Title: All Decks / ETH_Sem6_old / QMII / Chapter04

Q: Probability density for two fermion/boson system:
A: \(d P_{\pm}({\bf x}_{1},{\bf x}_{2})\equiv d^{3}{\bf x}_{1}\,d^{3}{\bf x}_{2}\,|\phi_{\pm}({\bf x}_{1},{\bf x}_{2})|^{2}\)
Probability of finding one particle in \(x_1\) and the other in \(x_2\)
\(\begin{array}{l}{{d P_{\pm}({\bf x}_{1},{\bf x}_{2})=\left(d^{3}{\bf x}_{1}\left|\omega_{A}({\bf x}_{1})\right|^{2}\right)\left(d^{3}{\bf x}_{2}\left|\omega_{B}({\bf x}_{2})\right|^{2}\right)}}\\ {{\pm\,{\bf R e}\Big[\left(d^{3}{\bf x}_{1}\omega_{A}({\bf x}_{1})\omega_{B}^{*}({\bf x}_{1})\right)\Big(d^{3}{\bf x}_{2}\omega_{A}({\bf x}_{2})\omega_{B}^{*}({\bf x}_{2})\Big)\Big].}}\end{array}\)
With the last term being the exchange density

Q: QMII 4.5 more
A: cards

Q: Two particle scattering ansatz and image:
A: \(e^{i{\bf k}\cdot{\bf x}}+e^{-i{\bf k}\cdot{\bf x}}+\frac{e^{i k r}}{r}\left[f(\theta)+f(\pi-\theta)\right]\)

Q: Spin \(0\) scattering vs spin \(1/2\) scattering:
A: We need to consider the fact that we have \(3:1\) ratio of \(triplet: singlet\) and the fact that electrons are fermions

Q: What do we need to keep in mind when considering the fermi level of electrons?
A: We have a factor two because of spin degeneracy!