A report presents a theoretical study of a harmonic oscillator in homogeneous or nonhomogeneous externally applied electric and/or gravitational fields. The standard quantum-mechanical formalism for a simple harmonic oscillator, starting with the Hamiltonian and the associated creation and annihilation operators, is modified to incorporate the additional terms representing the external fields. The correspondingly modified solutions of the Schroedinger equation are derived.
This work was done by Igor Kulikov of Caltech for NASA’s Jet Propulsion Laboratory. To obtain a copy of the report, “Harmonic Oscillator in External Fields: Theory and Applications,” access the Technical Support Package (TSP) free on-line at www.nasatech.com/tsp under the Physical Sciences category. NPO-30262
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Quantum Mechanics of Harmonic Oscillator in External Fields
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Overview
The document presents a theoretical study by Igor K. Kulikov on the quantum mechanics of harmonic oscillators subjected to homogeneous and non-homogeneous external electric and gravitational fields. The research modifies the standard quantum-mechanical formalism of the harmonic oscillator, which typically involves the Hamiltonian and creation and annihilation operators, to account for the effects of these external fields.
Key findings indicate that while a homogeneous external field does not alter the frequency of the oscillator, it does shift its equilibrium position. In contrast, an inhomogeneous field affects both the frequency and the equilibrium position of the oscillator. This distinction is crucial for understanding the dynamics of systems influenced by varying field strengths.
The study further develops a formalism to calculate the partition function for a low-temperature gas of non-interacting fermions trapped in the harmonic oscillator potential. The interaction of these fermions with the external fields, while neglecting inter-particle interactions, allows for the derivation of equations related to the thermodynamic potential, internal energy, and the number of fermions in the gas. These equations elucidate how the thermodynamic properties of the Fermi gas are influenced by the external fields, providing insights into the behavior of quantum systems under varying conditions.
The document emphasizes the novelty of the approach, highlighting the compact structure of the Hamiltonian achieved through the reformulation of the creation and annihilation operator formalism. It also discusses the implications of the findings for simplifying the analysis of trapped fermions in gravitational fields, particularly in the Newtonian approximation.
Overall, this work contributes to the understanding of quantum harmonic oscillators in external fields, with potential applications in quantum computing and other advanced technologies. The research was conducted under the auspices of NASA's Jet Propulsion Laboratory, showcasing the intersection of fundamental physics and practical applications in space exploration and technology development. The findings are significant for both theoretical physics and practical applications, offering a deeper understanding of quantum systems in complex environments.
