Numerical calculation of tunneling probability

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SUMMARY

This discussion focuses on the numerical calculation of tunneling probability for a wave function |psi> in a double well potential. The user seeks guidance on determining the tunneling probability, specifically how to quantify the likelihood of a particle transitioning from the left well to the right well despite insufficient energy. The key concept involves integrating the square of the wave function over the classically forbidden region to obtain the tunneling probability. The user acknowledges their limited experience in Quantum Mechanics, having completed only two courses.

PREREQUISITES
  • Understanding of Quantum Mechanics principles, particularly wave functions and potential wells.
  • Familiarity with numerical integration techniques for calculating probabilities.
  • Knowledge of tunneling phenomena in quantum systems.
  • Experience with computational tools for simulating quantum systems, such as Python or MATLAB.
NEXT STEPS
  • Research numerical methods for integrating wave functions, focusing on techniques like the trapezoidal rule or Simpson's rule.
  • Study the concept of tunneling in quantum mechanics, specifically the mathematical formulation of tunneling probability.
  • Explore software tools for simulating quantum mechanics, such as QuTiP (Quantum Toolbox in Python).
  • Learn about the time evolution of wave functions and how to calculate expectation values in quantum systems.
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Students and researchers in Quantum Mechanics, physicists interested in quantum tunneling phenomena, and anyone involved in computational simulations of quantum systems.

chandanrupa
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Hi, I am trying to numerically calculate the tunneling probability for a wave function |psi> as a function of x. I have a double well potential. The wave function is initially in the left well. I do not know what exactly I have to find to show tunneling: the probability of being on the left well or the average value of x to see how it varies with time or something else. I need someone's help about this.
 
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it is my understanding that tunneling is basically if a particle gets to a classically forbidden region... i.e. jumping over a potential barrier that it doesn't have enough energy to get through. The probability of tunneling would then be the integral of the wave function squared over the region that it should not be in. So to find the tunneling probability to get to the right part of the well, you are trying to find the probability that the particle is on the right, somewhere. (that's of course given that it doesn't have enough energy to get there without tunneling)
that's my understanding, but I only took 2 courses in Quantum Mechanics, so I'm still very much a newbie :)
 

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