Questions and Answers of Quantum Mechanics A Graduate Course

How does translational invariance and rotational symmetry constrain the three-body potential coming from quantum field theory?
Consider the LCAO approximation for an atom with three electrons. Write the expansion, restricting yourself to four basis elements.
Write the coefficients of the expansion in exercise 2 as coefficients in terms of energy, \(C_{E_{1} \ldots}\), show explicitly their symmetry properties, and then rewrite them as coefficients in the
Consider five bosons with a one-body potential that can be approximated as harmonic oscillator. Write the wave function expansion in terms of harmonic oscillator states with \(n \leq 3\).
For the case in exercise 4, write the one-particle (kinetic) operator in the Hamiltonian, in the occupation number basis.Data From Exercise 4:-Consider five bosons with a one-body potential that can
For the case in exercise 4, consider a two-particle potential of the Coulomb ( \(\propto 1 / r)\) type. Write explicitly the first six nontrivial terms (those having an occupation number change in
Consider three fermions with a one-body potential that can be approximated as a harmonic oscillator potential. Write the first four one-particle terms in the Hamiltonian that acts on the Slater
Find the eigenvalue of the operator \(e^{\alpha b}\) on single-particle states.
If the Hamiltonian acting on the occupation number states of a system can be written as \(\hat{H}=\sum_{i} t_{i} b_{i}^{\dagger} b_{i}+V \sum_{i
Consider a system with seven one-particle states, and three bosons in the system. How many states are there in the Fock space? How many multiparticle states are possible?
Find the eigenvalue of the fermionic operator \(\exp \left[\left(\sum_{i} a_{i}\right) \alpha\right]\) in the fermionic Fock space.
Show the details of the proof of the Schrödinger equation for the occupation number states for fermions, and show the resulting absence of a sign.
Calculate the occupation number Hamiltonian for a delta function two-body interaction \(V\left(\vec{r}_{i}-\vec{r}_{j}\right)=V_{0} \delta\left(\vec{r}_{i}-\vec{r}_{j}\right)\).
Explain physically the Coulomb occupation number Hamiltonian (56.52) in terms of Feynman diagrams.Data From Equation 56.52:- - 1272 - + 2m koko + * + at (56.52)
Consider bosons instead of fermions, and set up the corresponding problem for a self-consistent approximation.
Continue with this method to find the equivalent of the Hartree-Fock potential \(\hat{V}_{H-F}\) for bosons.
Given the Hartree-Fock equation, rewrite the Hartree-Fock Hamiltonian.
Calculate the Hartree potential for the helium atom in its ground state.
Calculate the two-particle exchange term in the Hartree-Fock approximation for the lithium atom in its ground state.
Write explicitly the Hartree-Fock equation for the 2-point Green's function, in the coordinate ( \(\vec{r}\) and spin \(\sigma\) ) representation.
Write explicitly, in the coordinate ( \(\vec{r}\) and spin \(\sigma\) ) representation, the Dyson or Bethe-Salpeter equation.
The nonabelian Berry phase, by its very definition, changes the wave function, even though we return to the initial state in terms of \(K(t)\) (in the abelian case, the wave function changes by a
Show the steps in the proof of the relation (58.12), for constructing an anyon from a statistical gauge field.Data From Equation 58.12:- U(Pi-q(F))U = p- q(i) - q(Fi), (58.12)
Consider the (01) component of the equation of motion \(F_{\mu v}=\frac{2 \pi}{k} \epsilon_{\mu v ho} J^{ho}\), coming from the relativistic Chern-Simons action with a source. What physical property
Show that the Chern-Simons action (without a source) is equal to a (3+1)-dimensional action of a gauge invariant, topological (metric independent, and with discrete values) type.
When describing the FQHE in the text, we gave the option of solving for the statistical gauge field \(a_{\mu}\) and obtaining a Chern-Simons-like action for the electromagnetic field \(A_{\mu}\), or
When discussing the FQHE anyon as a statistical gauge delta function flux added to the quasiparticles, we argued that for the electromagnetic field \(F_{\mu u}\) to be a delta function is unphysical
Show that the factor \(\left(z_{i}-z_{j}\right)^{m}\) in the Moore-Read wave function (also present in the phenomenological "Laughlin wave function" for a theoretical description of the FQHE) implies
Consider a central potential of the spherical-step type, \(V=V_{0}>0\) for \(r \leq R\) and \(V=0\) for \(r>R\), and a solution with energy \(E>V_{0}\) and given angular momentum \(l>0\). If the
The decomposition of the free wave at infinity, (46.15), seems counterintuitive since the lefthand side is certainly nonzero (in fact, naively, it is of order 1!) if \(\vec{k}\) is other than
If the \(f_{\vec{k}}\left(\vec{n}_{r}\right)\) in (46.21) were independent of \(\vec{n}_{r}\), would that contradict unitarity or not?Data From Equation 46.21:- u (r) = eik. + f (n) eikr r = Uinc (F)
For the case in exercise 1, calculate the total cross section and the S-matrix.Data From Exercise 1:-Consider a central potential of the spherical-step type, \(V=V_{0}>0\) for \(r \leq R\) and
Consider the wave \(\psi=A e^{i k r / r}+B e^{-i k r / r}\), with \(A, B=(2 \pi / i k) \delta\left(\vec{n}_{k} \pm \vec{n}_{r}\right)+a, b\), where \(a, b\) are real constants. Can it be understood
Calculate the generalization of \(G_{0}^{ \pm}\left(\vec{r}, \vec{r}^{\prime}\right)\) on the complex plane for energy, \(G_{0}\left(z ; \vec{r}, \vec{r}^{\prime}\right)\).
Calculate the equivalent of (46.69) for the generalization \(G_{0}\left(z ; \vec{r}, \vec{r}^{\prime}\right)\) in exercise 6.Data From Exercise 6:-Calculate the generalization of \(G_{0}^{
Calculate the first two terms in the Born series for a delta function potential, \(V(r)=-V_{0} \delta^{3}(\vec{r})\).
Calculate the differential cross section for scattering in the Born approximation for a potential \(V(r)=A / r^{2}\).
Describe physically how it is possible that the Born approximation to quantum mechanical scattering in a Coulomb potential gives the classical-scattering Rutherford formula.
Calculate the first two terms in the Born series for a potential \(V(r)=A /\left(r^{2}+a^{2}\right)\), with \(a=\) constant.
Write down explicitly the Lippmann-Schwinger equation for the T-matrix in the Yukawa case of \(V(r)=V_{0} e^{-\mu r} / r\).
Write down and solve the Lippmann-Schwinger equation for the T-matrix in the case of the delta function potential \(V=-V_{0} \delta^{3}(\vec{r})\).
Is there a domain of validity of the Born approximation in the case of the Coulomb potential?
Consider scattering onto a delta function potential, \(V=-V_{0} \delta^{3}(\vec{r})\), in the Born approximation. Calculate the partial wave amplitudes \(a_{l}(k)\).
Consider a spherical well potential \(V=-V_{0}\) for \(r \leq R\), and \(V=0\) for \(r>R\), in the Born approximation, and waves with \(E>0\). Calculate the phase shifts \(\delta_{l}(k)\) for
In the case in exercise 2, calculate \(S_{l}(k)\) and the differential cross section.Data From Exercise 2:-Consider a spherical well potential \(V=-V_{0}\) for \(r \leq R\), and \(V=0\) for \(r>R\),
If \(\delta_{l}(k)\) is real, what do you deduce about \(T_{l}(k)\) ?
Calculate the scattering length for the case in exercise 1.Data From Exercise 1:-Consider scattering onto a delta function potential, \(V=-V_{0} \delta^{3}(\vec{r})\), in the Born approximation.
Is relation (48.72) well defined for complex \(k\) ? Why?Data From Equation 48.72:- 2ik [ (k) (k; r) + F (k) , (k;r)], (48.72)
If \(\lim _{k \rightarrow 0} \mathcal{F}_{l}^{+}(k) / k^{2}\) is constant for \(l\) even, find the \(k \rightarrow 0, r \rightarrow \infty\) behavior with \(k\) of \(\phi_{l}(k ; r)\) for even \(l\).
Use the Lippmann-Schwinger equation for \(\psi\) to write an expression for \(\sigma_{l}\) in terms of integrals involving the potential \(V(r)\) and the wave function.
Show the details of going from (49.19) to (49.20).Data From Equation 49.19 and 49.20:- 2mik Ri(k;r) ji(kr) - So dr'r' ji(kr
In the case of the radial delta function potential, prove that the radial wave function \(R_{l}(k ; r)\) is (49.31) and the Jost solutions are (49.32).Data From Equation 49.31 and 49.32:- (+) (E;r,
Expand the optical theorem first in angular momentum \(l\) and then in the Born series, for both \(f(\theta=0)\) and \(\left\langle\vec{k}|\hat{T}| \vec{k}^{\prime}\rightangle\).
For a potential with finite range, consider \(V=V_{0}>0\) inside \(rV_{0}\), imposing normalizability at \(r=0\).
For the hard sphere, calculate \(\sigma_{l}\) at general \(k\).
Calculate the high-energy limit \((k \rightarrow \infty)\) for the partial wave cross section \(\sigma_{l}\) for a hard sphere, and their relative weight in \(\sigma_{\text {tot }}\).
For a step potential, with energy \(E
Find the first correction to the approximate relation (50.30) between the binding energy \(I\) of the bound state close to zero and the large scattering length \(a\).Data From Equation 50.30:-
Find the equivalent of the relation replacing (50.30) if we still have \(r_{0} \ll a\), but \(E=V_{0} / 2>0\) and very small \(\left(k r_{0} \ll 1\right)\).Data From Equation 50.30:- I=-Ebounds k
Extend the analysis in complex space leading to \(S_{l}(k), a_{l}(k), \delta_{l}(k)\) and coming from the existence of a bound state, from the similar analysis for \(l=0\).
In the complex \(k\) plane do we still have real \(\delta_{l}(k)\) ?
Write down the Jost functions \(\mathcal{F}_{l}^{+}(k)\) and \(\mathcal{F}_{l}^{-}(k)\) in the case of a single bound state.
Derive the normalization constant (50.57).Data From Equation 50.57:- N = nl -4K F(-ikn) d(-ik)/dk (50.57) I=Kn
Write down the approximate value for the partial wave S-matrix \(S_{l}(k)\) for the case of two resonances and two bound states.
Consider the case where among the partial wave cross sections \(\sigma_{l}\) only \(\sigma_{1}\) is at (or very near) a resonance. What can you say about \(\sigma_{\text {tot }}\) ?
Consider the case where \(S_{l}(k)\) has a single pole (resonance), but \(k_{2}\) is comparable with \(k_{1}\). Calculate \(\tan \delta_{l}\) and the equivalent of the Breit-Wigner formula in this
If we have two resonances (complex poles) for \(\sigma_{l}\) close to each other, and we are at resonance on the real \(k\) line, write down the wave function in terms of the physical interpretation
In the case in exercise 4, calculate the asymptotic wave function in the metastable region of time, and in the out region.Data From Exercise 4:-If we have two resonances (complex poles) for
In the case of the Levinson theorem, for \(n_{b}^{l}\), do we need to count only the poles of \(S_{l}(k)\) that are exactly on the imaginary line, or can they be slightly away from the imaginary line?
Consider that \(S_{l}(k)\) for all \(l \in \mathbb{N}\) has a single resonance pole, very close to the real line. If we consider now complex momentum instead, what can we learn about the Regge
Consider a one-dimensional system with potential \(V(x)=V_{0} /\left(x^{2}+a^{2}\right)\). Write the WKB approximation for the wave function for scattering, for energy \(0
Consider a three-dimensional system with central potential \(V(r)=V_{0} /\left(r^{2}+a^{2}\right)\). Write the WKB approximation for the wave function for scattering of given angular momentum \(l\),
For the case in exercise 2, calculate the cross section as a formal sum over \(l\).Data From Exercise 2:-Consider a three-dimensional system with central potential \(V(r)=V_{0}
For the same potential as above, \(V(r)=V_{0} /\left(r^{2}+a^{2}\right)\), calculate \(\Delta(b)\) for the eikonal approximation. If the potential turns off \((V(r)=0)\) for \(r \geq r_{0}\),
For a Yukawa potential, \(V(r)=V_{0} e^{-\mu r} / r\), write an approximate value for the cross section in the eikonal approximation.
If we have a Coulomb potential that turns off \((V(r)=0)\) for a large \(r \geq r_{0}\), calculate the total (approximate) cross section in the eikonal approximation, and compare with the exact value
Are there any resonances for Coulomb scattering? Check that the analytical properties of \(S_{l}(k)\) in this case are satisfied.
Consider the mapping of the black disk eikonal into the general, \(\delta_{l}(k)\), representation. Calculate the total inelastic cross section in this representation.
Calculate the differential cross section \(d \sigma(\theta) / d \Omega\) for the black disk eikonal, integrate it, and compare with the total inelastic cross section.
Consider the scattering of a projectile \(A\) (nonidentical to the target components) off a hydrogenoid atom in the Born approximation, involving a jump of the electron from the ground state to the
Consider a system of \(N=4\) fundamental particles. Write down the channels for the scattering of the various fragments. Taking the in channel as one where each of the two fragments have two
Describe the general theory of scattering (as in the text) for the case when the target is a single fundamental particle (and the projectile is composite), so both in and out channels have a
How do you write the inelastic differential cross section for the case where the in channel has two parallel projectiles, and the out channel has a single outgoing projectile (fragment) of general
Specialize the scattering of the identical particle formalism for the case of an electron scattering off a helium atom to describe the differential cross sections of various spin projections in terms
Does a wave function satisfy the Klein-Gordon equation? What can you deduce about the sign of the rest energy \(m c^{2}\) in the Dirac equation?
Find the Dirac equation in 1+1 dimensions (one space dimension) and find a representation for the matrices involved.
Show that if we consider \(\gamma_{5}=i \gamma^{0} \gamma^{1} \gamma^{2} \gamma^{3}\) together with \(\gamma^{0}\) and \(\gamma^{i}\), they form a Clifford algebra in five dimensions.
Write down the Klein-Gordon equation with coupling to electromagnetism.
Consider a shell model for a nucleus, with the potential for a nucleon being approximated by the Yukawa potential. Calculate the first relativistic correction to the Hamiltonian for the nucleon.
Check the missing steps in the relativistic corrections to the hydrogenoid atom.
Calculate the relativistic corrections to a hydrogenoid atom in a magnetic field.
Consider a two-level system, with basis \(|1angle,|2angle\), and in this basis, a Hamiltonian with elements \(\left(\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right)\). Use the first form of the
Use the Ritz variational method for the harmonic oscillator, with trial wave functions \(\psi_{1}(x)=\) \(e^{-y^{2} / 2}, \psi_{2}(x)=e^{-y^{2}}, \psi_{3}(x)=e^{-2 y^{2}}\), where \(y=x \sqrt{m
Use the practical variational method for the same harmonic oscillator ground state energy, with trial wave function \(\psi_{a}(x)=e^{-a y^{2}}\).
Repeat exercise 3 for the third state (second excited state) \(|3angle\) of the harmonic oscillator, namely, for \(\psi_{a}(x)=y^{2} e^{-a y^{2}}\).Data From Exercise 3:-Use the practical variational
Consider the potential \(V=k x^{2}+\alpha|x|^{3}\), and a trial wave function \(\psi(x ;a, b)=|y|^{a} e^{-b y^{2}}\). Find an estimate of the ground state energy (write down the equations for the
Fill in the details in the text for minimization with the trial wave function \(R(r)=A e^{-r / a}\) for the hydrogenoid atom.
Fill in the details in the text for minimization with the trial wave function \(R(r)=A /\left(r^{2}+b^{2}\right)\).
Write down explicitly the \(4 s\) and \(4 p\) atomic orbital wave functions.
Use the shell model and the Hund rules to show how the orbitals are populated with electrons in the elements \(\mathrm{C}, \mathrm{O}\), and \(\mathrm{Mg}\).

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