Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 2.7

Lorentz invariance and second quantization — Schwartz M.D. z-lib.org

Physics — QFT · coherent-states-harmonic-oscillator

Problem

Coherent states of the simple harmonic oscillator. (a) Calculate ∂z(e−za†aeza†)\partial_z(e^{-za^\dagger} a e^{za^\dagger}) where zz is a complex number. (b) Show that ∣z⟩=eza†∣0⟩|z\rangle = e^{za^\dagger}|0\rangle is an eigenstate of aa. What is its eigenvalue? (c) Calculate ⟨n∣z⟩\langle n|z\rangle. (d) Show that these ``coherent states'' are minimally dispersive: ΔpΔq=12\Delta p \Delta q = \frac{1}{2}, where Δq2=⟨q2⟩−⟨q⟩2\Delta q^2 = \langle q^2\rangle - \langle q\rangle^2 and Δp2=⟨p2⟩−⟨p⟩2\Delta p^2 = \langle p^2\rangle - \langle p\rangle^2, where ⟨q⟩=⟨z∣q∣z⟩⟨z∣z⟩\langle q\rangle = \frac{\langle z|q|z\rangle}{\langle z|z\rangle} and ⟨p⟩=⟨z∣p∣z⟩⟨z∣z⟩\langle p\rangle = \frac{\langle z|p|z\rangle}{\langle z|z\rangle}. (e) Why can you not make an eigenstate of a†a^\dagger?

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