Advanced Quantum Mechanics Schwabl · Problem 1.8

Second Quantization

Physics — Quantum Mechanics · hamiltonian-momentum-representation

Problem

Express the Hamiltonian H=∑αpα22m+12∑α≠βV(xα−xβ)H = \sum_\alpha \frac{\mathbf{p}_\alpha^2}{2m} + \frac{1}{2}\sum_{\alpha \neq \beta} V(\mathbf{x}_\alpha - \mathbf{x}_\beta) in the momentum representation using creation and annihilation operators ak,σ†a_{\mathbf{k},\sigma}^\dagger and ak,σa_{\mathbf{k},\sigma} for particles with momentum ℏk\hbar\mathbf{k} and spin σ\sigma. Show that the result is H=∑k,σℏ2k22mak,σ†ak,σ+12V∑k,k′,q,σ,σ′V(q) ak+q,σ†ak′−q,σ′†ak′,σ′ak,σ,H = \sum_{\mathbf{k},\sigma} \frac{\hbar^2 k^2}{2m} a_{\mathbf{k},\sigma}^\dagger a_{\mathbf{k},\sigma} + \frac{1}{2V} \sum_{\mathbf{k},\mathbf{k}',\mathbf{q},\sigma,\sigma'} V(\mathbf{q})\, a_{\mathbf{k}+\mathbf{q},\sigma}^\dagger a_{\mathbf{k}'-\mathbf{q},\sigma'}^\dagger a_{\mathbf{k}',\sigma'} a_{\mathbf{k},\sigma}, where V(q)V(\mathbf{q}) is the Fourier transform of V(x)V(\mathbf{x}).

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