the+Morse+potential

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 * Homework 05 : The Morse Potential **


 * Don't hesitate to ask questions and make remarks on this wiki page.**

=-1- The Morse Potential= To model the interaction of two atoms in a diatomic molecule, Philip M. Morse proposed the following potential: math V(x) = \frac 12 \lambda^2 \Big(1-e^{-x}\Big)^2 math It is one of the few analytically solvable models of quantum mechanics: its eigenvalues are given by math E_n = -\frac{1}{8}(2n+1)(2n+1-4\lambda) \quad n=0,1,\dots math and the eigenfunctions by math \psi_n(x) \propto z^{\lambda - n - \frac 12} e^{-\frac 12 z}L_n(z;2\lambda-2n-1) \quad\text{ where }\quad z=2 \lambda e^{-x} math where //L n //(//z//;//α//) is a Laguerre polynomial which expresses as: math L_n(z;\alpha)=\frac{z^{-\alpha}}{n!}e^{z}\frac{d^n}{dz^n}(e^{-z} z^{n+\alpha}) math In this exercise, we recover some of these non-trivial results with the density matrices and path integrals.
 * 1) Determine analytically the ground state //ψ// 0 (//x//) and the first excited state //ψ// 1 (//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= >> (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).
 * 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 numpy.dot product. Plot the entire density matrix using matshow (part of pyplot in matplotlib ), and explain the two-dimensional figures for different temperatures, especially the "width" of the density matrix.
 * 5) At low temperature, and for several values of //λ//, compare:
 * 1) Explain why we can say that «the temperature is low» when those two quantities are equal.
 * 2) A good choice of values for parameters is //N//=100 slices in space, //λ// = 2, //β// initial = 2 −6, 9 iterations of the matrix squaring, with restricting //r// to the interval [-2,10]. Find other values of the parameters yielding consistent results.

=-3- Path-integral simulation for the Morse potential= Set up a naive path-integral simulation (as in SMAC algorithm 3.4) 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 of this histogram with the diagonal density matrix. Again compare with the analytic solution at low temperature.

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