Questions and Answers of Modern Physics

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

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