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Jan12-12, 08:03 AM
P: 7
Please help me, I need to derive exciton-photon interaction.
Here, we are using second quantization. Please refer to this paper
Hamiltonian of electron-photon is
[itex]H_{el-op}=\sum_k D_k c^+_{kc}c_{kv}(a+a^+)[/itex]

[itex]c^+_{kc}c_{kv}[/itex] are creation of electron to conduction band and annihilation electron in valence band, respectively. [itex](a+a^+)[/itex] are photon annihilation and creation operator.
Exciton wave function is
[itex] |\Psi^f\rangle=\sum_k Z^n_{k_{c},k_v}c^+_{k_{c}}c_{k_v}|0\rangle [/itex]
Where Z is weighting coefficient, kc is electron state (conduction band), kv is hole state (valence band), and [itex]|0\rangle [/itex] is ground state (all electrons occupy valence) band.

Matrix element of exciton-photon is defined as
[itex] M_{ex-op}=\langle\Psi^f|H_{el-op}|0\rangle[/itex]

[itex]M_{ex-op}=\sum_k Z^{n*}_{k_{c},k_v}D_k\langle 0|a+a^+|0\rangle[/itex]
My question is, how can we prove that [itex] \langle 0|a+a^+|0\rangle=1[/itex] to get

[itex]M_{ex-op}=\sum_k Z^{n*}_{k_{c},k_v}D_k [/itex]
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