关键词: Pauli-Fierz 哈密顿量, 规范变换, 光子, 坐标和动量表象

Abstract: When we discuss the interaction of atoms or molecules with quantized electromagnetic waves ( photons) ，the Hamiltonian in length gauge takes the advantage by replacing the vector potential A by electric filed multiplying the dipole. However，formulas in different papers seem to have different forms. In this paper，we start from Pauli-Fierz Hamiltonian in velocity gauge and obtain the length gauge Hamiltonian in coordinate and momentum 　representation，respectively，which proves the equivalence between different forms. We hope this could help students to understand the Hamiltonian in different forms.

Key words: Pauli-Fierz Hamiltonian, gauge transformation, photon, coordinate and momentum representation