Domain: Quantum chemistry

We consider a one-dimensional quantum system in which a particle of mass \(\mathrm{m}\) moves under the influence of an anharmonic potential that slightly deviates from the standard harmonic oscillator form. The complete potential is expressed as\(\mathrm{V(x) = \frac{1}{2} m \omega^2 x^2 \left( \frac{x}{a} \right)^{2\lambda}}\),where \(\mathrm{a}\) denotes a characteristic length-scale parameter and \(\mathrm{\lambda \ll 1}\) is a small, dimensionless perturbation parameter.The potential can be decomposed into two parts as\(\mathrm{V(x) = \frac{1}{2} m \omega^2 x^2 + \Delta V}\)with\(\mathrm{\Delta V(x) = \frac{1}{2} m \omega^2 x^2 \left[ \left( \frac{x}{a} \right)^{2\lambda} - 1 \right]}\).Treat \(\mathrm{\Delta V}\) as a small perturbation, determine the first-order energy correction to the ground-state energy of the system.

Grading rubric:
criterion: Correctly writes the total potential as an unperturbed harmonic term plus a perturbation V(x)=1/2 m ω^2 x^2 + ΔV(x).
weight: 10
criterion: Correctly constructs the full expectation-value integral with prefactor (mω^2/2)√(mω/πħ) and integrand e^{−mωx^2/ħ} x^2[(x/a)^{2λ}−1]. 
weight: 10
criterion: Correctly performs the scaling transformation y=√(mω/ħ) x and factors out all dimensional constants. 
weight: 10
criterion: Correctly defines the reduced integral J(λ)=∫_{−∞}^{∞} e^{−y^2}(y^2)^{1+λ} dy. 
weight: 20
criterion: Correctly evaluates the reduced integral exactly as J(λ)=Γ(λ+3/2). 
weight: 20
criterion: Correctly expands Γ(λ+3/2) to first order in λ as Γ(3/2)[1+λψ(3/2)]. 
weight: 20
criterion: Correctly derives and reports the final first-order correction E_0^(1)=λ(ħω/2)[1−γ/2+1/2 ln(ħ/(4mωa^2))].
weight: 10

Answer:
E0^(1) = λ(ℏω/2)[1 - γ/2 + (1/2) ln(ℏ/(4mωa^2))]