The anharmonic oscillator is the quantum system that is a modification to the harmonic oscillator, including a term quartic in position:

\[\begin{equation*}\hat{H}=\frac{\hat{p}^{2}}{2 m}+\frac{m \omega^{2}}{2} \hat{x}^{2}+\lambda \hat{x}^{4} \tag{10.128}\end{equation*}\]

where \(\lambda>0\) is a parameter that controls the strength of the quartic term. A plot of the anharmonic oscillator potential is shown in Fig. 10.7. In this problem, we will use perturbation theory for estimating the effect of this quartic term on the energy eigenvalues of the oscillator.

(a) Determine the first-order in \(\lambda\) correction to the energy eigenvalue of the \(n\) th-energy eigenstate of the harmonic oscillator using quantum mechanics perturbation theory.

It might help to express the quartic factor with the raising and lowering operators \(\hat{a}^{\dagger}\) and \(\hat{a}\).

(b) Now, let's briefly consider what happens if \(\lambda

(c) Back with \(\lambda>0\), determine the first correction in \(\lambda\) to the ground-state wavefunction of the harmonic oscillator.

(d) Calculate the variance \(\sigma_{x}^{2}\) of the ground state of the anharmonic oscillator using the perturbative result of part (c). Does the quartic term in the potential tend to increase or decrease the width of the ground-state position space wavefunction?

Transcribed Image Text:

V(x) x Fig. 10.7 An illustration of the anharmonic oscillator potential (solid), compared to the harmonic oscillator potential (dashed).