Coherent state

∣ α ⟩ = D (α) ∣0 ⟩

is normalized, satisfies

A ∣ α ⟩ = α ∣ α ⟩

, and evolves to a phase times

∣ α e^{- iω t} ⟩

.

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Coherent-state resolution of identity ( $\int d [\overset{ˉ}{ψ}, ψ] e^{- \overset{ˉ}{ψ} ψ} ∣ ψ ⟩ ⟨ ψ ∣ = I$ )

For an unnormalized bosonic coherent state, the measure is

d^{2} ψ / π

over the complex plane. Gaussian moments

\int d^{2} ψ e^{- ∣ ψ ∣^{2}} ψ^{n} \overset{ˉ}{ψ}^{m} / π = n! δ_{nm}

recover the identity in the Fock state basis. For a fermionic coherent state, use independent Grassmann variables and an ordered Berezin integral with

\int d \overset{ˉ}{ψ} d ψ \overset{ˉ}{ψ} ψ = - 1

. The coefficient of

- \overset{ˉ}{ψ} ψ

in the weighted projector is the identity. Products of these one-mode measures give the finite-mode result.

Unnormalized bosonic coherent state ( $∣ ψ ⟩ = e^{ψ a^{†}} ∣0 ⟩$ )

An unnormalized bosonic coherent state is

∣ ψ ⟩ = e^{ψ a^{†}} ∣0 ⟩

for a complex label

ψ

. The canonical commutation relation gives

a ∣ ψ ⟩ = ψ ∣ ψ ⟩

and overlap

⟨ ψ ∣ ψ^{'} ⟩ = e^{\overset{ˉ}{ψ} ψ^{'}}

. Multiplication by

e^{- ∣ ψ ∣^{2} /2}

produces a normalized coherent state. The unnormalized form makes the coherent-state resolution of identity and thermal time slicing particularly simple.

Fermionic coherent state ( $a ∣ θ ⟩ = θ ∣ θ ⟩$ )

For a single fermionic annihilation operator

a

and an independent odd Grassmann variable

θ

, a fermionic coherent state is an extended state obeying

a ∣ θ ⟩ = θ ∣ θ ⟩

. With

{a, θ} = 0

one may take

∣ θ ⟩ = exp (- θ a^{†}) ∣0 ⟩

. Its overlap is exponential in the Grassmann labels. A resolution of identity with a Berezin integral gives the time-slice construction of a fermionic path integral. These formal eigenstates use Grassmann-valued coefficients.

Fermionic coherent-state trace ( $Tr A = \int d [\overset{ˉ}{ψ}, ψ] e^{- \overset{ˉ}{ψ} ψ} ⟨ - ψ ∣ A ∣ ψ ⟩$ )

The ordinary trace of an even fermionic operator requires

- ψ

in the bra of its fermionic coherent state kernel. In one mode the diagonal matrix elements give

⟨ - ψ ∣ A ∣ ψ ⟩ = A_{00} - \overset{ˉ}{ψ} ψ A_{11}

, whose Gaussian Berezin integral is

A_{00} + A_{11}

. An untwisted bra gives

A_{00} - A_{11}

, the supertrace. This distinction produces the antiperiodic coherent-state thermal boundary conditions.

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Coherent state by

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A coherent state is a specific type of quantum state of a harmonic oscillator that exhibits properties reminiscent of classical oscillators. In quantum mechanics, a coherent state is often associated with the eigenstates of the annihilation operator in the context of quantum optics and quantum mechanics of harmonic oscillators. Key characteristics of coherent states include: 1. **Minimum Uncertainty**: Coherent states minimize the uncertainty relation between position and momentum; they achieve the Heisenberg uncertainty principle's lower bound.

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