Advanced Quantum Mechanics Schwabl · Problem 1.4

Second Quantization

Physics — Quantum Mechanics · single-particle-operators-second-quantization

Problem

Show that the second-quantized form of a single-particle operator

T = \sum_\alpha t_\alpha

for an

N

-particle system is given by

T = \sum_{i,j} t_{ij}\, a_i^\dagger a_j,

where

t_{ij} = \langle i | t | j \rangle

, by computing the action of

\sum_\alpha |i\rangle_\alpha \langle j|_\alpha

on an arbitrary symmetrized state.