Quantum Physics 2: Homework #6

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1 Quantum Physics : Homework #6 [Total 10 points] Due: (Mon) 1:30pm Exercises: (Tue)/11.6(Wed) 6:30 pm; 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.

2 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.

3 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

4 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

5 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 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 Also see Supplement I in J. J. Sakurai, Modern Quantum Mechanics (Revised Edition)