Lowest-order correction for the bendulum

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SUMMARY

The discussion focuses on calculating the lowest-order correction to the ground state energy of a pendulum under small angle approximation. The series expansion for cosine is utilized, specifically cos(x) = 1 - x²/2! + x⁴/4! + ..., to derive the correction term. The energy correction is expressed as δE ∝ <Ψ₀|x⁴|Ψ₀>, indicating the dependence on the fourth power of the position operator in the ground state wave function.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly perturbation theory.
  • Familiarity with the mathematical expansion of trigonometric functions.
  • Knowledge of the ground state wave function in quantum systems.
  • Basic concepts of expectation values in quantum mechanics.
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  • Research perturbation theory in quantum mechanics.
  • Study the mathematical derivation of energy corrections in quantum systems.
  • Explore the implications of the small angle approximation in classical and quantum mechanics.
  • Learn about expectation values and their calculations in quantum mechanics.
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Students and professionals in physics, particularly those studying quantum mechanics and perturbation theory, as well as researchers interested in the behavior of pendulum systems under quantum conditions.

eman2009
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what is the lowest-order correction to the ground state for the pendulum with small angle?
 
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cos(x)=1-x^2/2!+x^4/4!-...
\delta E\propto&lt;\Psi_0\mid x^4\mid\Psi_0&gt;
 

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