle H ^ {(0)}}$ Is the unperturbed Hamiltonian (moreover, the solution of the corresponding Schrödinger equation is known exactly), and $V$ ${\ displaystyle V}$ - small additive ( indignation ).

Stationary Perturbation Theory

The task is to find the eigenfunctions of the Hamiltonian ( stationary states ) and the corresponding energy levels. We will seek solutions to the Schrödinger equation for our system

H|\psi _{n}\rangle =E_{n}|\psi _{n}\rangle \qquad \qquad (**)

{\ displaystyle H | \ psi _ {n} \ rangle = E_ {n} | \ psi _ {n} \ rangle \ qquad \ qquad (**)}

in the form of expansion in a row

\psi _{n}=\psi _{n}^{(0)}+\psi _{n}^{(1)}+\psi _{n}^{(2)}+...

{\ displaystyle \ psi _ {n} = \ psi _ {n} ^ {(0)} + \ psi _ {n} ^ {(1)} + \ psi _ {n} ^ {(2)} +. ..}

E_{n}=E_{n}^{(0)}+E_{n}^{(1)}+E_{n}^{(2)}+...\qquad \qquad (***)

{\ displaystyle E_ {n} = E_ {n} ^ {(0)} + E_ {n} ^ {(1)} + E_ {n} ^ {(2)} + ... \ qquad \ qquad (* **)}

Where $\psi _{n}^{(0)}$ ${\ displaystyle \ psi _ {n} ^ {(0)}}$ and $E_{n}^{(0)}$ ${\ displaystyle E_ {n} ^ {(0)}}$ - wave functions and energy levels of the unperturbed problem

H^{(0)}|\psi _{n}^{(0)}\rangle =E_{n}^{(0)}|\psi _{n}^{(0)}\rangle ,

{\ displaystyle H ^ {(0)} | \ psi _ {n} ^ {(0)} \ rangle = E_ {n} ^ {(0)} | \ psi _ {n} ^ {(0)} \ rangle,}

and the number $n$ ${\ displaystyle n}$ numbers energy levels.

Substituting (***) in (**), up to first-order terms in perturbation, we obtain

(V-E_{n}^{(1)})|\psi _{n}^{(0)}\rangle =(E_{n}^{(0)}-H^{(0)})|\psi _{n}^{(1)}\rangle

{\ displaystyle (V-E_ {n} ^ {(1)}) | \ psi _ {n} ^ {(0)} \ rangle = (E_ {n} ^ {(0)} - H ^ {(0 )}) | \ psi _ {n} ^ {(1)} \ rangle}

Multiplying on the left by $\psi _{m}^{(0)}$ ${\ displaystyle \ psi _ {m} ^ {(0)}}$ , and given that $\psi _{m}^{(0)}$ ${\ displaystyle \ psi _ {m} ^ {(0)}}$ - ( orthonormalized ) eigenfunctions of the unperturbed Hamiltonian, we obtain

E_{n}^{(1)}=V_{nn}

{\ displaystyle E_ {n} ^ {(1)} = V_ {nn}}

\psi _{n}^{(1)}=\sum _{m\neq n}{\frac {V_{mn}}{E_{n}^{(0)}-E_{m}^{(0)}}}\psi _{m}^{(0)},

{\ displaystyle \ psi _ {n} ^ {(1)} = \ sum _ {m \ neq n} {\ frac {V_ {mn}} {E_ {n} ^ {(0)} - E_ {m} ^ {(0)}}} \ psi _ {m} ^ {(0)},}

Where $V_{mn}\equiv \langle \psi _{m}^{(0)}|V|\psi _{n}^{(0)}\rangle$ ${\ displaystyle V_ {mn} \ equiv \ langle \ psi _ {m} ^ {(0)} | V | \ psi _ {n} ^ {(0)} \ rangle}$ - matrix elements of perturbation.

The above procedure works if the undisturbed level $E_{n}^{(0)}$ ${\ displaystyle E_ {n} ^ {(0)}}$ non-degenerate . Otherwise, to find first-order corrections, it is necessary to solve the secular equation .

The corrections of the following orders are found in a similar way, although the formulas are very complicated.

Unsteady Perturbation Theory

In quantum field theory

Most of the calculations in quantum field theory, in particular in quantum electrodynamics (QED), are also done in the framework of perturbation theory. The unperturbed solution is free fields , and the interaction constant is a small parameter (in electrodynamics, the fine structure constant $\alpha =1/137$ ${\ displaystyle \ alpha = 1/137}$ ) To represent the members of a number of perturbation theory in visual form, Feynman diagrams are used.

Nowadays, many calculations in QED are not limited to the first or second order of perturbation theory. So, the anomalous magnetic moment of an electron is currently (2015) calculated up to 5th order from $\alpha$ ${\ displaystyle \ alpha}$ ^[1] .

Nevertheless, there is a theorem that a series of perturbation theory in QED is not convergent, but only asymptotic . This means that, starting from a certain (in practice, very large) order of the perturbation theory, the agreement between the approximate and exact solution will no longer improve but worsen ^[2] .

Examples of the inapplicability of perturbation theory

Despite its apparent universality, the perturbation theory method does not work in a certain class of problems. Examples are instanton effects in a number of problems of quantum mechanics and quantum field theory. Instanton contributions have significant features at the decomposition point. A typical example of an instanton contribution is:

T_{inst}=A\exp(-1/g)

{\ displaystyle T_ {inst} = A \ exp (-1 / g)}

where

g

{\ displaystyle g}

Is a small parameter.

This function is non-analytic at the point $g=0$ ${\ displaystyle g = 0}$ , and therefore can not be laid out in a series of Maclaurin $g$ ${\ displaystyle g}$ .

Notes

↑ E. de Rafael. Update of the Electron and Muon g-Factors // arXiv: 1210.4705 [hep-ph]
↑ Akhiezer A. I., Berestetsky V. B. Quantum electrodynamics. - M .: Nauka, 1981. - S. 210-212.

Literature

Physical Encyclopedia / A.M. Prokhorov (Ch. Ed.). - M .: Great Russian Encyclopedia, 1988-99.
Landau, L.D. , Lifshits, E.M. Quantum mechanics (nonrelativistic theory). - 4th edition. - M .: Nauka , 1989 .-- 768 p. - (“ Theoretical Physics ”, Volume III). - ISBN 5-02-014421-5 .
Messiah A. Quantum Mechanics: Trans. with fr. - Vol. 2, 1979. - 584 p.
J. Zinn-Justin and UD Jentschura. Multi-Instantons and Exact Results I: Conjectures, WKB Expansions, and Instanton Interactions // Ann. Phys. - 2004. - Vol. 313. - P. 197-267.
J. Zinn-Justin and UD Jentschura. Multi-Instantons and Exact Results II: Specific Cases, Higher-Order Effects, and Numerical Calculations // Ann. Phys. - 2004. - Vol. 313. - P. 269-325.
Dzhakalya G. E. O. Methods of perturbation theory for nonlinear systems. - M., Nauka, 1979. - 320 p.

[1] E. de Rafael. Update of the Electron and Muon g-Factors // arXiv: 1210.4705 [hep-ph]

[2] Akhiezer A. I., Berestetsky V. B. Quantum electrodynamics. - M .: Nauka, 1981. - S. 210-212.