Autumn · project 1 of 20

The quantum harmonic oscillator

Topic: Quantization and normal modes

Goal: Study the energy spectrum and the dynamics of a one-dimensional chain of quantum harmonic oscillators, including an analysis of its symmetries and a visualization of its evolution.

  1. The Hamiltonian: Consider a one-dimensional chain of \(N\) identical atoms with spacing \(a\) between neighboring atoms. The Hamiltonian of the system is \[\widehat{H} = \sum_{i=1}^N \frac{\widehat{P}_i^2}{2m} + \frac{k}{2} \sum_{i=1}^N (\widehat{X}_{i+1} - \widehat{X}_i)^2,\] where

    • \(m\) is the mass of an atom,

    • \(k = m\omega^2\) is the stiffness of the bond between neighboring atoms,

    • \(\widehat{P}_i\) and \(\widehat{X}_i\) are the momentum and coordinate (displacement from equilibrium) operators of the \(i\)th atom.

    Explain why only interactions between neighboring atoms are taken into account.

  2. Normal-mode operators: Define the normal-mode operators \[\begin{aligned} \widehat{Q}_k &= \frac{1}{\sqrt{N}} \sum_{n=1}^N e^{ikan} \widehat{X}_n, \\ \widehat{\Pi}_k &= \frac{1}{\sqrt{N}} \sum_{n=1}^N e^{-ikan} \widehat{P}_n, \end{aligned}\] where \(k = \frac{2\pi\ell}{Na}\), \(\ell = 0, 1, \dots, N-1\).

    1. Compute the commutators \([\widehat{Q}_k, \widehat{\Pi}_{k'}]\), \([\widehat{Q}_k, \widehat{Q}_{k'}]\), and \([\widehat{\Pi}_k, \widehat{\Pi}_{k'}]\).

    2. Write the Hamiltonian \(\widehat{H}\) in terms of \(\widehat{Q}_k\) and \(\widehat{\Pi}_k\).

  3. Periodic boundary conditions: Assume that the chain obeys periodic boundary conditions: \[\widehat{X}_{N+1} = \widehat{X}_1.\]

    1. Find the eigenfrequencies \(\omega_k\) of the system, where \(k\) is the wave vector.

    2. Show that the system is equivalent to a set of independent harmonic oscillators, one for each mode \(k\).

  4. Dynamics of the atoms:

    1. Compute the wave function \(\psi(t)\) of the system.

    2. Make an animation of the mean displacements of the atoms \(\langle X_n\rangle(t)\) for different initial conditions:

      • one atom displaced: \(X_i(0) \neq 0\), all other \(X_n(0) = 0\);

      • two atoms displaced: \(X_i(0) \neq 0\), \(X_k(0) \neq 0\), all other \(X_n(0) = 0\).

  5. Symmetries and conserved charges:

    1. Find all symmetries of the problem (for example, translation symmetry and invariance under time shifts).

    2. Using Noether’s theorem, compute the conserved charges associated with each symmetry.

    3. Explain the physical meaning of these charges (for example, energy, momentum, phase correlations of the modes).

Autumn: all projects, PDF

← All projects