- Prove that in Eq. (34.1) we may take as independent fermion fields $\psi_{\mathrm{L}}$ and $\psi_{\mathrm{L}}^{\mathrm{c}}$, instead of $\psi_{\mathrm{L}}$ and $\psi_{\mathrm{R}}$, because the charge
- Repeat Example 34.3 to relate the quark and leptonic charges for the $\mathbf{5}$ and $\mathbf{1 0}$ representations of Eq. (34.2).Data from Eq. 34.2Data from Example 34.3Data from Table 9.1 5(3,
- The anomaly $A(R)$ of a fermion representation $R$ is given by \[\operatorname{Tr}\left(\left\{T^{a}(R), T^{b}(R)\right\} T^{c}(R)\right)=\frac{1}{2} d^{a b c} A(R)\]where $T^{a}(R)$ is a
- The normal (not accidental; see Box 29.2) degeneracies in a quantum system result from symmetry. Show that if a Hamiltonian $H$ is invariant under transformation by a unitary symmetry operator $S$,
- Construct the braid group products(a)(b)using the algorithm of Fig. 29.16 .Data from Fig. 29.16
- Show that the two braids are mutual inverses under braid multiplication. XX XX
- For states $|\alphaangle$ and $|\betaangle$, define time-reversed states by $|\tilde{\alpha}angle=\Theta|\alphaangle$ and $|\tilde{\beta}angle=\Theta|\betaangle$. Show that the overlap of the
- Demonstrate that the operator $\Theta=i \sigma_{2} \mathscr{K}$ defined in Box 29.1 meets all the criteria given there for a fermion time-reversal operator: it preserves the norm of a wavefunction,
- Reproduce the plots in Fig. 29.10 by deriving formulas for the eigenvalues and eigenfunctions of the Hamiltonian (29.4). Hint: See the solution of Problem 14.14 .Data from Fig. 29.10 4 2- E OF -2
- Prove Kramers' theorem (Box 29.1 ) for spin- $\frac{1}{2}$ fermions by showing that a state and its time reverse are degenerate and orthogonal, so they are two independent states with the same
- Prove that for $6 J$ symbols\[\left\{\begin{array}{lll}a & b & c \\d & e & 0\end{array}\right\}=\frac{(-1)^{a+b+c}}{\sqrt{(2 a+1)(2 b+1)}} \delta_{a e} \delta_{b d}\]Note that since the $6
- Prove that\[\left\{\begin{array}{ccc}j_{1} & j_{2} & J \\j_{1}^{\prime} & j_{2}^{\prime} & J^{\prime} \\k & k^{\prime} & 0\end{array}\right\}=\frac{(-1)^{j_{2}+j_{1}^{\prime}+k+J}}{\sqrt{(2 J+1)(2
- For orbital angular momentum $L$, spin angular momentum $S$, total angular momentum $J$, and projection $M$ of the total angular momentum, use the Wigner-Eckart theorem to express $\left\langle L S J
- Two-body matrix elements for particles moving in central potentials are important in many areas of physics. These typically involve the matrix elements of Legendre polynomials, which can be written
- Evaluate the matrix element $\left\langle j_{1} j_{2} J\left|T_{k q}(1)\right| j_{1}^{\prime} j_{2}^{\prime} J^{\prime}\rightangle$, where the tensor operator $T_{k q}(1)$ operates only on the part
- Use Eq. (30.12) to evaluate the reduced matrix element of a spherical harmonic between states that are $l-s$ coupled to good total angular momentum $j$; thus obtain Eq. (30.13). The reduced matrix
- Use tensor methods to evaluate the reduced matrix element of the spin-orbit interaction \(\left\langle J\|\boldsymbol{l} \cdot \boldsymbol{s}\| J^{\prime}\right\rangle\) between states of good
- Shell models are important in various fields of physics. Consider a shell of fermions consisting of $(2 j+1)$ degenerate levels of angular momentum $j$, with each level labeled by a projection
- For the quasispin model of Problem 31.1 , find the eigenvalues of $s_{0}^{(m)}$ for the levels labeled by $m$. Show that the system has a total quasispin $S$ that is the vector sum of quasispins for
- Show that for a Hamiltonian of the form\[H=-G \sum_{m, m^{\prime}>0} a_{m^{\prime}}^{\dagger} a_{-m^{\prime}}^{\dagger} a_{-m} a_{m}\]the energy eigenvalues for the quasispin model of Problem 31.1
- Seniority quantum numbers typically measure how many fermions are in some sense "not paired" with another fermion. For the quasispin model of Problem 31.3 , define the Racah seniority $v$
- Consider a system described by the quasispin model of Problem 31.3 for $G>0$ with two identical fermions in a single $j$-shell. Show that the allowed seniorities are $v=0,2$, the allowed angular
- Prove that the binding energy of the ground state for the quasispin model in Problem 31.3 is linear in the number of pairs of particles $N$ for small $N$.Data from Problem 31.3Show that for a
- Use Eqs. (31.7)-(31.12) to verify the irreducible representations, quantum numbers, and spectrum of Fig. 31.5 . Data from Fig. 31.5Data from Eqs. (31.7)-(31.12) 72 8 42 20 0 6 CO K = 0 + 20 K = 0
- Highlight the propensity of cuprate antiferromagnetic Mott insulator states to condense a superconductor in the presence of small hole doping by showing that even the AF Mott insulator limit of SU(4)
- Show that the AF Mott insulator symmetry $\mathrm{SU}(4) \supset \mathrm{SO}(4)$ described in Section 32.3 .5 is locally isomorphic to $\mathrm{SU}(2) \times \mathrm{SU}(2)$, if new generators are
- Derive the commutator $\left[Q_{i}, Q_{j}\right]=i \epsilon_{i j k} Q_{k}$ for the charge defined in Eq. (33.4). Use the charge (33.4) to write the commutator, displaying explicit matrix
- Prove that the charges (33.7) obey the commutators in Eq. (33.8).Data from Eq. 33.7Data from Eq. 33.8 Q = Q = Qi-Qis = QR = Q Qi + Qis = 2 2
- Show that the generators of the algebra (33.8) are related by parity. For a Dirac wavefunction the action of parity is $P \psi(\boldsymbol{x}, t) P^{-1}=\gamma_{0} \psi(-\boldsymbol{x}, t)$, up to a
- Verify that the potential $V(\pi, \sigma)$ can be written as Eq. (33.11), and that if $\epsilon=0$ and the symmetry is implemented in the Wigner mode the masses for the $\pi$ and $\sigma$ fields are
- Demonstrate that the shift operators \(\tilde{A}_{i j}\) obey the commutation relations (10.8).Data from Eq. 10.8 [j, kl] = 8jk 8kj
- Prove that the 3D harmonic oscillator orbital angular momentum operators are given by \(\boldsymbol{L}=i \boldsymbol{a} \times \boldsymbol{a}^{\dagger}\). Show that the components \(L_{k}\) obey the
- Calculate the non-vanishing matrix elements of the six SU(3) raising and lowering operators \(U_{ \pm}, V_{ \pm}\), and \(T_{ \pm}\)between states of the octet representation. The required
- Beginning with Eq. (11.16), prove thatData from Eq. 11.16Data from Eq. 11.21 where we have defined D8 = - 3 2 F = FiFi T = F + F + F Y = F8. 3 Show that this leads to Eq. (11.21) with the Gell-Mann,
- Find the commutators of the spherical operators \(a_{0, \pm 1}^{\dagger}\) defined in Eq. (11.23) with the angular momentum operators. Thus show that they transform as rank-1 spherical tensors under
- Show that in 3D the translation operators \(P_{j}\) and rotation operators \(L_{j}\) given by Eq. (12.9) generate the non-abelian Lie group \(\mathrm{E}_{3}\), with the commutators (12.10).Data from
- Prove that the translations form an abelian invariant subgroup of the euclidean group \(\mathrm{E}_{2}\) by deriving the result (12.19).Data from Eq. 12.19 g(b,)T(a)g(b,) = T (R()a).
- Show that (12.20) gives the momentum content of \(R(\phi)\left|\boldsymbol{p}_{0}\right\rangle\).Data from Eq. 12.20 PR() Po) = R()\Po) Pk-
- Prove the useful identity employed in Eq. (14.20) that \(e^{i L \phi}=\cosh \phi+i L \sinh \phi\). Expand the exponential in a power series and compare the odd and even terms to the power series
- (a) Verify the entries given in Table 5.2 for the multiplication table of \(\mathrm{C}_{3 \mathrm{v}}\).(b) Show that the multiplication table for \(\mathrm{C}_{3 \mathrm{v}}\) can be put into one to
- Find the classes and their members for \(\mathrm{C}_{3 \mathrm{v}}\) as in Section 2.11 by forming for each group element \(q\) the conjugate elements \(g_{i}^{-1} q g_{i}\) for all elements
- (a) Demonstrate that Eq. (6.10) defines a generator of \(\mathrm{SO}(2)\) by examining the \(2 \mathrm{D}\) rotation matrix (6.3) for an infinitesimal rotation \(d \phi\).(b) Show that Eqs. (6.3) and
- Use Table 7.1 and Theorems 7.1 -7.2 to construct root diagrams for the rank-2 compact algebras \(\mathrm{SU}(3)\) and \(\mathrm{SO}(5)\).Data from Table 7.1Data from Theorem 7.1If α is a
- For an operator \(A=a_{\mu} X_{\mu}\) corresponding to a linear combination of generators \(X_{\mu}\) for a Lie algebra, use Eq. (7.10) and the Jacobi identity (3.6) to prove
- For coordinates \(\left(x^{1}, x^{2}\right)\) and metric \(g=\operatorname{diag}\left(g_{11}, g_{22}\right)\), the Gaussian curvature isFor a sphere with coordinates defined in the following
- Consider the holonomic basis defined in Box 26.1 . Using that the tangent vector for a curve can be written \(t=t^{\mu} e_{\mu}=\left(d x^{\mu} / d \lambda\right) e_{\mu}\), show thatThus, \(g_{\mu
- The Lie bracket of vector fields \(A\) and \(B\) is defined as their commutator, \([A, B]=\) \(A B-B A\). The Lie bracket of two basis vectors vanishes for a coordinate basis but not for a
- Prove the result of Eq. (26.10) that a path-dependent representation of a gauge group is sensitive to a gauge transformation only at the endpoints of the path.
- Demonstrate that for a closed path \(\operatorname{Tr} U_{\gamma}(x, x)\) is gauge invariant, where \(U_{\gamma}\left(x_{0}, x_{1}\right)\) is defined by Eq. (26.9).
- In Eq. (26.7), invert \(A^{\mu}(x)=A_{i}^{\mu}(x) \tau^{i}\) to obtain \(A_{i}^{\mu}(x) \equiv 2 \operatorname{Tr}\left(\tau_{i} A^{\mu}\right)\).
- Show that the vector potentials given in Eq. (27.1) imply the magnetic fields given in Eq. (27.2) by evaluating \(\boldsymbol{B}=\boldsymbol{abla} \times \boldsymbol{A}\) in the cylindrical
- Use Stokes' theorem [Eq. (27.5)] to prove that Eq. (27.4) leads to Eq. (27.6).Data from Eq. 27.4Data from Eq. 27.5Data from Eq. 27.6 = z ) ); A dr. - SA dr) = f A dr.
- Demonstrate that in the Aharonov-Bohm effect the scalar function \(\chi\) corresponding to the vector potential \(\boldsymbol{A}=\boldsymbol{abla} \chi\) outside the solenoid is given by Eq.
- Use the results of Table 24.1 to show formally that for the Aharonov-Bohm effect the mapping discussed in Section 27.1 .4 from the electromagnetic gauge group manifold \(\mathrm{U}(1)\) to the plane
- Show that solution of the eigenvalue problem \(H|\psiangle=E|\psiangle\) for the Hamiltonian (27.28) gives the eigenvalues (27.29) and the eigenfunctions (27.30).Data from Eq. 27.28Data from
- Prove that for a spin- \(\frac{1}{2}\) particle in a magnetic field, Eqs. (27.29) and (27.30) imply the Berry phase (27.32).Data from Eq. 27.29Data from Eq. 27.30Data from Eq. 27.32 1 E(B)=B+B + B
- Show that Eqs. (27.13) and (27.11) imply Eq. (27.14). Writeand evaluate the resulting derivatives of products.Data from Eq. 27.13Data from Eq. 27.14 a2 a a (R) |n, R) == OR aR (D(R) , R))) AR
- Prove using the definitions (27.16) that Eq. (27.15) is equivalent to Eq. (27.17), which has the form of a gauge coupling to a vector potential \(\boldsymbol{A}_{n}(\boldsymbol{R})\).Data from
- Show that under a local gauge transformation \(|n, \boldsymbol{R}angle \rightarrow e^{i \chi(\boldsymbol{R})}|n, \boldsymbol{R}angle\), the Berry connection \(\boldsymbol{A}_{n}(\boldsymbol{R})\) is
- Show that the Berry curvature (27.24) can also be written as\[\Omega_{\mu v}^{n}(\boldsymbol{R})=i\left(\left\langle\partial_{\mu} n(\boldsymbol{R}) \mid \partial_{v}
- Demonstrate that both the Berry phase \(\gamma_{n}\) and the Berry curvature \(\boldsymbol{\Omega}_{n}(\boldsymbol{R})\) are invariant under a local gauge transformation.
- Show that under a local gauge transformation (14.5) the vector potential A and the form of the wavefunction (28.7) are changed, but no observable is affected.Data from Eq. 28.7Data from Eq. 14.5 4nk
- Evaluate the formula for the Gauss-Bonnet theorem in Box 28.3 for a 2-sphere and show that this leads to the usual relation for the area of a sphere. The local curvature for a 2-surface is the
- (a) Show for a 2D Hall bar of length \(L\) and width \(w\) that \(j=\sigma E\) (where \(j\) is the current density, \(E\) is the electric field, and \(\sigma\) is the conductivity) is equivalent to
- Verify the commutation relations for the \(\mathrm{H}_{4}\) algebra displayed in Eq. (21.7).Data from Eq. 21.7 [, at ] = at [a,1]=0 [n, 1] = 0 [a,at]= 1 [n, a]=-a, [a, 1] = 0.
- For the coherent state of atoms in Section 21.4 , prove thatstarting from Eq. (21.29) and that \(J_{ \pm} \equiv J_{1} \pm i J_{2}\).Data from Eq. 21.29Data from Section 21.4...... () = exp : 0
- Show that the fermion operator set \(\left\{a, a^{\dagger}, a^{\dagger} a-\frac{1}{2}\right\}\) obeysand that this is equivalent to the \(\mathrm{SU}(2)\) Lie algebra of Eq. (3.18).Data from Eq. 3.18
- Construct the Hilbert space corresponding to the Lie algebra for a single fermion in Problem 21.3 . Remember that for a fermion the Pauli principle must be obeyed, which greatly restricts allowed
- For the single-fermion example worked out in Problems 21.3 and 21.4 , take as a reference state the minimal weight \(\mathrm{SU}(2)\) state \(\left|\frac{1}{2}-\frac{1}{2}\rightangle\) corresponding
- Show that the coset representative is\[\Omega(\xi)=e^{\xi J_{+}-\xi^{*} J_{-}}=e^{\xi a^{\dagger}-\xi^{*} a}\]for the generalized coherent state approximation corresponding to the single-fermion
- Show that the coherent state corresponding to the single-fermion problem worked out in Problems 21.3 through 21.6 iswhere \(\theta\) and \(\phi\) are angular variables parameterizing a sphere
- Prove that the projection operatordefined in Box 22.3 commutes with the rotation operator \(D(\Omega)\). Use the property \(D^{\dagger}(\Omega)=D(-\Omega)\) given in Eq. (6.35a).Data from Eq.
- Show that a variational calculation with a BCS wavefunction as the variational state,where \(\hat{H}\) is the Hamiltonian and \(\lambda\) is the variational parameter, leads to \(\lambda=d E / d N\),
- Show that the transformation (22.29) can be inverted to givefor the bare fermion operators \(\left\{c, c^{\dagger}\right\}\) in terms of the quasiparticle operators \(\left\{\alpha,
- Demonstrate that the Bogoliubov quasiparticle creation and annihilation operators obey the anticommutators \(\left\{\alpha_{k}, \alpha_{k^{\prime}}\right\}=\left\{\alpha_{k}^{\dagger},
- Show that the mean square deviation of the particle number from the actual particle number for a BCS wavefunction is given bywhere \(\hat{N}\) is the particle number operator and
- Show that the parity operator \(\Pi\) is its own inverse and is hermitian, so it is unitary.
- Prove that the parity operator \(\Pi\) does not commute with the position operator \(\hat{x}\) but it does commute with \(\hat{x}\)2 .
- Discuss the parity selection rule for electric dipole transitions of a single-electron atom (the Laporte rule). The electric dipole matrix element is \(\left\langle n^{\prime} l^{\prime}
- This problem and Problem 23.2 give a qualitative feeling for the difference between a phase dominated by classical thermal effects and one dominated by quantum effects by considering the
- Use the classical and thermal velocities derived in Problem 23.1 to estimate the ratio of the corresponding "quantum pressure" and classical "thermal pressure." Assume thermal electrons to obey an
- Assume a 2D lattice with islands of superconductivity surrounded by regions of insulating behavior. Within each SC island patch \(i\), assume a set of electron Cooper pairs having the same
- Show that the two terms in the Ising model Hamiltonian (23.9) do not commute and thus represent competing, incompatible tendencies in the corresponding system.Data from Eq. 23.9 H=-(80+001),
- Consider a set M={a, b, c, d, e} and the collection of subsetswhere \(\emptyset\) is the empty set. Prove that \(\tau\) defines a topology on the set \(M\). T= (0, M, {a}, {c, d), {a, c, d}, {b, c,
- Prove that an interval without endpoints is homeomorphic to the real number line \(\mathbb{R}\). Thus, boundedness is not a topological invariant. Take \(X=\left(-\frac{\pi}{2},
- What is the first homotopy group of a two-dimensional torus \(T^{2}\) ? For the 2-torus, \(T^{2}=S^{1} \times S^{1} .
- Using an open covering corresponding to the family of concentric open diskswhere \(\alpha=1,2, \ldots\), show that the open unit disk of Fig. 24.5 (b) is not compact.Data from Fig. 24.5 (b) Sa =
- Show that the winding number \(Q\) of Eq. (24.7) gives \(Q=n\) for the mapping (24.6).Data from Eq. 24.6Data from Eq. 24.7 An() = no,
- From Problems 2.9 and 2.11, the group D2= {e, a, b, c} has a factor (quotient) group with respect to the abelian invariant subgroup \(H=\{e, a\}\),\[\mathrm{D}_{2} / H=H+M=\{e, a\}+\{b, c\}\]with a
- Are the following homeomorphic? (a) A closed interval and an open interval for the real numbers \(\mathbb{R}\) ? (b) A parabola and a hyperbola? (c) A circle \(S^{1}\) and \(\mathbb{R}\) ?
- Sketch the results of path multiplications (indicated by \(\times\) ) for these examples: O (a) (b) (c) where each loop is defined in the same 2D euclidean plane with a single hole.
- Consider the isospin subgroup chain, \(\mathrm{U}(2) \supset \mathrm{U}(1)_{B} \times \mathrm{SU}(2) \supset \mathrm{U}(1)_{B} \times \mathrm{U}(1)_{T_{3}}\), where subscripts distinguish the
- Show that \(P_{0}^{2}\) defined in Eq. (20.18) and \(N_{z}\) defined in Eq. (20.7c) are equivalent. Thus the expectation value of either serves as an antiferromagnetic order parameter.Data from Eq.
- Show that for graphene in a magnetic field, \(P_{0}^{0}\) defined in Eq. (20.18) satisfies Eq. (20.21). Thus, \(P_{0}^{0}, S_{0}\), and \(n\) defined in Eq. (20.20) can serve as number operators.Data
- The AF order parameter \(N_{z}\) is related to the coherent state order parameter \(\beta\) by \(\left\langle N_{z}\right\rangle=2 \Omega\left|b_{2}\right|\left(f-\beta^{2}\right)^{1 / 2} \beta\),
- The Euler-Lagrange equation (16.14) is used in a field theory context in this chapter, but it is applicable to a broad range of problems. Show that inserting a Lagrangian \(L(x, \dot{x})=\frac{1}{2}
- Show that for the Lagrangian density (16.7) of a complex scalar field, the field equation (16.14) reduces to the two Klein-Gordon equations given by Eq. (16.8).Data from Eq. 16.7Data from Eq.
- Poincaré invariance requires that the action of a scalar field be unchanged under an infinitesimal spacetime translation \(x_{\mu} \rightarrow x_{\mu}^{\prime}=x_{\mu}+a_{\mu}\). Show that this
- Show that Eq. (16.10) is invariant under global \(\mathrm{U}(1)\) rotations \(\psi(x) \rightarrow e^{i \alpha} \psi(x)\), where \(\alpha\) is assumed to be independent of the spacetime coordinate
- Show that the Lagrangian density (16.30) is invariant under \(\mathrm{U}(1)\) phase rotations, find the corresponding conserved Noether current, and show that the conserved current is equivalent to