How Do You Calculate Electron Tunneling Probability Through a Barrier?

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SUMMARY

The calculation of electron tunneling probability through a barrier involves determining the transmission probability (T) using the equations T = (1 + (V^2 (Sinh[k*L]^2))/(4*R (V - R)))^-1 and k = Sqrt[2*m (V - E)]/h. In this case, electrons with an energy of 0.201 eV encounter a barrier of 2.386 eV height and 0.383 nm width. The key to solving the problem was converting the energy values from electronvolts to joules by multiplying by the charge of an electron (1.6 x 10^-19). This adjustment corrected the calculations performed in Mathematica, leading to accurate results.

PREREQUISITES
  • Understanding of quantum mechanics principles, specifically electron tunneling.
  • Familiarity with the concepts of potential barriers and energy levels.
  • Proficiency in using Mathematica for computational physics problems.
  • Knowledge of unit conversions, particularly between electronvolts and joules.
NEXT STEPS
  • Study the derivation of the transmission probability equation T for quantum tunneling.
  • Learn about the implications of potential barriers in quantum mechanics.
  • Explore advanced Mathematica functions for solving quantum physics problems.
  • Investigate the relationship between energy levels and tunneling probability in different materials.
USEFUL FOR

Students and researchers in quantum mechanics, physicists working on tunneling phenomena, and anyone involved in computational physics using Mathematica.

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Homework Statement



Electrons with energies of 0.201 eV are incident on a barrier 2.386 eV high and 0.383 nm wide. Find the probability for these electrons to penetrate the barrier.

Homework Equations


Note: h = h-bar

k=Sqrt[2*m (V - E)]/h

T = (1 + (V^2 (Sinh[k*L]^2))/(4*R (V - R)))^-1

where
T= transmission probability
V= potential of the barrier
E= energy of the electrons
m- mass of electrons = 9.11*10^-31
h= (4.14*10^-15)/(2*pi)
L=.383*10^-9 m

The Attempt at a Solution



I've been using the equations for k and T found both in my textbook and on wikipedia and plugging them into mathematica, but they're both giving me wrong answers. Can anyone see what's wrong?
 
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Figured it out! I had to multiply each energy by e to make it a potential. So 2.386 eV becomes (2.386*(1.6*10^-19)) V and same for the other
 

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