The Geometry Underlying the Quantum Harmonic Oscillator

• URL: http://arxiv.org/abs/2604.21373v1
• Date: Thu, 23 Apr 2026 07:38:35 GMT
• Title: The Geometry Underlying the Quantum Harmonic Oscillator
• Abstract summary: We show that the eigenfunctions $_n$ of the quantum Hamiltonian correspond to complex radial coordinates in the reduced phase space $mathbb C2/mathbb Z_nsubsetmathbb C2$.<n>We show that in the Kepler/hydrogen atom problem there is a similar correspondence between classical and quantum states.
• Score: 51.56484100374058
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider two-dimensional harmonic oscillator in the complex Bargmann-Fock-Segal representation with $T^*{\mathbb R}^{2}={\mathbb C}^2$ as classical phase space. We show that the eigenfunctions $ψ_n$ of the quantum Hamiltonian correspond to complex radial coordinates in the reduced phase space ${\mathbb C}^2/{\mathbb Z}_n\subset{\mathbb C}^2$. They describe ${\mathbb Z}_n$-invariant motion of particle along a circle $S^1$ in lens space $S^3/{\mathbb Z}_n\subset{\mathbb C}^2/{\mathbb Z}_n$, where ${\mathbb Z}_n$ is the cyclic group of rotation by an angle $2π/n$ on the circle $S^1$, $n=1,2,...\,$. Thus the general solution of the Schrödinger equation carries information about an infinite number of admissible classical states $ψ_n$ that can be mapped to other states after lifting into the quantum bundle. We show that in the Kepler/hydrogen atom problem there is a similar correspondence between classical and quantum states.

Related papers

• Discrete symmetries in classical and quantum oscillators [51.56484100374058]
  We show the eigenfunctions $_n=zn$ of the quantum Hamiltonian in the complex Bargmann-Fock-Segal representation.<n>The superposition $=sum_n c_n_n$ arises only with incomplete knowledge of the initial data for solving the Schrdinger equation.
  arXiv  Detail & Related papers  (2026-01-05T10:04:39Z)
• Dirac particles, spin and photons [51.56484100374058]
  We describe relativistic particles with spin as points moving in phase space $X=T* R1,3times C2_Ltimes C2_R$.<n>We show that taking into account the charges $q_sfv=pm 1$ of the fields $Psi_pm$ changes the definitions of the inner products and currents.
  arXiv  Detail & Related papers  (2025-08-29T12:47:56Z)
• Klein-Gordon oscillators and Bergman spaces [55.2480439325792]
  We consider classical and quantum dynamics of relativistic oscillator in Minkowski space $mathbbR3,1$. The general solution of this model is given by functions from the weighted Bergman space of square-integrable holomorphic (for particles) and antiholomorphic functions on the K"ahler-Einstein manifold $Z_6$.
  arXiv  Detail & Related papers  (2024-05-23T09:20:56Z)
• Antiparticles in non-relativistic quantum mechanics [55.2480439325792]
  Non-relativistic quantum mechanics was originally formulated to describe particles.<n>We show how the concept of antiparticles can and should be introduced in the non-relativistic case without appealing to quantum field theory.
  arXiv  Detail & Related papers  (2024-04-02T09:16:18Z)
• (2+1)D topological phases with RT symmetry: many-body invariant, classification, and higher order edge modes [6.267386954898001]
  We consider many-body systems of interacting fermions with fermionic symmetry groups $G_f mathbbZf times mathbbZ$. We show that (2+1)D invertible fermionic phases with these symmetries have a $mathbbZ times mathbbZ_8$, $mathbbZ_8$, $mathbbZ2 times mathbbZ$, and $mathbbZ2
  arXiv  Detail & Related papers  (2024-03-27T18:00:00Z)
• Vacuum Force and Confinement [65.268245109828]
  We show that confinement of quarks and gluons can be explained by their interaction with the vacuum Abelian gauge field $A_sfvac$.
  arXiv  Detail & Related papers  (2024-02-09T13:42:34Z)
• Quantum connection, charges and virtual particles [65.268245109828]
  A quantum bundle $L_hbar$ is endowed with a connection $A_hbar$ and its sections are standard wave functions $psi$ obeying the Schr"odinger equation. We will lift the bundles $L_Cpm$ and connection $A_hbar$ on them to the relativistic phase space $T*R3,1$ and couple them to the Dirac spinor bundle describing both particles and antiparticles.
  arXiv  Detail & Related papers  (2023-10-10T10:27:09Z)
• Quantized charge polarization as a many-body invariant in (2+1)D crystalline topological states and Hofstadter butterflies [14.084478426185266]
  We show how to define a quantized many-body charge polarization $vecmathscrP$ for (2+1)D topological phases of matter, even in the presence of non-zero Chern number and magnetic field. We derive colored Hofstadter butterflies, corresponding to the quantized value of $vecmathscrP$, which further refine the colored butterflies from the Chern number and discrete shift.
  arXiv  Detail & Related papers  (2022-11-16T19:00:00Z)
• Bulk-boundary asymptotic equivalence of two strict deformation quantizations [0.0]
  The existence of a strict deformation quantization of $X_k=S(M_k(mathbbC))$ has been proven by both authors and K. Landsman citeLMV. A similar result is known for the symplectic manifold $S2subsetmathbbR3$.
  arXiv  Detail & Related papers  (2020-05-09T12:03:18Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.