Title: All Decks / ETH_Sem6_old / QMII / Chapter02

Q: Lippman Schwinger equation + intuition:
A: \(\ket {E_0} \) is for the inhomogenious part of the solution.
\((E-H_0 + i\delta)^{-1} V\ket E\) is from inverting the Eigenvalue problem and adding a convergence generating factor.
(\(\delta = \varepsilon \neq \delta(x)\))

Lippmann Schwinger in Position space

Q: Intuition Lippman Schwinger in position space:
A: Plane wave + Circular wave with source depending on the probability density and the energy of the object in the way.

Q: Expansion of \(|x-x'|\) and approximation:
A: \(=|\mathbf{x}|\left(1-2{\frac{\mathbf{x}\cdot\mathbf{x}^{\prime}}{|\mathbf{x}|^{2}}}+{\frac{\mathbf{x}^{\prime2}}{\mathbf{x}^{2}}}\right)^{\frac{1}{2}}\)
\(\approx |{\bf x}|\left(1-\frac{\mathrm{e_{r}}\cdot\mathrm{x^{\prime}}}{\left|{\bf x}\right|}+\mathcal O\left(\frac{\left|{\bf x^{\prime}}\right|^{2}}{\left|{\bf x}\right|^{2}}\right)\right)\)

Q: Scattering Amplitude + usage
A: \(f({\bf k},{\bf k^{\prime}})=-\frac{2M}{\hbar^{2}}\int\frac{d^{3}{\bf x^{\prime}}}{4\pi}e^{-i{\bf k^{\prime}}\cdot{\bf x^{\prime}}}V({\bf x}^{\prime})\left(2\pi\hbar\right)^{3/2}\psi_{E}({\bf x}^{\prime})\)
\(\psi_{E}({\bf x})=\frac{1}{\sqrt{2\pi\hbar^{3}}}\biggl\{{e}^{\frac{1}{\hbar}{\bf k}\cdot{\bf x}}+\frac{e^{i\vert{\bf k}\vert|\bf x|}}{\vert{\bf x}\vert}f({\bf k},{\bf k^{\prime}})\biggr\}\)

Q: Differential crossection from scattering amplitude:
A: \({\frac{d\sigma}{d\Omega}}=|f(\mathbf{k},\mathbf{k}^{\prime})|^{2}\)
\(d\sigma\equiv{\frac{j_{scattered}\cdot d\mathrm{S}}{\left|j_{\mathrm{incoming}}\right|}}\)

Q: Specific surface to solid angle:
A: \(d\mathbf{S}=\mathbf{e}_{r}r^{2}d\Omega\)

Q: Probability flux in scattering:
A: \(\mathrm{j}=\mathrm{Re}\left({\psi}^{*}\frac{\left(-i\hbar\nabla\right)}{\mathrm{M}}\psi\right)=\langle{\psi}|\frac{\mathrm{\hat{p}}}{\mathrm{M}}\ket{\psi}\)

Q: Born approximation, scattering amplitude and intuiton:
A: \(f(q)=-\frac{2M}{\hbar}\int_{0}^{\infty}\mathrm{d}r\left(r V(r)\right)\frac{\sin\left(\frac{q r}{\hbar}\right)}{q}\)
Scattering amplitude is FT of scattering potential.
We take \(\psi_{in}\) instead of \(\psi_{tot}\) for the scattering.

Q: Momenum transfer depending on scattering angle:
A: \(q=2\left|\mathbf{k}\right|\sin{\frac{\theta}{2}}\)

Q: Yukawa potential
A: \(V(r)=\frac{V_{0}}{r}e^{-\frac{m c}{h}r}\)

Q: General solution spherically symmetric Hamiltonian:
A: \(\psi_{\ell m}(r,\theta,\phi)\equiv R_{\ell}(r)Y_{\ell}^{m}(\theta,\phi)\equiv\frac{U_{\ell}(r)}{r}Y_{\ell}^{m}(\theta,\phi)\)

Q: Radial SE for spherical symetric Problem:
A: \(U_{\ell}^{\prime\prime}(r)-\left[\frac{2M}{\hbar^{2}}V(r)+\frac{\ell(\ell+1)}{r^{2}}-k^{2}\right]U_{\ell}(r)=0\)
With
\(k^{2}\equiv\frac{2M E}{\hbar^{2}}\)

Q: Trick to solving scattering in spherically symmetric potential:
A: Consider
 \(\lim_{r\to \infty}V(r) = 0\)
This simplifies the equation for \(U_l\)
\({ U}_{\ell}^{\prime\prime}(r)+k^{2}{ U}_{\ell}(r)=0\)
Which has the well known solutions:
\(U_{\ell}(r)=A_{\ell}\sin(k r+\tilde{\delta}_{\ell})\)
(Note that the phase \(\delta\) is non trivial and depends on \(k\))

Q: Relation spherical harmonics with \(m=0\) & ... polynomials
A: Related to Legendre polynomials:
\(Y_{\ell}^{0}(\theta,\phi)=\sqrt{\frac{2\ell+1}{4\pi}}P_{\ell}(\cos\theta)\)

Q: Differential equation for Legendre polynomials
A: What is this?
\((1-x^{2}){\frac{\partial^{2}}{\partial x^{2}}}P_{\ell}(x)-2x{\frac{\partial}{\partial x}}P_{\ell}(x)+\ell(\ell+1)P_{\ell}(x)=0\)

Q: Orthonormality Legendre Polynomials
A: \(\int_{-1}^{1}d\cos\theta\,P_{\ell}(\cos\theta)P_{\ell^{\prime}}(\cos\theta)=\frac{2\delta_{\ell\ell^{\prime}}}{2\ell+1}\)

Q: Optical theorem, statement & intuition:
A: \(\sigma_{scat,tot} = \frac{4\pi}{k}\Im(f(k, \theta = 0))\)
Total crossection (at infinity) is given as the imaginary part of the scattering amplitude.
Intuition:
The imaginary part is due to absorbtion being irrelevant for this kind of exchange. (\(\theta = 0 \implies q = 0)\). The larger the momentum is, the smaller the scattering crossection.

Q: Ampere Maxwell Law in Coulomb gauge for vectorpotential:
A: (as a reminder)

Q: Magnetic and Electric field given vector Potential:
A: \(\begin{array}{l}{{\mathbf{E}(\mathbf{r},t)=\,-\,\frac{1}{c}\frac{\partial}{\partial t}\mathbf{A}(\mathbf{r},t),}}\\ {{\mathbf{B}(\mathbf{r},t)=\,\nabla\times\mathbf{A}(\mathbf{r},t).}}\end{array}\)

Q: SE for \(e^-\) in Field:
A: \(i\hbar\frac{\partial}{\partial t}\psi({\bf r},t)=\left[\frac{1}{2m}\left(-i\hbar\nabla-\frac{e}{c}{\bf A}({\bf r},t)\right)^{2}+e\phi({\bf r},t)+U({\bf r})\right]\psi({\bf r},t)\)

Gague transforms \(A,\phi, \psi\)

Q: Electron in EM Field, coulomb gauge:
A: \(H_{k}=-{\frac{e}{m c}}{\hat{\mathbf{p}}}\cdot\mathbf{A}({\hat{\mathbf{r}}})+{\frac{e^{2}}{2m c^{2}}}\mathbf{A}^{2}({\hat{\mathbf{r}}})+e\phi({\hat{\mathbf{r}}}).\)

Q: Current density operator:
A: \(\mathbf{\hat{J}}(\mathbf{r})={\frac{1}{2m}}\left[\mathbf{\hat{p}}\delta(\mathbf{r}-\mathbf{\hat{r}})+\delta(\mathbf{r}-\mathbf{\hat{r}})\mathbf{\hat{p}}\right]\)

Q: Freespace simplified interaction hamiltonian:
A: \(H_{k}=\int d^{3}{\bf r}~\Bigl[-{\frac{e}{c}}\hat J({\bf r})\cdot{\bf A}({\hat{\bf r}})\Bigr]\;\)

Q: Scattering wavefunction in \(r\to \infty\) limit?
A: \(\psi_{\ell m}(r,\theta,\phi)=\frac{U_{\ell}(r)}{r}Y_{\ell}^{m}(\theta,\phi)=\frac{A_{\ell}}{r}\sin(k r+\tilde{\delta}_{\ell})Y_{\ell}^{m}(\theta,\phi)\)
Note that due to \(m=0\) we can get
\(\psi_{\ell}(r,\theta)\stackrel{r\rightarrow\infty}{\sim}\frac{A_{\ell}}{r}\sin(k r+\tilde{\delta}_{\ell})P_{\ell}(\cos\theta)$\)

Q: What can we get from the fact that most scattering problems are cylindrically symmetric?
A: \(m = 0\)

Q: Spherical harmonics for \(m =0\)?
A: \(Y_{\ell}^{0}(\theta,\phi)=\sqrt{\frac{2\ell+1}{4\pi}}P_{\ell}(\cos\theta).\)
With \(P_\ell\) the legendre polynomials

Q: Diff eq for legendre polynomials?
A: \((1-x^{2})\frac{\partial^{2}}{\partial x^{2}}P_{\ell}(x)-2x\frac{\partial}{\partial x}P_{\ell}(x)+\ell(\ell+1)P_{\ell}(x)=0\)

Q: Idea Optical Theorem?
A: Total cross section relates to scattering at \(\theta = 0 \) due to conservation of momentum.