Abstract
We consider the problem of a particle confined to a one-dimensional power-law potential of the form , but also subject to a constant force F. This might represent, for example, the application of an external electric field, , via the perturbing potential . The electric polarizability of the system in a given state is defined via the energy shift due to the external field. Exact expressions for are easily obtained for the two most familiar one-dimensional potentials, namely the harmonic oscillator (k = 2) and the symmetric infinite well . For the harmonic oscillator (infinite well) the polarizability scales as 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 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 as a function of k and (ii) the approximate value of k at which the crossover from to (for large n) occurs is . 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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.