Quantum Physics : Homework #6 [Total 10 points] Due: 014.1.1(Mon) 1:30pm Exercises: 014.11.5(Tue)/11.6(Wed) 6:30 pm; 56-105 Questions for problems: 민홍기 hmin@snu.ac.kr Questions for grading: 모도영 modori518@snu.ac.kr 1. (0pts) Selection rules involving l and l (a) (5pts) Derive [L, [L, r]] = ħ (rl + L r). (b) (5pts) From (a), show that n l m [L, [L, r]] nlm = ħ 4 [l (l + 1) + l(l + 1)] n l m r nlm. (c) (5pts) Show that n l m [L, [L, r]] nlm = ħ 4 [l (l + 1) l(l + 1)] n l m r nlm. (d) (5pts) From (b) and (c), obtain the condition that n l m r nlm is not zero.. (0pts) Consider a system whose Hamiltonian is given by H = p m + V(r). Let ε n be an eigenenergy of an eigenstate n in this system. (a) (pts) Show that [x, H] = iħp x m. (b) (3pts) For eigenstates n and i, show that n p x i = imω ni n x i. (ħω ni = ε n ε i ) (c) (10pts) Using the commutation relation [x, p x ] = iħ and completeness condition 1 = n n n, show that mω ni ħ n n x i =1. (d) (5pts) Define the dimensionless quantity f ni = mω ni n r i as the oscillator strength 3ħ for transition between i and n. Show that the following sum rule is satisfied: n f ni = 1. The sum rule presented here is called the Thomas-Reiche-Kuhn sum rule. This relation is useful when we calculate the absorption cross-section.
3. (30pts) Consider the Hamiltonian of a system H(R(t)) with an external time-dependent parameter R(t). Assume that an eigenstate n satisfies the eigenvalue relation at R(t): (a) (5pts) Prove the following relation: H(R(t)) n; R(t) = ε n (R(t)) n; R(t). n; R dh m; R = (ε dt m ε n ) n; R d m; R dt (m n). (b) (5pts) If an initial wave function at t = 0 is given by Ψ(0) = n; R(0), the wave function at t > 0 can be expressed as Ψ(t) = c m (t) m; R(t). Using the Schrödinger equation iħ d dt Ψ(t) = H(R(t)) Ψ(t), find the equation that that the coefficient c n(t) for a state n satisfies. (c) (10pts) Within the adiabatic approximation, for a wave function which started at t = 0 as Ψ(0) = n; R(0) can be approximated at t > 0 as Ψ(t) = c n (t) n; R(t). Using the result of (a), show that the condition for the adiabatic approximation is given by Interpret the meaning of this condition. m n; R dh m; R dt = 0 (n m). ε n ε m (d) (10pts) Let us express c n (t) as the following form: c n (t) exp (iγ n (t) i ħ dt ε n (R(t ))). What is the meaning of the second factor in the exponential? Within the adiabatic approximation, show that the additional phase factor γ n (t) is given by γ n (t) = i R(t) R(0) t 0 dr n; R R n; R. 4. (10pts) Problem 10.3 Apply the concept of the geometric phase to a familiar example such as the 1D infinite square well problem.
5. (30pts) Berry connection and curvature for a spin Imagine a particle of spin 1 under a uniform magnetic field along a direction R = (sin θ cos φ, sin θ sin φ, cos θ), then we can write down the Hamiltonian as the following form : H = R σ where R is a vector along the direction R and σ represents Pauli matrices. (a) (10 점 ) We can express the eigenstates of H as χ + (t) = ( cos θ e iφ sin θ ) and χ (t) = ( sin θ e iφ cos θ ) representing spin up and down along R, respectively. (See HW #5, Problem 1.) Find the Berry connection and Berry curvature for these eigenstates. (b) (10pts) Alternatively, we can express the eigenstates as χ (t) + = ( e iφ cos θ sin θ ) and χ (t) = ( e iφ sin θ cos θ ) in a different gauge. Find the Berry connection and Berry curvature for these eigenstates. (c) (10pts) Comparing the results obtained in (a) and (b), discuss the followings: 1) What is the region that the Berry connection or Berry curvature is not well defined? ) Among the Berry connection and curvature, what is gauge-dependent and what is not? 3) If gauge-dependent, discuss how expressions in different gauges are related with each other. 6. (0pts) Assume that a parameter R is a three-dimensional vector. For simplicity, let us define n n; R, ε n ε n (R), and H H(R). (a) (5pts) For the Berry connection A n (R) = i n R n, the Berry curvature is defined by F n (R) R A n (R). Show that F n (R) = i R n R n. (b) (5pts) Prove the following relation: m R H n = (ε m ε n ) R m n = (ε n ε m ) m R n. (c) (10pts) From (b), show that F n (R) can be expressed as the following form: F n (R) = i n RH m m R H n (ε n ε m ). m n
7. (40pts) Consider a spin S in an external magnetic field B(t) whose direction is changing adiabatically with the fixed magnitude B. Then the Hamiltonian is given by H(B) = gμ B ħ B(t) S where g is the Lande-g-factor and μ B is the Bohr magneton. If we set the direction of the magnetic field along the z-axis, then S z = ħm (m = S, S + 1,, S) thus the energy eigenvalue is given by ε m = gμ B mb. (a) (5pts) Prove the following relations: S, m ± 1 S x S, m = ħ (S m)(s ± m + 1), S, m ± 1 S y S, m = iħ (S m)(s ± m + 1). (b) (0pts) From (a) and the above HW Problem 6, show that the Berry curvature for a state m is given by F m (B) = m B R. (c) (5pts) Assume that B(t) is changing the direction along a closed loop C on a sphere of a radius B. Let Ω(C) be the solid angle for C. Then show that the Berry s phase is given by the following two forms: γ m (C) = mω(c) mod π, γ m (C) = m(4π Ω(C)) mod π. (d) (5pts) From (c), show that m should be quantized. (e) (5pts) The (first) Chern number C 1 is defined by the integration of the Berry s curvature over the sphere (or closed manifold) divided by π: C 1 = 1 da F m (B). Find C 1 for a state m. π B =B
8. (40pts) Berry curvature for a general two level system For a two level system, a general Hamiltonian which depends on a vector k can be expressed as the following form: H(k) = a 0 (k) + a(k) σ where a = (a 1, a, a 3 ) is a k-dependent vector and σ represents Pauli matrices. (a) (5pts) What are the eigenenergies and corresponding eigenstates? (b) (30pts) Show that the z-component of the Berry curvature for the state ± is given by F ± (z) (k) = 1 a 3 a xa y a where a = a and i k i. (c) (5pts) Show that the result of the above HW Problem 5 is consistent with (b). Use the spinor eigenstate in HW Problem 5 and the result of the Berry curvature in HW Problem 6.(c). Even though the final expression looks simple, the intermediate steps could be non-trivial. This result is frequently used when you deal with the Berry phase in a two-level system. If you want to study further on the Berry s phase, refer to the following references. References for the Berry s phase D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 8, 1959 (010) Berry phase effects on electronic properties http://rmp.aps.org/abstract/rmp/v8/i3/p1959_1 A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, and J. Zwanziger (Springer 003) The geometric phase in quantum systems : foundations, mathematical concepts, and applications in molecular and condensed matter physics http://library.snu.ac.kr/search/detailview.ax?cid=114415 Also see Supplement I in J. J. Sakurai, Modern Quantum Mechanics (Revised Edition)