Question

A coherent state $|\lambda\rangle$ of a one-dimensional simple harmonic oscillator is defined to be an eigenstate of the annihilation operator $a$: $a|\lambda\rangle = \lambda|\lambda\rangle$, (1) where $\lambda$ is, in general, complex number. Using the energy eigenstate $|n\rangle$ as basis, we can write $\infty$ $|\lambda\rangle = \sum_{n=0} C_n|n\rangle$. (2) (a) Show the recursion relation of the coefficient $C_n = \frac{\lambda}{\sqrt{n}}C_{n-1}$. (3) (b) Determine the coefficient $C_0$.

          A coherent state $|\lambda\rangle$ of a one-dimensional simple harmonic oscillator
is defined to be an eigenstate of the annihilation operator $a$:
$a|\lambda\rangle = \lambda|\lambda\rangle$,
(1)
where $\lambda$ is, in general, complex number. Using the energy eigenstate $|n\rangle$ as basis, we
can write
$\infty$
$|\lambda\rangle = \sum_{n=0} C_n|n\rangle$.
(2)
(a) Show the recursion relation of the coefficient
$C_n = \frac{\lambda}{\sqrt{n}}C_{n-1}$.
(3)
(b) Determine the coefficient $C_0$.

A coherent state |λ⟩ of a one-dimensional simple harmonic oscillator
is defined to be an eigenstate of the annihilation operator a:
a|λ⟩ = λ|λ⟩,
(1)
where λ is, in general, complex number. Using the energy eigenstate |n⟩ as basis, we
can write
∞
|λ⟩ = ∑n=0 Cn|n⟩.
(2)
(a) Show the recursion relation of the coefficient
Cn = (λ)/(√(n))Cn-1.
(3)
(b) Determine the coefficient C0.

Added by Consuelo G.

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