In quantum mechanics, a coherent state is a specific type of quantum state that exhibits properties most closely resembling classical harmonic oscillators. Coherent states are widely used in various fields of quantum physics due to their unique characteristics. ### Key Characteristics: 1. **Quantum Harmonic Oscillator**: - Coherent states are often associated with the quantum harmonic oscillator. They are the eigenstates of the annihilation (lowering) operator, which means they remain unchanged except for a phase factor when the annihilation operator acts on them [[1](https://en.wikipedia.org/wiki/Coherent_state)][[3](https://physics.stackexchange.com/questions/742288/coherent-state-basis)]. 2. **Minimal [[Uncertainty]]**: - These states minimize the [[Heisenberg Uncertainty Principle]], meaning the product of uncertainties in position and momentum is as small as quantum mechanics allows. This characteristic makes them particularly stable and well-suited for describing systems like lasers and other bosonic fields [[1](https://en.wikipedia.org/wiki/Coherent_state)]. 3. **Classical Analog**: - Coherent states exhibit a behavior that closely mirrors classical oscillatory motion. In phase space, they are represented as points with small, circular uncertainty regions, indicating well-defined amplitude and phase, akin to a classical harmonic oscillator [[2](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_(Chong)/06%3A_Appendices/6.05%3A_E-_Coherent_States)]. 4. **Mathematical Representation**: - A coherent state |α⟩ can be expressed as an exponential operator acting on the vacuum state. Mathematically, it is defined as: ![[Pasted image 20240711183737.png]]where α is a complex number representing the amplitude and phase of the state, and |n⟩ are the number states (Fock states) of the harmonic oscillator [[2](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_III_(Chong)/06%3A_Appendices/6.05%3A_E-_Coherent_States)]. 5. **Applications**: - Coherent states are used extensively in quantum optics, quantum information, and the study of quantum chaos. They provide a bridge between quantum and classical descriptions, facilitating the understanding of phenomena like laser light, quantum communication, and more [[4](https://www.studysmarter.co.uk/explanations/physics/quantum-physics/coherent-state/)].