SUMMARY
The coherent state defined as $$|\phi\rangle = \exp\left(\sum_{i}\phi_i \hat{a}^\dagger_i\right)|0\rangle$$ is indeed an eigenstate of the annihilation operator $$\hat{a}_i$$, satisfying the equation $$\hat{a}_i|\phi\rangle=\phi_i|\phi\rangle$$. To demonstrate this, one must apply the annihilation operator to the coherent state and utilize the properties of the exponential function, specifically expanding it in a Taylor series. This approach clarifies how the annihilation operator interacts with each term in the coherent state representation.
PREREQUISITES
- Understanding of quantum mechanics and quantum states
- Familiarity with the annihilation and creation operators in quantum optics
- Knowledge of Taylor series expansions
- Basic concepts of eigenstates and eigenvalues in linear algebra
NEXT STEPS
- Study the properties of annihilation and creation operators in quantum mechanics
- Learn about the mathematical formulation of coherent states
- Explore Taylor series and their applications in quantum state representations
- Investigate the implications of eigenstates in quantum systems
USEFUL FOR
Quantum physicists, students of quantum mechanics, and researchers focusing on quantum optics and coherent states.