Calculation of the ground state of one-dimensional harmonic oscillator using variational Monte Carlo method. The Hamiltonian can be written as ˆH = (-∇2 + x2)/2, where units were chosen so that mass = 1, ¯h = 1, and ω = 1.
(a) How to choose a mesh for -∞ < x < ∞?
Since the oscillator is more likely to be near the minimum (at x = 0) of the potential, we can restrict the range of the mesh to |x| < 3 and outside the mesh, ψ(x) ? 0. If ?x = 0.2 , x = -2.0 + (m - 1)?x where m = 1,....,M with M = 31.
(b) Initial trial function
We can simply use a normalized constant function in |x| < 3 , φ(x) = 1/√6.
(c) Random number generator for selecting m′of Φtest(m′)
m′ can be chosen using a uniform random number generator that will select a grid point on the mesh with equal probability. To improve the efficiency of the minimization, one may use a non-uniform random number generator to force computer to put more effort in certain region of space. For the harmonic oscillator, the region near x = 0 is more important to the ground state. A Gaussian random number generator could therefore be more efficient.
In this project, you need to complete the calculation with both the uniform and Gaussian random numbers and compare the computational efficiency of these two different random number generators for the minimization.
(d) How do you know the minimization is good enough (converged)?
Plot E[φ] as a function of the number of iterations for φ. It should monotonically decrease and converges to the ground state energy. From the plot, you can easily ?gure out when the minimization is good enough.
(e) Compare the numerically obtained energy and wave function with the exact solution. Plot the exact wave function and the numerical solution. Also plot the initial trail function and φ(x) at two intermediate steps of the minimization of E[φ].