chmodfree said:How can I show that the state is eigenstate of annihilation operator a?
Maybe one should add that the exponential function must be expanded in the Taylor series.PeterDonis said:What does the annihilation operator do to each term in the sum?
A coherent state is a quantum state that exhibits classical-like properties, such as a well-defined position and momentum. It is a superposition of different energy levels and is described by a wave function that is a Gaussian distribution in both position and momentum space.
An eigenstate of the annihilation operator is a state in which the annihilation operator acts as a scalar multiple of the state. In other words, the state is unchanged when the annihilation operator is applied to it.
A coherent state can be proven to be an eigenstate of the annihilation operator by showing that the annihilation operator acting on the coherent state results in the same coherent state multiplied by a complex number, known as the eigenvalue.
Proving that a coherent state is an eigenstate of the annihilation operator confirms that the coherent state is a stable and well-defined quantum state. It also allows for the calculation of the average number of particles in the coherent state, which is equal to the eigenvalue of the annihilation operator.
This proof is an important demonstration of the principles of quantum mechanics, specifically the concept of eigenstates and operators. It also has practical applications in fields such as quantum computing and quantum information processing.