Show that the two wave functions are eigenfunction

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SUMMARY

The discussion focuses on demonstrating that the wave functions ψ0(x)=e^(-x²/2) and ψ1(x)=xe^(-x²/2) are eigenfunctions of the dimensionless harmonic oscillator Hamiltonian H=½ P²+½ X², with eigenvalues ½ and 3/2, respectively. Additionally, it addresses finding the coefficient 'a' for the wave function ψ2(x)=(1+ax²)e^(-x²/2) to ensure orthogonality with ψ0(x) and confirms that ψ2(x) is an eigenfunction of H with an eigenvalue of 5/2. The solution requires applying definitions of orthogonality and eigenfunctions, rather than merely taking products or derivatives.

PREREQUISITES
  • Understanding of quantum mechanics principles, specifically harmonic oscillators.
  • Familiarity with eigenfunctions and eigenvalues in the context of differential equations.
  • Knowledge of wave function orthogonality and normalization techniques.
  • Proficiency in calculus, particularly in taking derivatives and evaluating integrals.
NEXT STEPS
  • Study the properties of eigenfunctions and eigenvalues in quantum mechanics.
  • Learn about the mathematical derivation of the harmonic oscillator Hamiltonian.
  • Explore techniques for ensuring orthogonality in quantum wave functions.
  • Investigate the implications of the quantum harmonic oscillator in physical systems.
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Students and professionals in quantum mechanics, particularly those studying wave functions and harmonic oscillators, as well as educators seeking to enhance their understanding of eigenvalue problems in physics.

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Homework Statement


Consider the dimensionless harmonic oscillator Hamiltonian
H=½ P2+½ X2, P=-i d/dx.
  1. Show that the two wave functions ψ0(x)=e-x2/2 and ψ1(x)=xe-x2/2 are eigenfunction of H with eigenvalues ½ and 3/2, respectively.
  2. Find the value of the coefficient a such that ψ2(x)=(1+ax2)e-x2/2 is orthogonal to ψ0(x). Then show that ψ2(x) is an eigenfunction of H with eigenvalue 5/2.

The Attempt at a Solution


For orthogonality the wave function product must equal to zero, and for eigenfunction we take the second derivative for both wave functions and make a comparison between the eigenvalues.
But I can't finalise the problem, so I appreciate any help in advance.
 
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Please show working.
Your method is sort of OK.
It's a bit more than just taking the product or the second derivative.
You have to apply the definitions.
 

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