QFT — Michael Luke Lectures · Problem 2

Example: Non-Relativistic Quantum Mechanics ("Second Quantization") — Michael Luke

Physics — QFT · second-quantization-canonical-quantization

Problem

Canonically quantize the theory with L0=iψ∗∂0ψ+b∇⃗ψ∗⋅∇⃗ψ\mathcal{L}_0 = i\psi^*\partial_0\psi + b\vec{\nabla}\psi^* \cdot \vec{\nabla}\psi. (HINT: You may be bothered by the fact that the momentum conjugate to ψ∗\psi^* vanishes. Don't be. Because the equations of motion are first-order in time, a complete and independent set of initial-value data consists of ψ\psi and its conjugate momentum alone. It is only on these that you need to impose the canonical quantization conditions.) Identify appropriately normalized coefficients in the expansion of the fields in terms of plane wave solutions with annihilation and/or creation operators, and write the energy, linear momentum and internal-symmetry charge in terms of these operators. (Normal-order freely.) Find the equation of motion for the single particle state ∣k⃗⟩|\vec{k}\rangle and the two particle state ∣k⃗1,k⃗2⟩|\vec{k}_1, \vec{k}_2\rangle in the Schr\"odinger Picture. What physical quantities do bb and the internal symmetry charge correspond to?

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