Abstract
We study a harmonic molecule confined to a one-dimensional box with impenetrable walls. We explicitly consider the symmetry of the problem for the cases of different and equal masses. We propose suitable variational functions and compare the approximate energies given by the variation method and perturbation theory with accurate numerical ones for a wide range of values of the box length. We analyze the limits of small and large box size.
Highlights
The interaction potential depends on the distance between the particles the problem is not separable and should be treated as a two-dimensional eigenvalue equation
It is almost separable for a sufficiently small box because the interaction potential is negligible in such a limit, and for a sufficiently large box where the boundary conditions have no effect
It is convenient to take into account the symmetry of the problem and classify the states in terms of the irreducible representations because it facilitates the discussion of the connection between both regimes
Summary
There has been great interest in the model of a harmonic oscillator confined to boxes of different shapes and sizes [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] Such a model has been suitable for the study of several physical problems, ranging from dynamical friction in star clusters [4] to magnetic properties of solids [6] and impurities in quantum dots [23]. This model is rather artificial because the cause of the force is not specified It may, for example, arise from an infinitely heavy particle clamped at x0. In such a case we think that it is more interesting to consider that the other particle moves within the box
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