# Perturbation theory

I am puzzled with perturbation theory when studying quantum mechanics and solid theory.

What I learn about perturbation is, from my ignorant point of view, just mathematics, or even simpler, matrix theory, regardless of being conventional perturbation theory, degenerate pertubation, or quasi-degenerate pertubation.

The point is that I can only understand it as a math tool, rather than any physics-related concept. Does perturbation theory inherently build in some concepts in quantum physics?

Probably its hard to be complete on this topic. Can someone make sort of "broad brush stroke"?

• What exactly are you asking for? Physical intuition for the various forms of perturbation theory? – Michael Aug 27 '13 at 12:16
• In some sense, yes. I wonder if someone understand the pertubation theory as a math tool rather than a physical tool. – Lorniper Aug 27 '13 at 12:53
• Perturbation theory is not just for quantum mechanics. It applies to classical mechanics as well. – DanielSank Aug 1 '18 at 17:09

## 5 Answers

As far as it is a way of solving mathematical equations, it is math.

The physical content is in the physical meaning of the exact equations and (if it is catched well) in the initial approximation, which roughly describes the solutions. Perturbative corrections modify (slightly) the numerical values of the initial approximation.

I'd say perturbation theory is a generalisation of several mathematical tools you find in different fields -- e.g. Taylor series, momentary expansions of statistical variables, Fourier series and in general any convergent infinite sum. With the specific case of kernel perturbations, you can consider each perturbation to the kernel to be a result of a scattering process that affects the Lagrangian in some way.

• I just wonder if anyone interprets the perturbation theory as a math tool rather than a physical tool, as it seems to me. – Lorniper Aug 27 '13 at 14:24
• nevertheless, it looks to me only as linear operators. – Lorniper Aug 27 '13 at 14:49
• It's an approximation method. So in some sense, yes it's just a math tool: you come across a set of physically motivated equations that you want to solve, but there is no known analytical solution with a finite number of terms. Luckily, perturbation theory is there so that you can get an approximate solution by taking a solution with infinitely many terms and evaluating as many as you care to to get an estimate of the solution. – Kyle Oman Aug 27 '13 at 15:05
• Yes, of course. But even if I forget about physics I could come up with a perturbative solution to a set of equations... that's why I see it more as a math tool being applied to a physical problem. – Kyle Oman Aug 27 '13 at 15:10
• @Kyle You should write an answer so that people can upvote it. – joshphysics Aug 27 '13 at 22:24

By your definition, most of physics is "just mathematics." Classical mechanics is "just a certain branch of dynamical systems." Electromagnetism is "just PDEs involving tensors." Statistical mechanics is "just the statistical behavior of dynamical systems." Quantum mechanics is "just a description of a C*-algebra."

The distinction comes when you apply the mathematical tools to describe a physical phenomenon. When you apply perturbation theory to a physical situation, it is no longer "just mathematics."

We can imagine that the complicated array of moving things which constitute "the World" is something like a chess game being played by the Gods, and we are observers of the game. We do not know what the rules of the game are; all we are allowed to do is to watch the playing. Of course if we watch long enough, we may catch on to a few of the rules.The rules of the game are what we mean by fundamental physics. Even if we know every rule, however...what we really can explain in terms of these rules are very limited, because almost all situations are so enormously complicated that we can not flow the playsof the game using the rules, much less tell what is going to happen next. We must , therefore limit ourselves to the more basic questions of the rules. If we know the rules, we consider that we 'understand' the world. -Richard Feynman, The making of the Atomic Bomb (1980)

For most problems in Quantum Mechanics, it is extremely difficult to obtain exact solutions of Schrodinger equations and one has to resort to approximate methods. The three most important ones are

1. The perturbation method
2. The variational method
3. The JWKB approximation

The approximations must be using mathematical tools and one of the three methods can be chosen for the specific range of complexities.

There the physical picture of the problem comes in. For example, Variational methods can give good results for ground state and for an excited state the perturbation can be a better substitute and for smoothly varying potentials the JWKB gives a good result.

What I wish to underline is that the physical picture of the problem, the nature of interactions and the choice of perturbing potentials, do give a physical insight, of course with limitations, and it is not the 'mathematical tool's play' only.

The Quantum Mechanics as such are sometimes seen by students as "Mathematical Physics" but they forget that solutions of those partial differential equations can be a large set but only a few limited by the physical boundary conditions are to be taken in as 'real solutions'.

Everything can be regarded as a mathematical tool, after all. Physics just state the problem and interpret the results, but the developement is usually pure mathematics. This is like this for any problem in any physics branch.

However, you sometimes need to include some information in the middle of developement which is only physical, not mathematical. In quantum perturbations, for example, the fact that a global phase factor is irrelevant is a physical particularity. A mathematician could not say that. Or the fact than you can compare energies and say if first order is enough. As I see it, this kind of extra information needed makes it "physics" and not just "math".