### Abstract

We consider the problem of a particle confined to a one-dimensional power-law potential of the form V^{(k)}(x) = V_{0}|x/a|^{k}, but also subject to a constant force F. This might represent, for example, the application of an external electric field, ε, via the perturbing potential Ṽ(x) = Fx = qεx. The electric polarizability of the system in a given state is defined via the energy shift ΔE_{n} = α_{n}ε^{2}/2 due to the external field. Exact expressions for α_{n} are easily obtained for the two most familiar one-dimensional potentials, namely the harmonic oscillator (k = 2) and the symmetric infinite well (k → ∞). For the harmonic oscillator (infinite well) the polarizability scales as n^{0} (n_{-2}) and is positive (negative) for large values of n indicating a qualitatively different response of the system to an external field for large quantum numbers. In order to examine this problem for a more general power-law (2 < k < ∞) potential, we apply WKB techniques to evaluate the energy shifts for large n and we find that (i) the n-dependence of the polarizability scales as α_{n} ∝ n^{2(2-k)/(2+k)} as a function of k and (ii) the approximate value of k at which the crossover from α_{n} > 0 to α_{n} < 0 (for large n) occurs is k ≈ 6. This study provides a useful example of numerical WKB techniques, the use of scaling ideas in simple one-dimensional quantum mechanical systems, and a better 'feel' for the meaning of the polarizability which is an important physical quantity in more realistic atomic and molecular systems.

Original language | English (US) |
---|---|

Pages (from-to) | 31-39 |

Number of pages | 9 |

Journal | European Journal of Physics |

Volume | 19 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1998 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

### Cite this

}

*European Journal of Physics*, vol. 19, no. 1, pp. 31-39. https://doi.org/10.1088/0143-0807/19/1/006

**The polarizability of a particle in power-law potentials : A WKB analysis.** / Robinett, Richard Wallace.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The polarizability of a particle in power-law potentials

T2 - A WKB analysis

AU - Robinett, Richard Wallace

PY - 1998/1/1

Y1 - 1998/1/1

N2 - We consider the problem of a particle confined to a one-dimensional power-law potential of the form V(k)(x) = V0|x/a|k, but also subject to a constant force F. This might represent, for example, the application of an external electric field, ε, via the perturbing potential Ṽ(x) = Fx = qεx. The electric polarizability of the system in a given state is defined via the energy shift ΔEn = αnε2/2 due to the external field. Exact expressions for αn are easily obtained for the two most familiar one-dimensional potentials, namely the harmonic oscillator (k = 2) and the symmetric infinite well (k → ∞). For the harmonic oscillator (infinite well) the polarizability scales as n0 (n-2) and is positive (negative) for large values of n indicating a qualitatively different response of the system to an external field for large quantum numbers. In order to examine this problem for a more general power-law (2 < k < ∞) potential, we apply WKB techniques to evaluate the energy shifts for large n and we find that (i) the n-dependence of the polarizability scales as αn ∝ n2(2-k)/(2+k) as a function of k and (ii) the approximate value of k at which the crossover from αn > 0 to αn < 0 (for large n) occurs is k ≈ 6. This study provides a useful example of numerical WKB techniques, the use of scaling ideas in simple one-dimensional quantum mechanical systems, and a better 'feel' for the meaning of the polarizability which is an important physical quantity in more realistic atomic and molecular systems.

AB - We consider the problem of a particle confined to a one-dimensional power-law potential of the form V(k)(x) = V0|x/a|k, but also subject to a constant force F. This might represent, for example, the application of an external electric field, ε, via the perturbing potential Ṽ(x) = Fx = qεx. The electric polarizability of the system in a given state is defined via the energy shift ΔEn = αnε2/2 due to the external field. Exact expressions for αn are easily obtained for the two most familiar one-dimensional potentials, namely the harmonic oscillator (k = 2) and the symmetric infinite well (k → ∞). For the harmonic oscillator (infinite well) the polarizability scales as n0 (n-2) and is positive (negative) for large values of n indicating a qualitatively different response of the system to an external field for large quantum numbers. In order to examine this problem for a more general power-law (2 < k < ∞) potential, we apply WKB techniques to evaluate the energy shifts for large n and we find that (i) the n-dependence of the polarizability scales as αn ∝ n2(2-k)/(2+k) as a function of k and (ii) the approximate value of k at which the crossover from αn > 0 to αn < 0 (for large n) occurs is k ≈ 6. This study provides a useful example of numerical WKB techniques, the use of scaling ideas in simple one-dimensional quantum mechanical systems, and a better 'feel' for the meaning of the polarizability which is an important physical quantity in more realistic atomic and molecular systems.

UR - http://www.scopus.com/inward/record.url?scp=3643142853&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3643142853&partnerID=8YFLogxK

U2 - 10.1088/0143-0807/19/1/006

DO - 10.1088/0143-0807/19/1/006

M3 - Article

VL - 19

SP - 31

EP - 39

JO - European Journal of Physics

JF - European Journal of Physics

SN - 0143-0807

IS - 1

ER -