Hamiltonian of Minimum Uncertainty State

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SUMMARY

The Hamiltonian of a minimum uncertainty state is specifically associated with the harmonic oscillator Hamiltonian, which has Gaussian energy eigenstates in the position basis. This state is characterized by its wavefunction, which maintains its form under the Fourier transform, indicating a unique property of symmetry in both position and momentum representations. The discussion highlights the importance of understanding the physical implications of these eigenstates in quantum mechanics.

PREREQUISITES
  • Quantum mechanics fundamentals
  • Understanding of Hamiltonian mechanics
  • Familiarity with Gaussian distributions in quantum states
  • Knowledge of Fourier transforms in quantum physics
NEXT STEPS
  • Study the properties of harmonic oscillator Hamiltonians in quantum mechanics
  • Explore the implications of minimum uncertainty states in quantum theory
  • Learn about the role of Fourier transforms in quantum state representations
  • Investigate the physical interpretations of Gaussian wavefunctions
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Quantum physicists, students of quantum mechanics, and researchers interested in wavefunction properties and their implications in quantum theory.

Gza
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I was curious as to the form of the hamiltonian, whose energy eigenstate in the position basis is a gaussian distribution (or minimum uncertainty state, as I've heard from somewhere.) I haven't taken quantum for a few years, and remember studying the minimum uncertainty state as a wavefunction, without reference to its hamiltonian.
 
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Also, does the fact that it is an eigenfunction of the Fourier transform have any physical interpretation? (aside from the obvious fact that it looks the same in the position, as well as momentum basis)
 
A minimum uncertainty state that's also the eigenstate of an hamiltonian is a gaussian, and it corresponds to the harmonic oscillator hamiltonian.
 

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