Quantum Spectrum of a Hamiltonian with Linear Potential

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SUMMARY

The discussion centers on the quantum spectrum of a Hamiltonian with a linear potential, specifically represented as z x, where x is the variable and z is a parameter. When a particle is confined in a 1-D infinite potential well, the linearly-varying potential leads to Airy function solutions, resulting in eigenvalues that are inherently numerical. This was illustrated through a Mathematica simulation involving a stationary electric field perturbing an electron in a 1-D box. For finite potential well walls, the numerical solution becomes more complex.

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  • Understanding of Hamiltonian mechanics
  • Familiarity with Airy functions
  • Knowledge of quantum mechanics principles, particularly potential wells
  • Experience with Mathematica for numerical simulations
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  • Study the derivation of Airy functions in quantum mechanics
  • Explore numerical methods for solving Hamiltonians with linear potentials
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Suppose to have a hamiltonian with a linear potenzial like z x, where x is the variable and z a parameter. Which is the spectrum of the Hamiltonian of this sistem?
 
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Depends... if the particle is to be bounded in a 1-D infinite potential well then a linearly-varying potential gives rise to Airy function solutions and hence the eigenvalues are inherently numerical (I remember this distinctly from a recent Mathematica simulation of a stationary electric field perturbing an electron confined to a 1-D box). You can find a derivation in David Miller's Quantum book. If the walls are finite then the numerical solution is more difficult to find.
 

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