Title: All Decks / ETH_Sem6_old / QMII / Chapter03

Q: Change from sum to integral QMII
A: \(\frac{1}{L^{3}}\sum_{\bf k}\rightarrow\int\frac{d^{3}{\bf k}}{(2\pi)^{3}}\)

Q: QM: Dipole approx
A: \(\hat J(-k) \approx \sum_{particles} \frac{\hat p}{m} + \mathcal O(r/\lambda)\)
The probability flux is only due to classical momentum (no wave stuff)
This leads to a hamiltonian:
\(H \propto r\cdot E\)

Q: Second quantisation Field Hamiltonian
A: \(\hat{H}_{\mathrm{SF}}=\sum_{\mathbf{k},\lambda}\hbar\omega_{k}\Bigl\{\hat{\alpha}_{\mathbf{k},\lambda}^{\dagger}\hat{\alpha}_{\mathbf{k},\lambda}+\frac{1}{2}\Bigr\}\)
(same as harmonic osci but summed over all modes)
note \(\lambda \)  is the polarisation (1,2)

Q: Second quantisation number operator:
A: \(\check{N}({\bf k},\lambda):=\hat{\alpha}_{{\bf k},\lambda}^{\dagger}\hat{\alpha}_{{\bf k},\lambda}\)
(Anihilation, then creation) (r to l)

Q: What result is different in second quantisation to classical EM?
A: \(\frac{\Gamma_{n\to0}}{\Gamma_{0\to n}}=\frac{N_{\mathbf{k}_{k},\lambda_{k}}}{N_{\mathbf{k}_{k},\lambda_{k}}+1}\)
(at low occupations its easyer to populate photon modes than to depopulate them)
This is spontaneous emission!