Energy levels in quantum well structures

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SUMMARY

The discussion focuses on solving the eigenvalue equations for energy levels in quantum well structures using MATLAB. The specific equations provided are for even modes (n=2,4,6...) and odd modes (n=1,3,5...), involving parameters such as potential well depth (V), effective mass (m), and reduced Planck's constant (hbar). A suggested method for numerical solution is to reformulate the equations into a zero-finding problem using MATLAB's fzero command. This approach allows for the calculation of energy levels (E) within the defined quantum well structure.

PREREQUISITES
  • Understanding of quantum mechanics, specifically quantum well structures
  • Familiarity with eigenvalue equations and their significance in quantum physics
  • Proficiency in MATLAB programming, particularly with numerical methods
  • Knowledge of physical constants such as Planck's constant (hbar) and effective mass (m)
NEXT STEPS
  • Research MATLAB's fzero function for root-finding in numerical analysis
  • Explore quantum mechanics textbooks for deeper insights into potential wells and energy quantization
  • Learn about numerical methods for solving differential equations in quantum mechanics
  • Investigate existing MATLAB codes or libraries specifically designed for quantum well calculations
USEFUL FOR

Physicists, engineers, and researchers working in quantum mechanics, particularly those involved in semiconductor physics and nanotechnology, will benefit from this discussion.

starla
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Does anyone knows how to solve the following eigenvalue equation using matlab? so as to obtain the energy levels in a quantum well structure.. or does anyone has the codes or knows where to find the codes to do so?

Eigenvalue equation:

tan(sq_root(2mE/hbar^2)*(d/2)) = sq_root((V-E)/E) for even mode (n=2,4,6...)

cot(sq_root(2mE/hbar^2)*(d/2)) = -sq_root((V-E)/E) for odd mode (n=1,3,5...)

where V is the depth of the potential well and m is the effective mass. The eignvalue equation shown above can be numerically solved to yield the energy levels E in a potential well.
 
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starla said:
Does anyone knows how to solve the following eigenvalue equation using matlab? so as to obtain the energy levels in a quantum well structure.. or does anyone has the codes or knows where to find the codes to do so?

Eigenvalue equation:

tan(sq_root(2mE/hbar^2)*(d/2)) = sq_root((V-E)/E) for even mode (n=2,4,6...)

cot(sq_root(2mE/hbar^2)*(d/2)) = -sq_root((V-E)/E) for odd mode (n=1,3,5...)

where V is the depth of the potential well and m is the effective mass. The eignvalue equation shown above can be numerically solved to yield the energy levels E in a potential well.
Can't you just write, e.g.,

0 = tan(sq_root(2mE/hbar^2)*(d/2)) - sq_root((V-E)/E),

and use the fzero command.
 

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