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Questions and Answers of Modern Physics

Use Young diagrams to find the \(\mathrm{SU}(2)\) isospin content of the flavor \(\mathrm{SU}(3)\) irreducible representation \((1,2)\).
Use the methods to construct a proton flavor-spin wavefunction that is symmetric with respect to flavor-spin exchange.
If we assume a non-relativistic model, the magnetic moment of a point quark is given by \(\mu_{i}=q_{i} / 2 m_{i}\), where \(q_{i}\) is the charge and \(m_{i}\) the effective mass of quark \(i\).
Construct a quark wavefunction for the neutron and proton, and use the results of Problem 9.8 to show that for the ratio of magnetic moments for protons and neutrons, \(\mu_{p} /
Use Young diagrams to obtain the SU(6) Clebsch-Gordan series \(\mathbf{6} \otimes \mathbf{6} \otimes \mathbf{6}=\mathbf{5 6} \oplus\) \(\mathbf{7 0} \oplus \mathbf{7 0} \oplus \mathbf{2 0}\) of Eq.
Show that the if space, spin, and flavor are assumed to be the operative degrees of freedom for particles in the baryonic \(\mathbf{1 0}\) of Fig. 9.1 (d), the ground states for the spin-
Show that the set of permutations on \(n\) objects forms a group with a multiplication operation defined as the application of two successive permutation operations.
Verify that the mapping \(\{e,(123),(321)\} \rightarrow 1\) and \(\{(12),(23),(31)\} \rightarrow-1\) preserves the group multiplication operation for the permutation group \(\mathrm{S}_{3}\).
Sketch the Young tableaux for the permutation group \(\mathrm{S}_{5}\) and find their dimensionalities using the hook rule.
Use counting of Young tableaux to find the dimensionalities of the irreps for the group S4.
For the irreps \(\Gamma^{(1)}, \Gamma^{(2)}\), and \(\Gamma^{(3)}\) of the group \(\mathrm{S}_{3}\), determine the irrep content of the nine direct products \(\Gamma^{(n)} \otimes \Gamma^{(m)}\).
Write the Young operators for the \(\mathrm{S}_{4}\) tableauxand construct explicit wavefunctions having these permutational symmetries. 14 2 N 123 4 3
Construct the outer product [4] \(\times\) [3] for the partitions [4] of \(S_{4}\) and [3] of \(S_{3}\). Check the dimensionalities using Eq. (4.9).Data from Eq. (4.9) Dim ([f]x[f']) = (k + k')! ==
Determine the irrep content of
Prove that for a group the inverse of each group element and the identity are unique. Prove that the inverse of the product of two group elements is given by \((a \cdot b)^{-1}=\) \(b^{-1} a^{-1}\).
Write out the multiplication table for all possible products of elements in the group \(S_{3}\) (permutations on three objects). Use this to demonstrate explicitly that \(S_{3}\) is a group, that it
If the operator \(c_{4}\) rotates a system by \(\frac{\pi}{2}\) about a specified axis, demonstrate that the operator set \(\mathrm{C}_{4}=\left\{e, c_{4}, c_{4}^{2}, c_{4}^{3}\right\}\) constitutes
Show that the groupwith \(e\) the identity has one possible multiplication table; thus there is only one finite group of order three. G = {e, a, b}
Demonstrate that for the cyclic group \(\mathrm{C}_{4}\) with multiplication table given in Example 2.15 , the subgroup \(H=\left\{e, a^{2}\right\}\) is an abelian invariant subgroup.Data from
Prove that \(\{e,(123),(321)\}\) is an invariant subgroup of \(\mathrm{S}_{3}\) but \(\{e,(12)\}\) is not.
Show that the cyclic group \(\mathrm{C}_{4}\) is neither simple nor semisimple.
Prove that if a,b and \(c\) are elements of a group and class conjugation is indicated by \(\sim\), then (1) \(a \sim a\), (2) if \(a \sim b\), then \(b \sim a\), and (3) if \(a \sim b\) and \(b \sim
Show that the group {e, a, b, c} with multiplication table (b) below, is in one to one correspondence with the geometrical symmetry operations on figure (a) belowThis is called the 4-group or
Demonstrate that the identity, reflections about the three symmetry axes (dashed lines), and rotations by \(\frac{2 \pi}{3}\) and \(\frac{4 \pi}{3}\) about the center of the equilateral triangleform
Show that the quotient group of the 4-group \(\mathrm{D}_{2}\) defined in Problem 2.9 is \(\mathrm{C}_{2}\).
Prove that the angular momentum operator \(L_{z}\) generates rotations around the \(z\)-axis.
Show that for real numbers \(\alpha, \beta\), and \(\delta\) the matricesform a group under matrix multiplication. Show that the matrices \(G\) with \(\alpha=\beta=0\) form an invariant subgroup of
Prove that the direct product of two representations is a representation, and that the character of the direct product is the product of characters for the representations.
Determine the irrep content for the equivalent reducible \(S_{3}\) representations \(\Gamma^{(4)}\) and \(\Gamma^{(5)}\) of Fig. 2.4 .Data from Fig. 2.4 F(1) (2) 1 (3) (4) r(5) (6) 1 0 0 0 0 0 0 1
Demonstrate explicitly that for invariant subgroups the cosets form a group under the coset multiplication law (2.27).
Prove that for a finite group \(G\) with an invariant subgroup \(G \supset H\) and \(g_{i} \in G\), two cosets \(g_{i} H\) and \(g_{j} H\) have no elements in common if \(i eq j\).
Use cosets to show that a finite group with an order that is a prime number can have no proper subgroups. Show that a finite group with order equal to a prime number is isomorphic to a cyclic group.
Show that the set of matrices \(\{a, b, c, d\}\) given bycloses under multiplication and is a representation of the group \(\mathrm{D}_{2}\) in Problem 2.9 .Data from Problem 2.9Show that the group
Show that the set of functions \(\left\{f_{1}(x)=x, f_{2}(x)=-x, f_{3}(x)=x^{-1}, f_{4}(x)=-x^{-1}\right\}\) forms a group under the binary operation of substitution of one function into another.
The real numbers form a group under the binary operation of arithmetic addition. Show that for real numbers \(v\) the matricesform a 2D representation of this additive group of real numbers. Show
Show that for a matrix representation \(D(x)\) satisfying Eq. (2.8), a new set of matrices formed by performing the same similarity transform \(S^{-1} D(x) S\) for a fixed matrix \(S\) on all
Consider a cartesian \((x y z)\) coordinate system and define the following operations: \(R=\) rotation by \(\pi\) in the \(x-y\) plane, \(E=\) do nothing, \(I=\) inversion of all three axes, and
Verify that the mapping \(e \rightarrow 1\) and \(a \rightarrow-1\) gives a representation of the cyclic group \(\mathrm{C}_{2}\) described in Box 2.2 that preserves the group multiplication, as does
Show that the matricesconstitute a representation of the group \(\mathrm{C}_{2}\) described in Box 2.2 . Diagonalize this set of matrices and show that the 2D representation \(\left(t_{1},
Prove the trigonometric identitiesby the following group-theoretical means.1. Show that the complex numbers of unit modulus \(c=x+i y\) with \(|x|^{2}+|y|^{2}=1\) form a group under multiplication,
(a) Divide the integers up into four equivalence classes,Show that {e, a, b, c} form a group \(\left(\mathrm{Z}_{4}\right)\), under addition modulo 4 by constructing the multiplication table. Hint:
Show that the multiplication table for the 4-group {I, a, b, c\} given in Problem 2.9 follows from the algebraic requirements \(a^{2}=b^{2}=I\) and \(a b=b a=c\).Data from Problem 2.9Show that the
For finite groups each group element \(a\) must give the identity \(e\) when raised to some finite power: \(a^{p}=e\). The integer \(p\) is called the order of the element \(a\). Show that two
Prove that the group identity \(e\) is always in a conjugacy class of its own, and that no group element can be in two different conjugacy classes.
Show that the group \(Z_{2}\) of integers under addition modulo 2 is isomorphic to the cyclic group \(\mathrm{C}_{2}\) described in Box 2.2 . Show that the set \(\{1,-1\}\) is isomorphic to
Show that for a direct product group \(G=A \times B\), the groups \(A\) and \(B\) are invariant subgroups of G .
Verify that the representation \(T\) given by Eq. (2.9) obeys \(T\left(G_{i}\right) T\left(G_{j}\right)=T\left(G_{i} G_{j}\right)\), so it preserves the group multiplication law for \(G\) and is a
Use the highest-weight algorithm to show that \(\mathbf{2} \otimes \mathbf{2} \otimes \mathbf{2}=\mathbf{4} \oplus \mathbf{2} \oplus \mathbf{2}\), for the product of three fundamental
Suppose that \(D^{(\alpha)}\) and \(D^{(\beta)}\) are irreps of a compact simple group with dimensions \(n_{\alpha}\) and \(n_{\beta}\), respectively, and basis vectors \(x_{i}\) and \(y_{i}\),
By expanding the group elements written in the canonical exponential form about the origin, derive the Lie algebra \(\left[X_{a}, X_{b}\right]=i f_{a b c} X_{c}\). A commutator can be defined by
Show that the most general form for \(2 \times 2\) unitary matrices of unit determinant iswhere \(a\) and \(b\) are complex numbers. U = a |a|2 + |b| = 1, -b a
Prove that if matrices \(T_{a}\) with matrix elements \(\left(T_{a}\right)_{b c}\) proportional to the structure constants \(f_{a b c}\) are defined as in Eq. (3.8), these matrices satisfy the Lie
Use that the adjoint representation for \(\mathrm{SU}(2)\) is three-dimensional and that generators of the adjoint representation are the structure constants of the group to construct a 3D matrix
Prove Schur's lemma: a matrix that commutes with all generators of an irrep is a multiple of the unit matrix. Hint: Assume that \(\left[T_{a}, M\right]=0\) for all \(a\), where \(T_{a}\) is a group
Show that for a four-dimensional cartesian space \((x, y, z, t)\) the operatorswhich is the algebra associated with the group \(\mathrm{SO}(4)\). Show that this \(\mathrm{SO}(4)\) is locally
Show that the structure constants \(f_{a b c}\) appearing in Eq. (3.3) are real if the generators of the Lie algebra are hermitian.Data from Eq. (3.3) [Xa, Xb]=ifabe Xc = [A, B] AB BA,
Matrices are central to representation theory. If \(A, B, C\), and \(D\) are square, invertible, \(n \times n\) matrices with complex entries, prove the following useful properties.a. The trace of a
Show that for a continuous parameter \(\theta\) the set of matricesforms a one-parameter abelian Lie group under matrix multiplication, and that if the matrices \(G\) operate on a 2D cartesian space
Show that the Pauli matrices generate a matrix representation of Lie group elements having the formas required by Problem 3.4 .Data from Problem 3.4Show that the most general form for \(2 \times
Prove that if the Lie group \(G_{1}\), with generators \(J_{i}(1)\), is isomorphic to a group \(G_{2}\), with generators \(J_{i}(2)\), and \(J_{i}(1)\) and \(J_{i}(2)\) operate on independent degrees
Show that the group-element commutator \(R_{y}(\delta \theta) R_{x}(\delta \theta) R_{y}^{-1}(\delta \theta) R_{x}^{-1}(\delta \theta)\), is related to the generator commutator \(\left[J_{x},
(a) Confirm the validity of the Jacobi identity given in Eq. (3.6). (b) Use Eq. (3.6) to confirm the validity of Eq. (3.7b).Data from Eq. (3.6)Data from Eq. (3.7b) [[A, B], C]+[[B, C], A] + [[C, A],
If a local density operator is expressed by \(ho(r)=\sum_{i} \delta\left(r-r_{i}\right)\), where the sum is over particles and \(\delta\left(r-r_{i}\right)\) is the Dirac delta function, what is its
In \(\hbar=c=1\) natural units a particular hadronic cross section \(\sigma\) is estimated to bewhere the mass of the pion is \(M_{\pi} \sim 140 \mathrm{MeV}\). What is this cross section in more
In special relativity it is common to use units where the speed of light \(c\) is set to one. The world record in the 100 meter dash is about 9.6 seconds. What is this time expressed in \(c=1\)
In natural \((\hbar=c=1)\) units the mean life for the decay \(\Sigma^{0} \rightarrow \Lambda+\gamma\), where \(\Sigma^{0}\) and \(\Lambda\) are elementary particles and \(\gamma\) is a photon,
What is one Joule in \(c=1\) units? What is one atmosphere \(\left(10^{5} \mathrm{~N} \mathrm{~m}^{-2}\right)\) of pressure expressed in \(c=1\) units?
Using the commutators for \(J_{i}\) and \(K_{i}\) given in Problem 3.8 , argue that the \(\mathrm{SO}(4)\) Lie algebra is semisimple, but not simple. Argue that \(\mathrm{SO}(4)\) can be written as a

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