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Download 1 1. Determine if the following vector operators are Her
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1 1. Determine if the following vector operators are Hermitian. Separate those that are not into their Her⃗ × P⃗ , (ii) mitian and anti-Hermitan Parts: (i) R ⃗ × P⃗ , (iii) R ⃗ × L. ⃗ L ⃗ and W ⃗ be two vector operators of a certain 2. Let V quantum mechanical system, i.e., they satisfy the commutation relations ∑ ∑ [Ji , Vj ] = i εijk Vk [Ji , Wj ] = i εijk Wk k ⃗v = 1 ( ⃗ q ⃗) P− A . m c (b) Show that ⃗v × ⃗v = k with the components of the total angular momentum J⃗ of the system. ⃗ ·W ⃗ is a scalar with respect to (a) Show that V [ ] ⃗ V ⃗ ·W ⃗ . rotations, i.e., evaluate J, ⃗ =V ⃗ ×W ⃗ is a vector under rota(b) Show that U tions. ⃗ be a vector operator whose Cartesian compo3. Let V nents may or may not commute with one another. ⃗× (a) Show that the ith Cartesian component of V ⃗ ⃗ ⃗ ⃗ ⃗ (V × V ) and (V × V ) × V satisify the relations [ ] ∑ ⃗ × (V ⃗ ×V ⃗) = V Vj [Vi , Vj ] i j [ ] ∑ ⃗ ×V ⃗)×V ⃗ = [Vi , Vj ] Vj (V i 4. Determine the form taken by the components of ⃗ ×K ⃗ in the the angular momentum operator ⃗ℓ = R (Cartesian) wavevector representation. What form do they take in spherical wavevector coordinates (k, θ, ϕ)? What form do the common eigenfunctions ψlm (k, θ, ϕ) of ℓ2 and ℓz take in this representation? [Note: This spherical wavevector coordinates are related to their Cartesian counterparts in the usual way, i.e., ky = k sin θ sin ϕ, i~q ⃗ B. m2 c (c) Prove the relation m q d ⃗ −B ⃗ × ⃗v ⟩ + q⟨E⟩, ⃗ ⟨⃗v ⟩ = ⟨⃗v × B dt 2c ⃗ = −∇ϕ− where the electric field operator is E ⃗ (1/c)∂ A/∂t. 6. Consider an irreducible invariant subspace corresponding to j = 1. (a) Show that the matrix representing any component Ju = J⃗ · û of the angular momentum within this subspace obeys the relation Ju3 = Ju . (Feel free to use any appropriate symmetry arguments to simplify your calculation.) (b) Show using this result that the rotation matrix Ru (α) within this subspace can be written in the form j (b) Use this to deduce the general relation [ ] ⃗ ,V ⃗ ·V ⃗ =V ⃗ × (V ⃗ ×V ⃗ ) − (V ⃗ ×V ⃗)×V ⃗. V kx = k sin θ cos ϕ, (a) Show, by considering the evolution of the ⃗ mean position ⟨R(t)⟩ of such a particle, that the velocity operator takes the form kz = k cos θ. ⃗ = 5. In the presence of an electromagnetic field B ⃗ ⃗ ∇ × A, the classical Hamiltonian for a particle of charge q takes the form ]2 q⃗ 1 [ p⃗ − A(⃗ r, t) + qϕ(⃗r). H= 2m c Assume that this holds for a quantum particle with ⃗ and p⃗ → P⃗ . ⃗r → R Ru (α) = 1 − iJu sin α − Ju2 (1 − cos α). (c) Explicitly construct the matrix Ry (α) corresponding to rotations about the y axis. 7. Consider an angular momentum subspace with j = 1 ⃗ (Take ~ = σ denote the vector operator 2J. 2 . Let ⃗ 1.) (a) Show that the matrix representing any component σu = ⃗σ · û of the operator ⃗σ within this subspace obeys the relation [σu ]2 = 1, and use this relation to derive an expression for the matrix representing the operator ρu (α) = exp[−iασu ] similar to that given in the last problem. Use this result to obtain a similar expression for the matrix representing the rotation operator Ru (α). (b) Find the eigenvectors |j, mx ⟩ of J 2 and Jx as an expansion (or column vector) in the basis of eigenstates |j, mz ⟩ of J 2 and Jz . (c) In the |j, mz ⟩ basis construct the unitary matrix representing Rz (π/2). 2 (d) Use the matrix of the last part to transform the states |j, mx ⟩ that you found. Of what component of J⃗ should these transformed states be eigenstates? (e) Find the matrix representing the operator Jx after a rotation by Rz (π/2) and interpret your result. {Tqm |m 8. A set of 2q +1 operators = q, · · · , −q} form the spherical components of an irreducible tensor T of rank q if they transform under rotations according to the relation RTqm R+ = q ∑ ′ (q) Tqm Rm′ m . m′ =−q (a) Show by considering infinitesimal rotations Ru (ϵ) this reduces the transformation law to a set of characteristic commutation laws associated with the components of angular momentum. (b) Use the results of the last part, by considering rotations along the three cartesian axes and using the known form of the matrices representing the cartesian components of J⃗ in a standard representation to derive the following form of the relations derived in the last part: [Jz , Tqm ] = mTqm [J+ , Tqm ] = √ q(q + 1) − m(m + 1)Tqm+1 [J− , Tqm ] = √ q(q + 1) − m(m − 1)Tqm−1 . (c) Let {|k, J, M ⟩} denote the basis vectors of a standard representation of eigenstates of J 2 and Jz . Use the commutation relations above to show that Tqm |k, J, M ⟩ is an eigenstate of Jz with eigenvalue m + M , and hence deduce the “selection rule” that ⟨k ′ , J ′ , M ′ |Tqm |k, J, M ⟩ is zero unless ∆M = M ′ − M = m. 9. Let Vu denote the component of a vector opera⃗ along a given direction, Ju the component tor V of angular momentum along the same direction, and let |a⟩ and |b⟩ denote two arbitrary vectors lying within the same irreducible invariant subspace S(k, j). Show that ⃗⟩ ⟨J⃗ · V ⟨a|Vu |b⟩ = ⟨a|Ju |b⟩ j(j + 1) ⃗ ⟩ denotes the expectation value of this where ⟨J⃗ · V scalar operator taken with respect to any state in this subspace. 10. The total angular momentum of a particle with ⃗ +S ⃗ of its orspin s = 12 is the sum J⃗ = L bital and spin angular momentum. Let the states |n, l, s, ml , ms ⟩ = |n, l, ml ⟩ ⊗ |s, ms ⟩ form an ONB of (direct product) eigenstates of L2 , Lz , S 2 , and Sz . (a) Classify the irreducible invariant subspaces S(n, l, s, j) of the combined state space for such a particle. (That is, determine for given l the values of j associated with the irreducible invariant subspaces that actually exist, and the number of subspaces associated with each value of j.) (b) Describe how to construct, for each subspace S(n, l, s, j), the state vectors |n, l, s, j, mj = j⟩ having the maximum component of angular momentum along the z axis. (c) Explicitly construct the basis vectors associated with the subspaces of the form S(n, 1, s, j), having l = 1. Determine the Clebsch-Gordon coefficients which result from this construction. 11. Consider a particle of spin s = 12 moving in a central potential V (r). The relativistic theory of the electron reveals that there is a (small) interaction between the magnetic moment associated with the spin of the electron and the central potential in which it moves. To lowest order, this interaction is of the form ⃗ · S, ⃗ V = αL where α is generally a scalar function of r which we will, for simplicity, take to be a constant. The total Hamiltonian can then be written ⃗ ·S ⃗ H = H0 + αL where H0 = P 2 /2m+V (R). Assume that the states |n, l, s, ml , ms ⟩ referred to in the last problem are eigenstates of H0 with energy En . (a) Show explicitly that the basis states {|n, l, s, j, mj ⟩} whose construction was addressed in the last problem are eigenstates of this Hamiltonian, and determine the corresponding energies of the “full” Hamiltonian H. (b) Discuss how one would obtain the eigenstates of the system if α = α(r) were a spherically symmetric function of r. 12. Consider N particles of spin s = 12 . The total space of this system can be viewed as the tensor product ⃗ α }) and the spin space of the spatial space S({R 3 ⃗α }). Each of these factor spaces can separately S({S be decomposed into irreducible invariant subspaces. In this problem we focus only on the spin part of the space. Let ∑ ⃗= ⃗α S S α be the total spin angular momentum of the system. Find the values of s associated with the eigenvalues of S 2 , and the number of irreducible invariant subspaces associated with each value represented for a system containing N particle where (a) N = 3 (b) N = 4 (c) N = 5 (d) Discuss briefly any general information that you can deduce about the structure of the irreducible invariant subspaces of the spin space of N particles for arbitrary N. 13. A particle of spin s = 1 moves in a state of orbital angular momentum l = 7. (a) What are the irreducible invariant subspaces (i.e., eigenspaces of J 2 ) associated with this system? (b) If the system is in the state |j, m⟩ = |7, 7⟩ and a measurement of Lz is made, what values can be obtained and what are the probabilities of obtaining those values? Repeat the question assuming that the operator being measured is Sz . (c) If the system is in the state |ml , ms ⟩ = |6, 1⟩, and a measurement of J 2 is made, what values can be obtained and what are the probabilities of obtaining those values? 14. A single spinless particle of mass µ is constrained to move on the surface of a sphere of radius a. The Hamiltonian for the system is just the rotational kinetic energy H= ~2 ℓ̂2 2µa2 The particle is in a state characterized by the wave function [ ] 2 ψ(θ, ϕ) = A 2 + (sin θ cos ϕ + i sin θ sin ϕ) where A is a normalization constant. (a) If the energy is measured on this state, what values can be obtained and with what probability will each be found? (b) What is the state of the system immediately after a measurement which obtains each of the possible values that can be obtained? ⃗1 , and S ⃗2 , 15. Consider two particles of spin 12 , and let S denote the corresponding spin angular momentum operators. Let {|m1 , m2 ⟩} be an ONB of direct products states for the combined { }spin space of this system, with m1 , m2 ∈ + 12 , − 12 = {↑, ↓}. These states are eigenstates of S1z and S2z . Suppose the particles interact through a Hamiltonian ⃗1 · S ⃗2 H = −ε0 S where ε0 is a positive constant. Find the energy eigenvalues and their degeneracies for this system. Construct an orthonormal basis of eigenstates for this Hamiltonian as linear combination of direct product states. 16. Consider two particles of spin s = 1. Let {|m1 , m2 ⟩} be an ONB of direct products states for the combined spin space of this system, with m1 , m2 ∈ {−1, 0, 1} . These states are eigenstates of S1z and S2z . (a) According to the angular momentum addtion theorem, what irreducible invariant subspaces ⃗ = associated with the total spin operator S ⃗ ⃗ S1 + S2 occur for this system? (b) Explicitly construct the eigenstates {|s, m⟩} of S 2 and Sz for this spin-space, as linear combinations of the direct product states. (c) From the states that you have produced, construct a table of the Clebsch-Gordon coefficients required to combine two angular momenta with j1 = j2 = 1. 17. An atomic nucleus of spin s1 = 1/2 is in an orbital angular momentum state with ℓ = ℓ1 = 0. A single electron (spin s2 = 1/2) is bound to this nucleus in an orbital angular momentum state with ℓ = ℓ2 = 2. (a) What values are possible for the quantum numbers j1 and j2 associated with the to⃗1 + S ⃗1 and tal angular momentum J⃗1 = L ⃗ ⃗ ⃗ J2 = L2 + S2 of the nucleus and the electron, respectively. (b) What values are possible for the quantum numbers ℓ and s associated with the total or⃗ =L ⃗ 1 +L2 bital and spin angular momentum L ⃗=S ⃗1 + S ⃗2 , respectively. and S (c) What values are possible and how many irreducible subspaces exist for each value of the quantum number j associated with the total ⃗ +S ⃗ = J1 + J2 of angular momentum J⃗ = L this system? 4 18. Two distinguishable spinless particle of mass M move freely on the surface of a sphere of radius r = a . The two particle state is described by the angular position space wave function ψ (θ1 , ϕ1 , θ2 , ϕ2 ) = Y22 (θ1 , ϕ1 ) Y10 (θ2 , ϕ2 ) . ⃗ = ⃗ℓ1 + ⃗ℓ2 be the total angular momentum Let L of the system. If L2 and Lz are measured on this system, what values can be obtained and with what probability will they occur? [Hint: Feel free to use an online table of CG coefficients, which you can find, e.g., on Wikipedia. On a test such a table would provided. 19. Consider a particle of mass M moving on the surface of a sphere of radius a, in the presence of a weak electric field, with total Hamiltonian H = H0 + H (1) = ~2 ℓ̂2 − eE0 Z 2M a2 where ℓ̂2 = ⃗ℓ · ⃗ℓ and where on the surface of the sphere, in the position representation Z → z = a cos θ. (a) What are the energies and degeneracies of the system in the absence of the applied field? What are the energy eigenfunctions in the angular position representation (in which χ (θ, ϕ) = ⟨θ, ϕ|χ⟩, for example. (b) Find the energy of the ground state of the unperturbed Hamiltonian to 2nd order in the field. [Hint: use the Wigner-Eckart theorem to determine which matrix elements of H (1) are not zero. 20. For a particle moving on the surface of a sphere, let |ℓ, m⟩ denote a standard representation of eigen⃗ be a vector operator for states of ℓ̂2 and ℓ̂z . Let V this system which has the property that the matrtix element ⟨1, 0|Vx |1, 1⟩ = 3v0 where v0 is a positive constant. Construct the 3 × 3 submatrices [Vx ] , [Vy ] , [Vz ], representing the Cartesian components of this vector operator within the irreducible invariant subspace S (1) , spanned by the states {|1, m⟩|m = −1, 0, 1}.