[Quantum Mechanics] Anharmonic oscillator

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SUMMARY

The discussion focuses on computing the expectation value for an anharmonic oscillator in quantum mechanics. The user suggests starting with the position operator ξ expressed as ξ = (a+ + a-) * sqrt(mw/2h), where a+ and a- are the ladder operators. The solution involves manipulating the ladder operators acting on the wave function φ to derive the expectation value. This approach simplifies the computation of the expectation value in the context of quantum harmonic oscillators.

PREREQUISITES
  • Understanding of quantum mechanics principles, specifically harmonic oscillators.
  • Familiarity with ladder operators (a+ and a-).
  • Knowledge of expectation values in quantum mechanics.
  • Basic grasp of operator algebra in quantum systems.
NEXT STEPS
  • Study the derivation of expectation values in quantum mechanics.
  • Learn about the properties and applications of ladder operators in quantum systems.
  • Explore the mathematical formulation of anharmonic oscillators.
  • Investigate the implications of the position operator in quantum mechanics.
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Students and researchers in quantum mechanics, physicists working on quantum systems, and anyone interested in advanced topics related to anharmonic oscillators.

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how to compute <n|ξ^4|n>?

The problem is above
I guess the ladder operator becomes some very ugly term. There should be a trick to compute <n|ξ^4|n>. Could anyone tell me?
 
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I found a solution : )

just start with ξ = (a+ + a-) * sqrt(mw/2h)
and keep dealing with (a+)φ and (a-)φ.
 

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