page_title: Quantum LC circuit
section_title: Hamiltonian and energy eigenstates
hierarchical_section_title: Quantum LC circuit / One-dimensional harmonic oscillator / Hamiltonian and energy eigenstates
caption_attribution_description: English: Picture of wavefunctions (and energy levels) of quantum harmonic oscillator, using color scale for probability density.
context_page_description: An LC circuit can be quantized using the same methods as for the quantum harmonic oscillator. An LC circuit is a variety of resonant circuit, and consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electric current can alternate between them at the circuit's resonant frequency:
  where L is the inductance in henries, and C is the capacitance in farads. The angular frequency has units of radians per second. A capacitor stores energy in the electric field between the plates, which can be written as follows:
  Where Q is the net charge on the capacitor, calculated as
  Likewise, an inductor stores energy in the magnetic field depending on the current, which can be written as follows:
  Where is the branch flux, defined as
  Since charge and flux are canonically conjugate variables, one can use canonical quantization to rewrite the classical hamiltonian in the quantum formalism, by identifying
  and enforcing the canonical commutation relation
context_section_description: Like the one-dimensional harmonic oscillator  problem, an LC circuit can be quantized by either solving the Schrödinger equation or using creation and annihilation operators.  The energy stored in the inductor can be looked at as a "kinetic energy term" and the energy stored in the capacitor can be looked at as a "potential energy term".
  The Hamiltonian of such a system is:
  where Q is the charge operator, and  is the magnetic flux operator.  The first term represents the energy stored in an inductor, and the second term represents the energy stored in a capacitor.  In order to find the energy levels and the corresponding energy eigenstates, we must solve the time-independent Schrödinger equation,
  Since an LC circuit really is an electrical analog to the harmonic oscillator, solving the Schrödinger equation yields a family of solutions (the Hermite polynomials).
caption: Probability densities |ψn(x)|2  for the bound eigenstates, beginning with the ground state (n = 0) at the bottom and increasing in energy toward the top. The horizontal axis shows the position x, and brighter colors represent higher probability densities.
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