Coherent States of the Harmonic Oscillator

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Coherent states of a harmonic oscillator are wavepackets that have the shape of the ground state probability distribution but undergo the motion of a classical oscillator of arbitrary energy. Schrödinger first considered these in the context of minimum-uncertainty wavepackets. More recently (1963), Roy Glauber exploited coherent states in quantum-mechanical descriptions of oscillating electromagnetic fields in quantum optics and in connection with the Hanbury-Brown and Twiss experiment. Glauber shared the 2005 Nobel Prize in Physics for this work.

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A coherent state, also known as a Glauber state or a "squeezed quantum state", is an eigenfunction of the harmonic oscillator annihilation operator , where for simplicity. The eigenstates of (a nonhermitian operator) are given by , where are the harmonic-oscillator eigenstates. These coherent states are solutions of the eigenvalue equation with energy expectation values . The eigenvalues can be complex numbers but we restrict them here to real values. This average energy has a form analogous to the harmonic oscillator eigenvalues . The latter are represented by horizontal lines within the potential-energy parabola. The probability density for the wavepacket representing works out to , shown as a blue Gaussian. As the time variation is run, the wavepacket oscillates so that its peak moves between the classical turning points. The wavepacket remains localized in the lowest energy state, when .

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Contributed by: S. M. Blinder (March 2011)
With contribution by Jeremy Michelson
Open content licensed under CC BY-NC-SA


Snapshots


Details

Snapshot 1: wavepacket

Snapshots 2,3: oscillation of wavepacket



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