Numerical Calculation of Energies

The energy eigenvalues and eigenfunctions of the harmonic oscillator (HO) are analytically known. Deviations from the quadratic potential lead to shifted eigenvalues and modified eigenfunctions. One can calculate them approximately using perturbation theory if the deviation from the HO potential are small. In any case, one can calculate the changes using a "lattice" in space. The discretized Hamilton operator is a nxn matrix and the eigenfunctions are the n-dimensional eigenvectors.

Harmonic Oscillator potential (red dashed line) and anharmonic oscillator (blue solid line).

Energy shifts

This picture reveals a few general properties: The anharmonic potential is larger for larger |x| than the HO one. States are therefore more "confined" to a smaller region in space, leading to an increase of the energy ( $\Delta E>0$ ). We also observe that the corrections up to second order (1+2) are closer to the (almost) exact lattice solutions (solid lines) than the first-order corrections (1).

Modification of wave functions

As mentioned before, the chosen potential "squeezes" the wave functions into a smaller space. We illustrate this for the second excites state (n=2), for which this effect is larger than for n=0 and n=1:

Second excited state for HO (dotted line) vs. anharmonic oscillator, calculated with the lattice method. The picture demonstrate that the wave function corresponding to the anharmonic oscillator is "squeezed" together compared to the HO, due to its chosen properties (see potential plot above).

Next, we show the changes of wave functions with respect to the harmonic oscillator, calculated in perturbation theory and with the lattice method:

Changes of the wave functions compared to the HO ones, calculated in first-order perturbation theory (dashed lines) and lattice eigenstates (solid lines).

Finite-volume effects

While the discretization of the Hamiltonian allows in principle for an exact calculation, in practice the calculation can only be carried out on a lattice of finite size L. In three dimensions, this would be a cube of side length L, hence we will refer to this as finite-volume effects from here on, even if we stay in one dimension. The impact is illustrated below. For the picture, we choose Dirac boundary conditions, i.e., an infinite potential setting in at $|x|\cdot m=1.3$ . This forces the wave function to zero at the boundary.

Ground state harmonic oscillator wave function (red dotted) and energy (1.0). The blue solid line shows the solution of the HO in a very small finite volume with *Dirac boundary conditions*, i.e., an infinite potential setting in at *|x|=1.3/m*.

The finite-volume effects play an important role in lattice Quantum Chromodynamics (lattice QCD). The latter is a numerical method to solve the Hamiltonian of the strong interaction in a small volume. In the above example, the difference to infinite volume for the energy eigenvalues is small ( $\Delta E/m\approx 0.1$ ) but non-negligible. The main property of finite-volume effects for bound states is that they are exponentially suppressed. We check this behavior for the above example: