Show that the Hamiltonian of the quantum elastic chain can be represented as a sum of harmonic oscillator Hamiltonians


 * 1) Find the Hamiltonian that describes an elastic chain of length \(L\): \(\mathbf{\hat{H}} = \int dx \left(\frac{1}{2m}\hat{\pi}^2 + \frac{k_s a^2}{2}\left(\partial_x \phi\right)^2\right)\) where \([\hat{\pi}(x),\hat{\phi}(x\prime)] = -i\hbar\delta\left(x-x\prime\right)\).
 * 2) Show that this commutation relation holds after a Fourier transform.
 * 3) Transform \(\hat{\pi}\) and \(\hat{\phi}\): \(\hat{\phi}_k = \frac{1}{\sqrt{L}}\int_0^L dxe^{- ikx} \phi(x)\) \(\hat{\pi}_k = \frac{1}{\sqrt{L}}\int_0^L dxe^{+ ikx} \pi(x)\)
 * 4) Find the commutator of \([\hat{\pi}_k,\hat{\phi}_{k\prime}]\): \([\hat{\pi}_k,\hat{\phi}_{k\prime}] = [\,\int dx\frac{1}{\sqrt{L}}e^{ikx}\hat{\pi}(x), \int dy\frac{1}{\sqrt{L}}e^{ik\prime y}\hat{\phi}(y)\,]\)
 * 5) Move commuting elements outside commutator: \([\hat{\pi}_k,\hat{\phi}_{k\prime}] = \int \int dx \, dy \frac{1}{L}e^{ikx} e^{ik\prime y}[\hat{\pi}(x), \hat{\phi}(y)]\)
 * 6) Substitute the commutation relation: \([\hat{\pi}_k,\hat{\phi}_{k\prime}] = - \int \int dx \, dy \frac{1}{L}e^{ikx} e^{ik\prime y}i\hbar\delta\left(x-y\right)\)
 * 7) Integrate: \([\hat{\pi}_k,\hat{\phi}_{k\prime}] = -i\hbar\delta_{kk\prime}\)
 * 8) Find the Hamiltonian in Fourier representation.
 * 9) Substitute the Fourier transformed operators into the Hamiltonian: \(\mathbf{\hat{H}} = \frac{1}{L} \int dx \left(\frac{1}{2m}\left(\sum_k e^{-ikx}\hat{\pi}_k\right)\left(\sum_{k\prime} e^{-ik\prime x}\hat{\pi}_{k\prime}\right) + \frac{k_s a^2}{2}\left(ik\sum_k e^{-ikx}\hat{\phi}_k\right)\left(-ik\prime\sum_{k\prime} e^{-ik\prime x}\hat{\phi}_{k\prime}\right)\right)\)
 * 10) Move the summation signs outside the integral: \(\mathbf{\hat{H}} = \sum_{k, k\prime}\frac{1}{L} \int dx \left(\frac{1}{2m}e^{-ikx}\hat{\pi}_k e^{-ik\prime x}\hat{\pi}_{k\prime} + \frac{k_s a^2}{2}e^{-ikx}\hat{\phi}_k -ik\prime e^{-ik\prime x}\hat{\phi}_{k\prime}\right)\)
 * 11) Integrate: \(\mathbf{\hat{H}} = \sum_{k, k\prime}\left[\frac{1}{2m}\delta_{k,-k}\prime\hat{\pi}_k\hat{\pi}_{k\prime} + \frac{k_s a^2}{2}\left(-kk\prime\right)\delta_{k,-k\prime}\hat{\phi}_k\hat{\phi}_{k\prime}\right]\)
 * 12) Use delta functions to simplify: \(\mathbf{\hat{H}} = \sum_{k}\left[\frac{1}{2m}\hat{\pi}_k\hat{\pi}_{-k} + \frac{k_s a^2}{2}k^2\hat{\phi}_k\hat{\phi}_{-k}\right]\)
 * 13) Identify the ladder operators for this Hamiltonian: \(\hat{a} = \sqrt{\frac{m\omega}{2\hbar}} \left ( \hat{\phi}_k + i\frac{\hat{\pi}_{-k}}{m\omega} \right ) \) \(\hat{a}^{\dagger} = \sqrt{\frac{m\omega}{2\hbar}} \left ( \hat{\phi}_{-k} - i\frac{\hat{\pi}_k}{m\omega} \right ) \)
 * 14) Write the Hamiltonian in terms of ladder operators.
 * 15) Find \(\hat{\phi}\) and \(\hat{\pi}\) in terms of ladder operators: \(\hat{\phi}_k=\sqrt{\frac{2\hbar}{m\omega}} \left ( \hat{a}_k + \hat{a}^{\dagger}_{-k} \right ) \) \(\hat{\pi}_k=i\sqrt{\frac{m\omega}{2\hbar}} \left ( \hat{a}_k - \hat{a}^{\dagger}_{-k} \right ) \)
 * 16) Substitute the ladder operators into the Hamiltonian: \(\mathbf{\hat{H}} = \sum_{k}\left[\frac{1}{2m}\frac{m\omega}{2\hbar} \left(\left ( \hat{a}_k - \hat{a}^{\dagger}_{-k} \right )\left ( \hat{a}_{-k} - \hat{a}^{\dagger}_{k} \right )+\left ( \hat{a}_{-k} - \hat{a}^{\dagger}_{k} \right )\left ( \hat{a}_{k} - \hat{a}^{\dagger}_{-k} \right )\right) + \frac{k_s a^2}{2}\frac{2\hbar}{m\omega}k^2\left(\left ( \hat{a}_k + \hat{a}^{\dagger}_{-k} \right )\left ( \hat{a}_{-k} + \hat{a}^{\dagger}_{k} \right )+\left ( \hat{a}_{-k} + \hat{a}^{\dagger}_{k} \right )\left ( \hat{a}_{k} + \hat{a}^{\dagger}_{-k} \right )\right)\right]\)
 * 17) Simplify: \(\mathbf{\hat{H}} = \sum_{k}\hbar\omega_k \left(\hat{a}^{\dagger}_k\hat{a}_k+1/2\right)\)