Breakdown of perturbation expansion

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SUMMARY

The discussion focuses on the perturbation expansion of the simple harmonic oscillator with a perturbation term of Lambda*(x)^4. The first-order correction to the n-th eigenstate is shown to be proportional to (1 + 2n + 2n^2). Furthermore, it is established that regardless of the size of Lambda, the perturbation expansion will eventually break down for sufficiently large n due to the increasing influence of higher-order terms, which dominate the behavior of the system.

PREREQUISITES
  • Understanding of quantum mechanics principles, specifically the Schrödinger equation.
  • Familiarity with perturbation theory in quantum mechanics.
  • Knowledge of raising and lowering operators in quantum harmonic oscillators.
  • Basic mathematical skills for handling polynomial expressions and limits.
NEXT STEPS
  • Study the implications of perturbation theory in quantum mechanics.
  • Learn about the mathematical derivation of the Schrödinger equation.
  • Explore the role of raising and lowering operators in quantum mechanics.
  • Investigate the conditions under which perturbation expansions break down.
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Students and researchers in quantum mechanics, particularly those studying perturbation theory and its applications to harmonic oscillators.

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Homework Statement

consider a perturbation to the simple harmonic oscillator problem Lambda* (x)^4
question a) show tht the first order correction to n-th eigenstate is proportional to (1+2n+2n^2)
b) argue that no matter how small lambda is ,the perturbation expansion will break down for some large enough "n."
what is the physical reason?


Homework Equations

relations are scrodinger equation and raising and lowering operators.



The Attempt at a Solution

a)i have worked out.
but i do not know how to proceed for part b.
any help will be highly appreciated. many thanks in advance.:P
 
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You might want to compute the ratio of the correction to the unperturbed energy.
 

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