Quantum Tunneling: Estimating Probability

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SUMMARY

The forum discussion focuses on estimating the tunneling probability of a 10eV electron encountering a potential barrier of 25eV and 1nm width. The relevant equation used is T=16(E/V)(1-(E/V))exp(-2α), where α is calculated using α=√(2m(V-E))/ħ. The user initially calculated α as 3.157x10^9 but received conflicting answers for the tunneling probability, with values of 9.2x10^-9 and 4.95x10^-13. A critical error identified was the use of Planck's constant (h) instead of the reduced Planck's constant (ħ), which is essential for accurate calculations.

PREREQUISITES
  • Understanding of quantum mechanics concepts, specifically quantum tunneling.
  • Familiarity with the Schrödinger equation and potential barriers.
  • Knowledge of Planck's constant and its reduced form (ħ).
  • Basic proficiency in mathematical manipulation of exponential functions.
NEXT STEPS
  • Review the derivation and applications of the quantum tunneling equation T=16(E/V)(1-(E/V))exp(-2α).
  • Learn about the significance of Planck's constant (h) and reduced Planck's constant (ħ) in quantum mechanics.
  • Explore the implications of potential barrier width on tunneling probability.
  • Investigate numerical methods for estimating tunneling probabilities in various quantum systems.
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Students and researchers in quantum mechanics, particularly those studying quantum tunneling phenomena and their applications in fields such as semiconductor physics and quantum computing.

fredrick08
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Homework Statement


A 10eV electron is incident on a potential barrier of height 25eV and width 1nm. Estimate (order of magnitude) the probability that the electron will tunnel through the barrier, repeat your calculations for a barrier of 01.nm

Homework Equations


T=16(E/V)(1-(E/V))exp(-2\alphaa)
\alpha=\sqrt{}(2m(V-E))/\hbar

The Attempt at a Solution


E=10eV=1.6x10^-18J, V=25eV=4x10^-18J

\alpha=\sqrt{}(2*9.1x10^-31(4x10^-18-1.6x10^-18))/6.62x10^-34=3.157x10^9

T=3.84exp(-2*3.157x10^9*1x10^-9)=7x10^-3?

the answer is meant to be 9.2x10^-9 or 4.95x10^-13? sorry, my book says 4.95 and the lecturers notes say 9.2... doesn't matter, coz I am way way off...

please can someone help me where i am going wrong?
 
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sorry about the latex the x is multiply
 
please anyone?
 
Well, the only thing I see is that instead of hbar you used h in your calculations. hbar is a factor of 2pi smaller than h.
 
omg oops, thanks
 
still have wrong answer, anyone know y?
 

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