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Physics · Glossary

What is Coherent states?

Definition 9.6 University Physics — Year 3 · Chapter 9 — The Quantum Harmonic Oscillator

A coherent state ∣α⟩\ket\alpha is an eigenvector of the annihilation operator, a^∣α⟩=α∣α⟩\hat a\ket\alpha = \alpha\ket\alpha, with α\alpha any complex number. Expanded on the ladder,

∣α⟩=e−∣α∣2/2∑n=0∞αnn! ∣n⟩:\ket\alpha = \eu^{-|\alpha|^2/2}\sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}}\,\ket n :

the quantum count is Poisson-distributed with mean ⟨N^⟩=∣α∣2\langle\hat N\rangle = |\alpha|^2 and spread ΔN=∣α∣\Delta N = |\alpha|.

Phase-space portrait (compare ): the ground state is a minimal uncertainty blob at the origin; a coherent state is the same blob displaced, circling at without deforming — quantum mechanics’ best imitation of a classical oscillation.
Phase-space portrait (compare Chapter 2): the ground state is a minimal uncertainty blob at the origin; a coherent state is the same blob displaced, circling at ω\omega without deforming — quantum mechanics’ best imitation of a classical oscillation.
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