Average value of position for the anharmonic oscillator: Classical versus quantum results

The evaluation of the average value of the position coordinate, (x), of a particle moving in a harmonic oscillator potential (V(x) = kx²/2) with a small anharmonic piece (V′(x)= - λkx³) is a standard calculation in classical Newtonian mechanics and statistical mechanics where the problem has relevance to thermal expansion. In each case, the calculation is most easily done using a perturbative expansion. In this note, we perform the same computation of (x) in quantum mechanics using time-independent perturbation theory and the ladder operator formalism to show how similar results are obtained. We also indicate how a semiclassical calculation using a classical probability distribution can also be used to obtain the same result.