Abstract
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.
| 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 |
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All Science Journal Classification (ASJC) codes
- Physics and Astronomy(all)
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The polarizability of a particle in power-law potentials : A WKB analysis. / Robinett, Richard Wallace.
In: European Journal of Physics, Vol. 19, No. 1, 01.01.1998, p. 31-39.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.
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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 -