Why perturbation theory uses power series?

[Phylosopher]

TL;DR Summary

Why do we use power series in perturbation theory? What is its mathematical justification?

I am revising perturbation theory from Griffiths introduction to quantum mechanics.
Griffiths uses power series to represent the perturbation in the system due to small change in the Hamiltonian. But I see no justification for it! Other than the fact that it works.

I searched on the internet a bit, but I saw no digestible answer.

This: Ref1, Ref2 are good to read, but they do not answer why use a power series to begin with.

Interestingly, Ref2 uses perturbation on a simple algebraic equation and claims at the end that "regular" perturbation cannot generate all the possible answers.

 

[hilbert2]

When you add a perturbation to a Hamiltonian $H_0$ to give $H_0 + \lambda H'$, the new energy eigenvalue is assumed to be

$E = E^{(0)} + \lambda E^{(1)} + \lambda^2 E^{(2)} + \dots$

and the new eigenvector

$\left|\psi\right.\rangle = \left|\psi^{(0)}\right.\rangle + \lambda \left|\psi^{(1)}\right.\rangle + \lambda^2 \left|\psi^{(2)}\right.\rangle + \dots$

If $\lambda \ll 1$, the terms of order $\lambda$, $\lambda^2$, $\lambda^3$ and so on are of decreasing significance, and you can include only one or two of them when $\lambda$ is small enough. To me it seems that it's just this division of the effect of perturbation to parts of decreasing significance that makes this effective. And it's usually also much easier to solve the terms $E^{(k)}$ and $\left|\psi^{(k)}\right.\rangle$ than it would be to find the whole solution.

Perturbation theory can also be applied to classical orbital mechanics, for instance changing the gravitational potential energy

$\displaystyle V(r) = -\frac{Gm_1 m_2}{r}$

to a perturbed one

$\displaystyle V(r) = -\frac{Gm_1 m_2}{r} + \lambda \frac{m_1 m_2}{r^2}$.

Another application is in the solving of perturbed algebraic equations like $x^2 + (2+\lambda )x + 1 = 0$ where the case $\lambda =0$ has an expectionally simple solution. In the case of a quantum system with a finite-dimensional Hilbert space, the energy eigenvalues are found as solutions of the characteristic equation

$\det (H-E \mathbb{I}) = 0$

which is also an algebraic equation, so it's not very surprising that the method is valid for both eigensystems and zeros of polynomials.

 

[HomogenousCow]

For most applications of perturbation theory the desired solutions actually have no convergent power series expansions so not only is there no justification in this matter but actually the entire assumption is false. This however is not important in practice since one can obtain asymptotic series which are formally divergent but give useful results if truncated.

 

[atyy]

If the power series is convergent or asymptotic, the error of the approximation can be estimated from the size of the first term omitted. As @hilbert2 said above, in many cases it can be shown that the successive terms are of decreasing significance.

 

[FactChecker]

For most applications of perturbation theory the desired solutions actually have no convergent power series expansions so not only is there no justification in this matter but actually the entire assumption is false. This however is not important in practice since one can obtain asymptotic series which are formally divergent but give useful results if truncated.

This surprises me. If the power series does not converge, then the linearized approximation from it is no good. Why would a conclusion from it any good?

 

[vanhees71]

The rule of thumb is to stop taking into account the terms of order $\geq (n+1)$ if they become larger than the term of order $n$.

 

[FactChecker]

The rule of thumb is to stop taking into account the terms of order $\geq (n+1)$ if they become larger than the term of order $n$.

OK. But in that case, I don't see how there can be any validity in the truncated series. So I am surprised, but I have no experience in this subject.

 

[atyy]

OK. But in that case, I don't see how there can be any validity in the truncated series. So I am surprised, but I have no experience in this subject.

This is what the "asymptotic series" used in several posts above means. Convergent series are asymptotic, but asymptotic series are not necessarily convergent. These series have the property that the error can be estimated by the size of the first term omitted. One of the most famous uses of asymptotic series is to obtain Stirling's approximation: https://en.wikipedia.org/wiki/Stirling's_approximation [see the part that starts "Stirling's formula is in fact the first approximation to the following series (now called the Stirling series )"]

 

[George Jones]

OK. But in that case, I don't see how there can be any validity in the truncated series. So I am surprised, but I have no experience in this subject.

Expanding about (pun intended) @atyy 's and @vanhees71 's comments, consider the following example.

Define

$$ f\left( x \right) = \int_0^\infty \frac{e^{-t}}{1 + \frac{t}{x}} dt $$

This function has an asymptotic expansion series expansion

$$ 1 - x + \frac{2!}{x^2} - \frac{3!}{x^3} + ... $$

which is a divergent series. If the number of terms is fixed, however, the series does a nice job of approximating $f\left(x\right)$ for large enough $x$. For example, take $x=100$. Then,
$$f\left(x\right) = 0.9901942...$$
The sum of the first four terms of the divergent asymptotic series is 0.9901940..., which is very close. It takes several hundred terms to see that the series blows up for $x=100$.

On the other hand, taking the first 20 terms of the convergent Taylor series expansion about $x = 50$ of $f$ gives

$$0.995284$$

when $x=100$, which is not so good.

 

[FactChecker]

This is what the "asymptotic series" used in several posts above means. Convergent series are asymptotic, but asymptotic series are not necessarily convergent. These series have the property that the error can be estimated by the size of the first term omitted. One of the most famous uses of asymptotic series is to obtain Stirling's approximation: https://en.wikipedia.org/wiki/Stirling's_approximation [see the part that starts "Stirling's formula is in fact the first approximation to the following series (now called the Stirling series )"]

Thank you. This is an entirely new idea to me.

 

[hilbert2]

"Borel summability" is another search term to look for if you want to find rigorous treatises of perturbation theory.

 

[Auto-Didact]

For most applications of perturbation theory the desired solutions actually have no convergent power series expansions so not only is there no justification in this matter but actually the entire assumption is false. This however is not important in practice since one can obtain asymptotic series which are formally divergent but give useful results if truncated.

On the other hand, taking the first 20 terms of the convergent Taylor series expansion about $x = 50$ of $f$ gives

$$0.995284$$

when $x=100$, which is not so good.

To add to these two wonderful posts:
If the function in question can be reimagined as a dynamical system, then there automatically arises the possibility to use non-perturbative methods to overcome the inherent limitations of perturbation theory; an example of a familiar and relatively simple non-perturbative method is the WKB method from mathematical physics.

 

[HomogenousCow]

I've always thought that divergent asymptotic series from perturbation theory was one of the most interesting subjects in mathematical physics, it's as if the mathematical machinery is trying its best to produce a power series but simply fails beyond some order. Interesting stuff.

 

[Phylosopher]

Perturbation theory can also be applied to classical orbital mechanics, for instance changing the gravitational potential energy

V(r)=−Gm1m2r\displaystyle V(r) = -\frac{Gm_1 m_2}{r}

to a perturbed one

V(r)=−Gm1m2r+λm1m2r2\displaystyle V(r) = -\frac{Gm_1 m_2}{r} + \lambda \frac{m_1 m_2}{r^2}.

Another application is in the solving of perturbed algebraic equations like x2+(2+λ)x+1=0x^2 + (2+\lambda )x + 1 = 0 where the case λ=0\lambda =0 has an expectionally simple solution.

Ref1 in my original post claims that using "regular" purterbation is not enough to find all the possible solutions for the following equation $x^{2}-1=\epsilon e^{x}$
But it is enough to find the solutions of $x^{2}-1=\epsilon x$