Introduction to Quantum Mechanics · Problem 5.38

Identical Particles — David J. Griffiths

Physics — Quantum Mechanics · harmonic_chain_phonons

Problem

The harmonic chain consists of NN equal masses arranged along a line and connected to their neighbors by identical springs: H^=−ℏ22m∑j=1N∂∂xj2+∑j=1N12mω2(xj+1−xj)2,\hat{H} = -\frac{\hbar^2}{2m} \sum_{j=1}^{N} \frac{\partial}{\partial x_j^2} + \sum_{j=1}^{N} \frac{1}{2} m\omega^2 (x_{j+1} - x_j)^2, where xjx_j is the displacement of the jjth mass from its equilibrium position. This system (and its extension to two or three dimensions—the harmonic crystal) can be used to model the vibrations of a solid. For simplicity we will use periodic boundary conditions: xN+1=x1x_{N+1} = x_1, and introduce the ladder operators a^k±≡1N∑j=1Ne±i2πjk/N[mωk2ℏ xj∓ℏ2mωk ∂∂xj]\hat{a}_{k\pm} \equiv \frac{1}{\sqrt{N}} \sum_{j=1}^{N} e^{\pm i 2\pi j k/N} \left[ \sqrt{\frac{m\omega_k}{2\hbar}}\, x_j \mp \sqrt{\frac{\hbar}{2m\omega_k}}\, \frac{\partial}{\partial x_j} \right] where k=1,…,N−1k = 1, \ldots, N-1 and the frequencies are given by ωk=2ωsin⁡(πkN).\omega_k = 2\omega \sin\left( \frac{\pi k}{N} \right). (a) Prove that, for integers kk and k′k' between 1 and N−1N-1, 1N∑j=1Nei2πj(k−k′)/N=δk′,k\frac{1}{N} \sum_{j=1}^{N} e^{i 2\pi j(k-k')/N} = \delta_{k',k} 1N∑j=1Nei2πj(k+k′)/N=δk′,N−k.\frac{1}{N} \sum_{j=1}^{N} e^{i 2\pi j(k+k')/N} = \delta_{k',N-k}. Hint: Sum the geometric series. (b) Derive the commutation relations for the ladder operators: [a^k−,a^k′+]=δk,k′and[a^k−,a^k′−]=[a^k+,a^k′+]=0.[\hat{a}_{k-}, \hat{a}_{k'+}] = \delta_{k,k'} \quad \text{and} \quad [\hat{a}_{k-}, \hat{a}_{k'-}] = [\hat{a}_{k+}, \hat{a}_{k'+}] = 0. (c) Using Equation 5.75, show that xj=R+1N∑k=1N−1ℏ2mωk(a^k−+a^N−k,+) ei2πjk/Nx_j = R + \frac{1}{\sqrt{N}} \sum_{k=1}^{N-1} \sqrt{\frac{\hbar}{2m\omega_k}} (\hat{a}_{k-} + \hat{a}_{N-k,+})\, e^{i 2\pi j k/N} ∂∂xj=1N∂∂R+1N∑k=1N−1mωk2ℏ(a^k−−a^N−k,+) ei2πjk/N\frac{\partial}{\partial x_j} = \frac{1}{N} \frac{\partial}{\partial R} + \frac{1}{\sqrt{N}} \sum_{k=1}^{N-1} \sqrt{\frac{m\omega_k}{2\hbar}} (\hat{a}_{k-} - \hat{a}_{N-k,+})\, e^{i 2\pi j k/N} where R=∑jxj/NR = \sum_j x_j / N is the center of mass coordinate. (d) Finally, show that H^=−ℏ22(Nm)∂2∂R2+∑k=1N−1ℏωk(a^k+a^k−+12).\hat{H} = -\frac{\hbar^2}{2(Nm)} \frac{\partial^2}{\partial R^2} + \sum_{k=1}^{N-1} \hbar\omega_k \left( \hat{a}_{k+} \hat{a}_{k-} + \frac{1}{2} \right). Comment: Written in this form above, the Hamiltonian describes N−1N-1 independent oscillators with frequencies ωk\omega_k (as well as a center of mass that moves as a free particle of mass NmNm). We can immediately write down the allowed energies: E=−ℏ2K22(Nm)+∑k=1N−1ℏωk(nk+12)E = -\frac{\hbar^2 K^2}{2(Nm)} + \sum_{k=1}^{N-1} \hbar\omega_k \left( n_k + \frac{1}{2} \right) where ℏK\hbar K is the momentum of the center of mass and nk=0,1,…n_k = 0, 1, \ldots is the energy level of the kkth mode of vibration. It is conventional to call nkn_k the number of phonons in the kkth mode. Phonons are the quanta of sound (atomic vibrations), just as photons are the quanta of light. The ladder operators ak+a_{k+} and ak−a_{k-} are called phonon creation and annihilation operators since they increase or decrease the number of phonons in the kkth mode.

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