Calculate the energy, corrected to first order, of a harmonic oscillator with potential 1 V(x) =kx + ox + ox°

icon
Related questions
Question
Calculate the energy, corrected to first order, of a harmonic oscillator with potential:
1
V(x) =- kx + @x* + w°x°
4
6
Transcribed Image Text:Calculate the energy, corrected to first order, of a harmonic oscillator with potential: 1 V(x) =- kx + @x* + w°x° 4 6
Expert Solution
Step 1

Given,

Vx=12kx2+ωx4+ω2x6The 1st excited state wave function for harmonic oscillator potential is given byψx=mωπh1/42mωhxe-mω2hx2The paturbation in Hamiltonian is given byH'=ωx4+ω2x6 , over harmonic potentialNow the 1st order correction isψ1H'ψ1=ωx4ψ12dx+ω2x6ψ12dx=ωmωπh1/42mωhe-mω2hx2x6dx+ω2mωπh1/42mωhe-mω2hx2x8dxTherefore the correction termψ1H'ψ1=ωmωπh1/42mωhx6 e-mω2hx2dx. ω2mωπh1/42mωhx8 e-mω2hx2dx=ω3mωπh1/22mωh12Γ1+622mωh7/2×12Γ1+822mωh9/2=ω3mωh3/21π×122mωh-8×52×32×π2×72×52×32×π2=π×0.024×ω3mωh-132

steps

Step by step

Solved in 2 steps

Blurred answer
Similar questions
  • SEE MORE QUESTIONS