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physics
quantum mechanics a graduate course

Quantum Mechanics A Graduate Course 1st Edition Horatiu Nastase - Solutions

Consider the element \(\mathrm{O}\) (oxygen). Write down the LS and \(j j\) couplings for the electrons, and show explicitly that the splitting of energy levels is the same for each type of coupling.
Consider the element O. Write down explicitly the method of atomic orbitals for it (for electronelectron interactions), using the basis of only the filled orbitals.
Write down explicitly the method of molecular orbitals for the \(\mathrm{O}_{2}\) molecule.
Write down explicitly the LCAO method for the \(\mathrm{O}_{2}\) molecule, applied to each of the atomic orbitals in \(\mathrm{O}\).
Write down the Bohr quantization for the common electron in benzene, and find the energy levels as a function of the "radius" of the benzene molecule (the distance between the center and a \(\mathrm{C}\) atom).
If a nucleus is spinning around a given axis, assuming the liquid droplet model how would you modify the law \(R=r_{0} A^{1 / 3}\) from the static case?
Calculate the eccentricity, \(\Delta R / R\), of the spinning nucleus in exercise 1 , in a classical physics approximation.Data From Exercise 1:-If a nucleus is spinning around a given axis, assuming the liquid droplet model how would you modify the law \(R=r_{0} A^{1 / 3}\) from the static case?
If the (effective) central potential is replaced with an (effective) azimuthal potential, \(V(r, z)\), depending independently on the polar radius \(r\) in a plane and on the height \(z\) along the axis transverse to the plane, is it possible to have a shell model? Why? If so, consider the harmonic
In the case of a spherical square well with \(V=0\) for \(r \geq R\), write down the solutions in the regions \(r \leq R\) and \(r \geq R\), the gluing (continuity) conditions at \(r=R\), and the resulting quantization condition.
Consider the nuclear central potential \(\left(R_{2} \gg R_{1}\right.\), but not by too large a factor)What can you (qualitatively) infer about the modification of the energy levels with respect to the levels of the spherical harmonic oscillator approximation in the text? VN (r) = -Vo 1 - R -r/R
In the case in exercise 5, calculate the spin-orbit interaction correction. What can you (qualitatively) infer about the modification of the various states from the uncorrected case?Data From Exercise 5:-Consider the nuclear central potential \(\left(R_{2} \gg R_{1}\right.\), but not by too large a
Calculate the pairing energy for the two-particle delta function potential, for two nucleons in the ground state, \(1 s_{1 / 2}\).
Show that (44.11) is the solution to the Bloch equations (44.8).Data From Equation 44.11 and 44.8:- dt d dt d dt 1 N(t) = / [Q(t) (V(t) V* (t)) + iP(t)(V(t) + V* (t))] ih Q(t) = woP(t) + N(t)(V(t) V* (t)) (().A - P(t) = woQ(t) + = N(t)(V(t) + V*(t)), h (44.8)
Show that \(P(t), Q(t)\) in (44.13) are the corresponding solutions to the Bloch equations.Data From Equation 44.13:- Q(t) =Q'(t) cos wt + P'(t) sin wt P(t) = P'(t) cos wt - Q'(t) sin wt 2Vo P'(t) = P(0) cost + N(0). - Wo-w 20(0)] si sin ot Wo-w Wo-w2Vo Q'(t) = Q(0) + -P(0) sin t N(0)- wo-Q(0)
Consider an exponentially decaying and oscillating potential,for the interaction with classical radiation. What is the equivalent of Fermi's golden rule in this case? V(t) = (Voeiat + Voeiwi)eyt
Find a limit in which the first-order perturbation theory for a harmonic potential is consistent with the exact, but two-level, calculation of Section 44.1 (giving the same result).Data From Equation 44.1:- H12 (44.1) H21 H22 C2(1 d ith 1/4 (c)()) = (c)(0)) = ( dt C(t)) C(t) Hu
Consider a possible quantum term, of the type \(\alpha \vec{A}^{2}\), for light-light interaction in the case of quantized radiation. Analyze and interpret the resulting terms in the potential \(V(t)\) in terms of photons, as was done in the text for the linear term.
If there are no photons (no radiation field), \(N_{j}(\vec{k})=0\), how do we interpret the rate of spontaneous emission (what is the physical situation it describes, and what are its limitations)?
In deriving the Planck formula, we used the classical Boltzmann distribution for the atoms. Why is this allowed, given that we are considering quantum interactions?
Consider the potential \(V(x)=V_{0} e^{-\alpha x^{2}}\). Calculate perturbatively the first two terms in the solutions \(\psi_{1}(x)\left(\sim e^{i k x}\right.\) at \(x=-\infty\), incoming) and \(\psi_{2}(x)\left(\sim e^{i k x}\right.\) at \(x=+\infty\), outgoing), if \(E \leq V_{0}\).
Show that \(G_{2}\left(x, x^{\prime}\right)\) is also a Green's function for the Schrödinger operator, as is \(G_{1}\left(x, x^{\prime}\right)\), and that their difference is a solution of the homogenous Schrödinger operator.
Write down the Lippmann-Schwinger equation in the momentum representation, and the corresponding first few terms in its perturbative solution.
Consider a barrier potential, \(V=V_{0}\) for \(|x| \leq L / 2\) and \(V=0\) for \(|x|>L / 2\), and an energy \(E \leq V_{0}\). Calculate \(\psi_{1}(x), \psi_{2}(x)\), and the corresponding transfer matrix \(K\).
In the case in exercise 4, calculate the S-matrix and the Hermitian matrix \(\hat{\Delta}\), with \(\hat{S}=e^{i \hat{\Delta}}\).Data From Exercise 4:-Consider a barrier potential, \(V=V_{0}\) for \(|x| \leq L / 2\) and \(V=0\) for \(|x|>L / 2\), and an energy \(E \leq V_{0}\). Calculate
Substitute the two-magnon ansatz into the Schrödinger equation for the Heisenberg spin \(1 / 2\) spin chain, and find that the energy of the two-magnon state is just the sum of the energies of the single magnons, and that (pi) S(P1, P2)= (P1) (p2)+i (p2)-i' (p) = P (45.99)
Solve the Bethe ansatz equations for two magnons for the case \(n=0\) in the quantization condition (45.91), and write down the corresponding explicit wave functions.Data From Equation 45.91:- P1 + P2 = 2 n mod(2/L). (45.91) L
Consider the bipartite state in \(\mathcal{H}_{A} \otimes \mathcal{H}_{B}\)where \(|1angle,|2angle,|3angle,|4angle\) are orthonormal states.Calculate the mixed state obtained by taking the trace over system \(A\), or system \(B\). 14) = = = [(1) | 1) + (2) > 2) + (3) (4)+(2) (3)], (32.82)
If the states \(|1angle,|2angle,|3angle,|4angle,|5angle\) are (orthonormal) eigenstates of the Hamiltonian of energies \(E_{1}, E_{2}, E_{3}, E_{4}\), and \(E_{5}\), respectively, and at time \(t=0\) the statistical operator isthen find the time evolution of \(\hat{ho}\) at small times. p
Consider a thermodynamic system of \(N\) (of the order of the Avogadro number \(N_{A}\) ) spins 1/2. Calculate their classical entropy, for a (quasi-)microcanonical ensemble. Describe a quantum statistical operator \(\hat{ho}\) for the same system now using quantum mechanics, that gives the same
Consider \(N\) harmonic oscillators of frequencies \(\omega_{i}, i=1, \ldots, N\), connected to a reservoir of temperature. Calculate (in the canonical ensemble) the free energy and the heat capacity \(C_{V}\).
Consider the harmonic oscillators from exercise 4, connected to a reservoir of temperature \(T\) and chemical potential \(\mu\). Calculate (in the grand canonical ensemble) the thermodynamic potential \(\Omega\) and the heat capacity \(C_{V}\).Data From Exercise 4:-Consider \(N\) harmonic
Consider a distribution of free, nondegenerate, relativistic fermionic particles of mass \(m\), of arbitrarily large three-dimensional momentum. Calculate the number density and energy density as a function of temperature. Write down an explicit analytical form at large temperatures.
Consider a system \(A \cup B\), with total Hamiltonian diagonalized by states \(|1angle_{A} \otimes|1angle_{B},|2angle_{A} \otimes\) \(|2angle_{B},|3angle_{A} \otimes|4angle_{B},|4angle_{A} \otimes|3angle_{B}\), of energies \(E_{1}, E_{2}, E_{3}, E_{4}\), respectively. Calculate the thermal
Show the details of proving the formula (33.5) for the Shannon entropy in (33.6).Data From Equation 33.6:- H(X) k - p(x) log2 p(x). x=1 (33.6)
Show that the mutual information is zero if and only if, in the probability of occurrence of the message \(x\) given the message \(y, p(x, y)\), the presence of the message \(y\) is irrelevant (so we have independence of \(y\) ).
Show the details of the proof of the concavity of the von Neumann entropy.
Give simple examples of when the von Neumann entropy satisfies \(S(A+B)
Give an example of a finite quantum circuit made up of a very large number of infinitesimal unitary gates acting on the same two qubits (and calculate the circuit).
Prove the quantum teleportation relation (33.30).Data From Equation 33.30:- |4c|+AB = [10+AC|B+|+AC1|4B + |4)ac(i2)|4)b + |Q->AC3|4)B] (33.30)
If a quantum computer can factorize a large number in polynomial time, thus breaking banking and internet security encryption based on the same, does a quantum computer mean the end of banking safety? Explain. If not, what problems do you see that need solving?
Consider the following classical computation on six-qubit space, \((1,0,1,0,1,0) \rightarrow\) \((0,0,1,1,1,1)\). Choose your favorite universal set of gates, and find two examples of circuits (products of gates) giving this computation. Therefore deduce a bound on the complexity of the computation.
Consider the following quantum computation on five qubit space, from reference state \(\left|\psi_{R}\rightangle\) to final state \(\left|\psi_{T}\rightangle\),with tolerance \(\epsilon=1 / 10\). Using the universal set of gates in the text, find one example of a circuit (products of gates) giving
Consider a quantum computation on four-qubit space in the Nielsen approach. Find a basis \(\mathcal{O}_{I}\) for the computation, and write down explicitly the Schrödinger equation for \(U(s)\) in this basis.
Consider the function \(F\left(Y^{I}\right)=\sum_{I}\left|Y^{I}\right|\). Is it a good cost function (Finsler function)? Why?
Consider a Heisenberg spin chain Hamiltonian, \(H=-J \sum_{i} \vec{\sigma}_{i} \cdot \vec{\sigma}_{i+1}\), where \(\vec{\sigma}_{i}\) are the Pauli matrices at site \(i\) on a spin chain or, equivalently, on the data for a quantum computation, and the operator \(W=\sigma_{k}^{1}\), where \(k\) is a
Is there a classical limit of the measure of quantum chaos described here? How would you compare the classical and quantum chaos descriptions?
The conjectured bound (34.39) implies that at zero temperature \(\lambda_{L} \rightarrow 0\). Does that imply that the scrambling time \(t_{*} \rightarrow \infty\) ? Explain.Data From Equation 34.39:- 2 2kgT (34.39)
In the Schrödinger's cat Gedanken experiment, ignoring the issue of observers (which, admittedly, was what concerned Schrödinger), where would you simplify it, by eliminating pieces in its construction? Think then about how that relates to the quantum computation issue, as alluded to in the text.
At the end of decoherence, from the point of view of the system \(\mathcal{S}\), we no longer have a pure quantum state. But how is this result consistent with a classical picture (for instance, if we consider the Schrödinger cat Gedanken experiment)?
In the quantitative decoherence general model, calculate what will happen to \(\left\langle\phi\left|ho_{\mathcal{S}}\right| \phi\rightangle\) if we did not have \(\left\langle\epsilon_{i} \mid \epsilon_{j}\rightangle=\delta_{i j}\), and explain why that would be unsatisfactory.
Give an example of an integrable quantum mechanical model, and explain (without any calculation) why this is not expected to thermalize. One way to define an integrable quantum mechanical model is by saying that the three-point and higher-point scatterings can be reduced to just two-point
If the dimension of the Hilbert space \(D\) in (35.30) is infinite, is the formula still true, and is this still a GGE? Can we deduce something about the system's behavior then?Data From Equation 35.30:- Pa = exp m " m=1 (35.30)
Diagonalize the Hamiltonianwhere \(a_{i}, a_{i}^{\dagger}\) are harmonic-oscillator creation/annihilation operators. +aja H 2 #1 = { {a}a; +ao] + [(a; +a}) (am + a}.}]}}, - (35.59) i=1
The thermal particle creation in (35.58) arises from a state that was in a vacuum in region \(A\). Considering energy conservation, what can we deduce about the space? Can this process happen in Minkowski space? Why?Data From Equation 35.58:- a0bb10a=Bij ~ exp j j i (35.58) kBT
Consider a one-dimensional harmonic oscillator perturbed by a linear potential, \(\lambda \hat{H}_{1}=\lambda x\). Calculate the first-order perturbation theory corrections to the energy and wave functions, and compare with the exact solution for \(\hat{H}_{0}+\lambda \hat{H}_{1}\) (which is
Consider a hydrogenoid atom perturbed by a decaying exponential potential \(\lambda \hat{H}_{1}=\lambda e^{-\mu r}\). Calculate the first-order perturbation theory corrections to the energy and ground state wave function.
Write down an explicit formula for the second-order perturbation contribution to the ground state energy in exercise 2 , without calculating the terms and summing them.Data From Exercise 2:-Consider a hydrogenoid atom perturbed by a decaying exponential potential \(\lambda \hat{H}_{1}=\lambda
Calculate the first-order perturbation contribution to the energy of the first excited state of the hydrogenoid atom in exercise 2.Data From Exercise 2:-Consider a hydrogenoid atom perturbed by a decaying exponential potential \(\lambda \hat{H}_{1}=\lambda e^{-\mu r}\). Calculate the first-order
Formally, to obtain the perturbative solution (36.59), we did not need the self-consistent equation (36.58), we just needed to expand the definition of \(G(z)\) in \(\lambda\) : show this.Data From Equation 36.59:- ="o(o)". n=0 (36.59)
Prove the relation (36.73), used in the analysis of the Stark effect.Data From Equation 36.73:- (l, ml cos 011, m) = (1-1, m| cos 0|l, m) = 12-m 4/12 - 1' (36.73)
Use first-order perturbation theory to find the Stark effect splittings for the second excited state of the hydrogenoid atom, \(n=3\).
Consider \(\hat{H}_{0}\) corresponding to the hydrogen atom, with the initial state \(i\) being the ground state, and \(\lambda H_{1, m i}=\lambda e^{-m A+\tilde{\lambda} t}\), for \(t
Consider \(\hat{H}_{0}\) corresponding to the hydrogen atom, the initial state \(i\) being the ground state, and \(\hat{H}_{1}=K r\), with \(K\) constant, a perturbation introduced suddenly at \(t=0\). Calculate the transition probability to the state \(n=2, l=0\) as a function of time. What
Prove the formula (37.32) for the limit giving the delta function.Data From Equation 37.32:- 1 sin ax a00 ax lim = d(x), (37.32)
Consider an electron in an atom, with a potential that can be approximated by a (radial) square well of radius \(r_{0}\) above zero, in its ground state \(i=0\). Introduce a small perturbation \(H_{m i}=H=\) constant. Calculate the rate for the electron to transition out of the square well and into
Calculate the differential cross section for scattering from a delta function potential, \(V=V_{0} \delta(\vec{r})\).
Prove the formula (37.57).Data From Equation 37.57:- 2z Ze 1 V(p) = lim 1 1-0 2hh p/h + 12 (37.57)
Consider the adiabatic introduction of the perturbation \(H_{1}\) withfor \(t (t) = 1+2+2>
In general, can we have a zero-energy shift if we have a nonzero decay width (finite lifetime)? How about a zero decay width (infinite lifetime) for a nonzero energy shift?
In a transition, if the density of final states \(ho\left(E_{n}\right)\) is an increasing exponential, \(ho\left(E_{n}\right)=A e^{\alpha E_{n}}\), \(\alpha>0\), what does the (physical) condition of a finite energy shift imply for the transition Hamiltonian matrix \(H_{1, i n}\), if \(E_{i}>0\) ?
If \(H_{n i}\) is independent of \(E_{n}=E_{i}\), what physical condition can we impose on \(ho\left(E_{n}\right)\) ?
Argue for the energy-time uncertainty relation, \(\Delta E \Delta t \sim \hbar\), on the basis of the Breit-Wigner distribution.
Calculate the interaction picture operator \(\hat{U}_{I}\left(t, t_{0}\right)\) and \(b_{n}(t)\) for the interaction in the sudden approximation.
Show that the first-order formula for time-dependent perturbation theory can be recovered from (38.41).Data From Equation 38.41:- bn(t) = 8ni + (-i/h) {mk) S=1 s! dt. dts T exp (ti to)] H.,S.m H,s,nm, (11) exp Em (t1-to) (cxp + Em (12-10)] H.8mm (12) expm (210)]). [+ to) m2 i (exp(-10)]()
Show that the second-order formula for time-dependent perturbation theory can be recovered from (38.41).Data From Equation 38.41:- bn(t) = 8ni + (-i/h) {mk) S=1 s! dt. dts T exp (ti to)] H.,S.m H,s,nm, (11) exp Em (t1-to) (cxp + Em (12-10)] H.8mm (12) expm (210)]). [+ to) m2 i (exp(-10)]()
Consider the first relativistic correction to the energy of a particle, and introduce the coupling to the electromagnetic field. Write down the resulting extra terms that are linear and quadratic in \(\vec{A}\).
For the terms considered in exercise 1, find expressions for the first relativistic corrections to the linear and quadratic Zeeman effects (ignore any extra couplings to spin).Data From Exercise 1:-Consider the first relativistic correction to the energy of a particle, and introduce the coupling to
Show that the splittings of energy levels according to the Zeeman or Paschen-Back effects lead to the same number of lines.
Calculate the transition element \(H_{1, f i}\) for the interaction of a hydrogenoid atom with a monochromatic electromagnetic wave, in the electric dipole approximation, from the \(n=1, l=\) 0 state to the \(n=2, l=1\) state.
Consider a hydrogenoid atom in the state \(n=3, l=2, m=0\). What are the possible transitions induced by a monochromatic electromagnetic wave, to both first and second order?
Calculate the absorption cross section for the transition in exercise 4.Data From Exercise 4:-Calculate the transition element \(H_{1, f i}\) for the interaction of a hydrogenoid atom with a monochromatic electromagnetic wave, in the electric dipole approximation, from the \(n=1, l=\) 0 state to
Calculate the differential cross section for the photoelectric effect for a monochromatic electromagnetic wave of energy of \(14 \mathrm{eV}\) incident on a hydrogen atom in the ground state.
Use the WKB method, and Bohr-Sommerfeld quantization, to estimate the matrix element \(\left\langle 2\left|\exp \left(-a \partial_{x}\right)\right| 1\rightangle\) for a harmonic oscillator.
Consider the potential \(V=-A /\left(x^{2}+a^{2}\right)\), and states \(|1angle,|2angle\), the first two eigenstates in this potential. Using the methods in the text, estimate the transition probability \(f_{12} \equiv\langle 1|x| 2angle\).
Can the instanton be associated with the motion of a real particle, albeit nonclassical (perhaps a quantum path)?
Consider a theory with only two minima of the potential, at \(x_{1}\) and \(x_{2}\). Sum the series of noninteracting multi-instantons that start at \(x_{1}\) and end at \(x_{2}\), having been through a continuous motion.
Consider the Higgs-type potential \(V=\alpha\left(x^{2}-a^{2}\right)^{2}\), and a transition between the two vacua (minima of the potential) \(x_{1}=-a\) and \(x_{2}=+a\). Calculate the transition probability in the classical one-instanton approximation.
Consider the periodic potential \(V=A \cos ^{2}(a x)\), and a transition between the first two vacua (minima) of \(x>0\). Calculate the transition probability in the classical one-instanton approximation.
Calculate the (formal) fluctuation correction to the instanton calculation in exercise 6.Data From Exercise 6:-Consider the periodic potential \(V=A \cos ^{2}(a x)\), and a transition between the first two vacua (minima) of \(x>0\). Calculate the transition probability in the classical
Consider a helium atom with one electron in the \(n=1, l=0\) state, and another in the \(n=2, l=0\) state. Calculate the energy of the ortho-helium state (the lowest energy state for these quantum numbers), using the two approximations in the text (Approximation 1 and Approximation 2).Data From
Consider a helium atom with two electrons in the \(n=2, l=0\) state. Calculate their ground state, using the two approximations in the text.
Apply the variational method (Approximation 3) to the case in exercise 1.Data From Exercise 1:-Consider a helium atom with one electron in the \(n=1, l=0\) state, and another in the \(n=2, l=0\) state. Calculate the energy of the ortho-helium state (the lowest energy state for these quantum
Apply the variational method (Approximation 3 ) to the case in exercise 2.Data From Exercise 2:-Consider a helium atom with two electrons in the \(n=2, l=0\) state. Calculate their ground state, using the two approximations in the text.Data From Approximation 3:- We can use a variational method to
Argue that, for the \(\mathrm{H}_{2}\) molecule, \(E_{+}(R)-2 E_{0}\) tends to infinity from below, and \(E_{-}(R)\) tends to infinity from above.
Consider a potential \(\mathrm{H}_{3}\) molecule, in an equilateral triangle of side \(R\). Give an argument why this molecule would be unstable.
If we have four \(\mathrm{H}\) atoms, write down the Hamiltonian, and describe two potentially stable configurations.
The relativistic Lagrangian for particle is \(L=-m c^{2} \sqrt{1-\vec{v}^{2} / c^{2}}\). Does minimal coupling at the classical level based on it work? Justify.
Write down formally the quantum level version for exercise 1 , without bothering about what \(\psi\) means now.Data From Exercise 1:-The relativistic Lagrangian for particle is \(L=-m c^{2} \sqrt{1-\vec{v}^{2} / c^{2}}\). Does minimal coupling at the classical level based on it work? Justify.
Consider a conductor forming a closed loop \(C\) in a magnetic field, with nonzero flux \(\Phi=\) \(\int_{S} \vec{B} \cdot d \vec{S}\) through it. Show that there is a nonzero probability current loop \(\oint_{C} d \vec{l} \cdot \vec{I}\).
In the London-London theory, show that by writing down the energy of the magnetic field and the kinetic term for the superconducting electrons, we obtain the free energyand from it we obtain the London equation \(\lambda_{L}^{2} \vec{abla}^{2} \vec{B}=\vec{B}\). f = 2440 5600 (B+)2rdr, (22.64)

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