Title: ETH_Sem3 / Physik_III / Woche_13

Q: Parity operator in 3D:
\(\hat\Pi\Psi(x,y,z)=[...]\)
\(\hat\Pi\Psi(r,\theta,\phi)=[...]\)
A: \Psi(-x,-y,-z); \Psi(r,\pi-\theta,\pi+\phi)

Q: What does even/odd under parity mean?
A: Even: \(\hat\Pi^\ast \,\hat O\; \hat\Pi=\hat O\)
Odd: \(\hat\Pi^\ast\,\hat O\; \hat\Pi=-\hat O\)

Q: Are \(\hat r, \, \hat p\) and \(\hat L\) even or odd?
A: \(\hat r\) and \(\hat p\) are odd. \(\hat L = \hat r \times \hat p\) is therefore even.

Q: Spherical harmonics under parity\[\hat\Pi Y_{lm} = [...]\]
A: (-1)^l \;Y_{lm}

Q: Rotation operator for z-axis:
\(\hat R_z(\alpha) \; \Psi(r,\theta,\phi) = [...]\)
A: \Psi(r,\theta,\phi-\alpha)

Q: Rotation operator (arbitrary direction \(\hat n\)):
\[\hat R_{\hat n} (\alpha) = [...]\]
A: e^{-\frac{i\alpha}{\hbar}\hat n \cdot \hat L}

Q: Selection rule for \(l\) for an interaction with an electric field
A: \(l' - l= \pm1\)

Q: Selection rule for \(m\)
A: Linearly polarized light in z direction: \(m'-m=0\)
Circularly polarized light in z direction: \(m'-m=\pm1\)

Q: Fermi-Dirac distribution
A: \[n_j = \frac{g_j}{e^{\beta(\epsilon_j-\mu)}+1}\]
Where \(\beta = \frac{1}{k_bT}\) and \(\mu={(\frac{\partial U}{\partial N})}_{p,\,S}\)

Q: Bose-Einstein distribution
A: \[n_j = \frac{g_j}{e^{\beta(\epsilon_j-\mu)}-1}\]
Where \(\beta = \frac{1}{k_bT}\) and \(\mu={(\frac{\partial U}{\partial N})}_{p,\,S}\)

Q: Exchange operator:
\(\hat X \Psi(r_1,r_2) = [...]\)
A: \Psi(r_2,r_1)

Q: What are the eigenvalues of \(\hat X\)? How are the corresponding eigenstates called?
A: As \(\hat X^2=\mathbb{1}\), \(\hat X\) can have eigenvalues \(\pm 1\).
Eigenstates with eigenvalue \(+1\) are called symmetric.
Eigenstates with eigenvalue \(-1\) are called antisymmetric.

Q: If a system contains two indistinguishable particles, then\[[\hat O, \hat X] = [...]\]for any physical observable \(\hat O\).
A: 0

Bosons have integer spin values and are symmetric under exchange. Example: photon.
Fermions have half-integer spin values and are antisymmetric under exchange. Example: electron.

Q: Hamlitonian for atmos with multiple electrons.
A: \[\hat H = \sum_{j=1}^Z [-\frac{\hbar^2}{2m}\nabla_j^2-(\frac{1}{4\pi\epsilon_0})\frac{Ze^2}{r_j}] + \frac{1}{2}\sum_{j\neq k}^Z[(\frac{1}{4\pi\epsilon_0})\frac{e^2}{|\vec{r_j}-\vec{r_k}|}]\]

Q: What happens to the Bohr radius and the energy levels when \(Z\) increases, ignoring the electron-electron interaction?
A: new Bohr radius \(a'=\frac{a}{Z}\)
new energies \(E'_n=Z^2E_n\)