page_title: Coherent state
section_title: The wavefunction of a coherent state
hierarchical_section_title: Coherent state / The wavefunction of a coherent state
caption_attribution_description: English: Animation of the quantum wave function of a coherent state of initial α=3 in a Quantum harmonic oscillator. The probability distribution is drawn along the ordinate, while the phase is encoded by color. The gaussian wave packet oscillates sinusoidally in the harmonic potential.
context_page_description: In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator and hence, the coherent states arise in the quantum theory of a wide range of physical systems. For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well. The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be related to classical solutions by a particle oscillating with an amplitude equivalent to the displacement.
  These states, expressed as eigenvectors of the lowering operator and forming an overcomplete family, were introduced in the early papers of John R. Klauder, e. g.
context_section_description: To find the wavefunction of the coherent state, the minimal uncertainty Schrödinger wave packet, it is easiest to start with the Heisenberg picture of the quantum harmonic oscillator for the coherent state .  Note that
  The coherent state is an eigenstate of the annihilation operator in the Heisenberg picture.
  It is easy to see that, in the Schrödinger picture, the same eigenvalue 
  occurs,
   .
  In the coordinate representations resulting from operating by ,  this amounts to the  differential equation,
  which is easily solved to yield
  where θ(t) is a yet undetermined phase, to be fixed by demanding that the wavefunction satisfies the Schrödinger equation.
  It follows that
  so that σ is the initial phase of the eigenvalue.
  The mean position and momentum of this "minimal Schrödinger wave packet"  ψ are thus oscillating just like a classical system, 
  The probability density remains a Gaussian centered on this oscillating mean,
caption: Time evolution of the probability distribution with quantum phase (color) of a coherent state with α=3.