Quantum harmonic oscillator tunneling puzzle

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SUMMARY

The forum discussion centers on the quantum harmonic oscillator's tunneling probabilities, specifically the surprising behavior of these probabilities as the quantum number n increases. The classical oscillator's amplitude A relates to energy E through the formula A² = 2E, while the quantum oscillator's energy levels are quantized as 2E(n) = 2n + 1. Calculations using Wolfram's Mathematica reveal that the tunneling probability decreases slowly with increasing n, contradicting expectations from classical mechanics. The asymptotic behavior of the tunneling probability is approximated as c n^(-1/3), where c is a constant derived from the analysis of the wave function.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly the quantum harmonic oscillator.
  • Familiarity with Mathematica for symbolic calculations.
  • Knowledge of Hermite polynomials and their applications in quantum mechanics.
  • Basic grasp of probability density functions and normalization in quantum systems.
NEXT STEPS
  • Explore the derivation of tunneling probabilities in quantum mechanics using Hermite polynomials.
  • Learn about the Airy function and its applications in quantum tunneling problems.
  • Investigate the implications of the correspondence principle in quantum mechanics.
  • Study the numerical methods for calculating quantum probabilities and their convergence properties.
USEFUL FOR

Physicists, quantum mechanics students, and researchers interested in quantum tunneling phenomena and the behavior of quantum harmonic oscillators.

  • #31
Avodyne: Please check the new file .
The paper is at the final stages. there is still an "anonymous helper" in the acknowledgments.
 
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  • #33
So would I be right in thinking that this is no longer a puzzle but just a slight surprise? :biggrin:

Actually I found the result a little surprising myself. I think it was because the probability density is a function of both x and n. I naturally expected an exponential dependence on x and I guess this spilled over to expecting a steep dependence on n as well. But as soon as I realized this, I started wondering whether there was any intuitive reason to expect it to fall off with n at all - it would not, AFAIK, go against any fundamental principle if the probability of finding a system outside of its classical range asymptotically approached some constant.
 
  • #34
The old problem of large quantum numbers and the correspondence principle is still being discussed. Cabrera and Kiwi, Large quantum-number states and the correspondence principle, Phys. Rev. A 36, 2995(R) September 1987 show how it can be violated for the harmonic oscillator.
 

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