### Abstract

The evaluation of the average value of the position coordinate, (x), of a particle moving in a harmonic oscillator potential (V(x) = kx^{2}/2) with a small anharmonic piece (V′(x)= - λkx^{3}) 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.

Original language | English (US) |
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Pages (from-to) | 190-194 |

Number of pages | 5 |

Journal | American Journal of Physics |

Volume | 65 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1997 |

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### All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

### Cite this

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*American Journal of Physics*, vol. 65, no. 3, pp. 190-194. https://doi.org/10.1119/1.18747

**Average value of position for the anharmonic oscillator : Classical versus quantum results.** / Robinett, Richard Wallace.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Average value of position for the anharmonic oscillator

T2 - Classical versus quantum results

AU - Robinett, Richard Wallace

PY - 1997/1/1

Y1 - 1997/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0031482645&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031482645&partnerID=8YFLogxK

U2 - 10.1119/1.18747

DO - 10.1119/1.18747

M3 - Article

VL - 65

SP - 190

EP - 194

JO - American Journal of Physics

JF - American Journal of Physics

SN - 0002-9505

IS - 3

ER -