Advanced Quantum Mechanics Schwabl · Problem 1.7

Second Quantization

Physics — Quantum Mechanics · field-operators-commutation-relations

Problem

Starting from the bosonic commutation relations

[a_i, a_j^\dagger] = \delta_{ij}

, derive the commutation relations for the field operators

[\psi(\mathbf{x}), \psi^\dagger(\mathbf{x}')] = \delta^{(3)}(\mathbf{x} - \mathbf{x}')

and similarly for fermions (anticommutator). Use the definition

\psi(\mathbf{x}) = \sum_i \varphi_i(\mathbf{x}) a_i

where

\{\varphi_i(\mathbf{x})\}

is a complete orthonormal set.