The Morse Potential
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The Morse potential for λ = 2.

-1- The Morse Potential

To model the interaction of two atoms in a diatomic molecule, Philip M. Morse proposed the following potential:

It is one of the few analytically solvable models of quantum mechanics: its eigenvalues are given by

and the eigenfunctions by

where Ln(z;α) is a Laguerre polynomial which expresses as:

In this exercise, we recover some of these non-trivial results with the density matrices and path integrals.
  1. Determine analytically the ground state ψ0(x) and the first excited state ψ1(x) up to a constant factor.
  2. Determine explicitly the normalisation of these wavefunctions. What happens if λ is too small? Comment. In particular: discuss in details the number of bound states , especially in relation (i) to the expression of the eigenvalues given above and (ii) to the shape of the Morse potential.

-2- Density matrix approach to the Morse potential

  1. Produce a few pictures of the Morse potential, for different parameters λ > 3/2.
  2. Justify that one can restrict the space to some finite interval.
  3. At high temperature, set up the density matrix as a numpy array, on a grid of points chosen with numpy linspace.
  4. Perform the matrix-squaring procedure using the product. Plot the entire density matrix using matshow (part of pyplot in matplotlib), and comment on the dependency on temperature of the two-dimensional plots of the density matrix — especially regarding the «width» of the density matrix.
  5. At low temperature, and for several values of λ, compare:
    • (i) the diagonal density matrix, determined numerically from the matrix squaring algorithm, and
    • (ii) the known expression of those diagonal elements in terms of the groundstate wavefunction (see equation (3.5) in SMAC keeping only the ground state n=0 in the sum).
  6. Explain why we can say that «the temperature is low» when those two quantities are equal.
  7. A good starting choice of values for parameters is N=100 slices in space, λ = 2, βinitial = 2−6, 9 iterations of the matrix squaring, with restricting r to the interval [-2,10]. Find other values of the parameters, also yielding consistent results.

-3- Path-integral simulation for the Morse potential

Set up a naive path-integral simulation (as in SMAC algorithm 3.4, page 151) for a single particle in the Morse potential. Take a moderate number of time slices, and produce a histogram of the particle positions. Explain the exact relationship between this histogram and the diagonal elements of the density matrix. Compare again your numerical results with the analytical solution at low temperature.

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