Eigenfunctions corresponding to a particular energy value

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SUMMARY

The discussion focuses on obtaining eigenfunctions of a One-Particle Hamiltonian corresponding to a specific energy value, E, using free particle eigenfunctions, specifically plane waves. It is established that the effectiveness of this approach is contingent upon the Hamiltonian in question, contrasting it with the variational principle where trial wave-functions can yield reasonable energy estimates. The discussion highlights that using plane waves as approximations can lead to significant inaccuracies, particularly noting that they are inadequate for approximating the ground state wave-function of the harmonic oscillator, which is Gaussian in nature.

PREREQUISITES
  • Understanding of One-Particle Hamiltonians in quantum mechanics
  • Familiarity with eigenvalues and eigenfunctions
  • Knowledge of the variational principle in quantum mechanics
  • Basic concepts of wave-functions, particularly Gaussian and sinusoidal forms
NEXT STEPS
  • Study the properties of One-Particle Hamiltonians in quantum mechanics
  • Explore the variational principle and its applications in estimating ground state energies
  • Investigate the differences between Gaussian and sinusoidal wave-functions
  • Learn about the limitations of using plane waves in quantum mechanical approximations
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Quantum physicists, students of quantum mechanics, and researchers interested in the mathematical foundations of wave-functions and Hamiltonians.

hokhani
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Suppose we want to get eigenfunctions of a One-Particle Hamiltonian corresponding to one of its eigenvalues, say E, in bases of free particle eigenfunctions (plane waves). Can we use the plane waves corresponding to energies near E to get a reasonable solution?
 
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No it depends entirely on the Hamiltonian. This isn't like the variational principle where just about any trial wave-function gives you a good estimate of the ground state energy. You're trying to go backwards and it can certainly fail to give a good approximation in general. For example, the plane wave is an absolutely terrible approximation of the ground state wave-function of the harmonic oscillator. You can't approximate a Gaussian with a sinusoid.
 

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