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Simple variational approaches to quantum wells

We discuss two simple variational approaches to quantum wells. The trial harmonic functions analyzed in an earlier paper give reasonable results for all well depths and are particularly suitable for deep wells. On the other hand, the exponential functions proposed here are preferable for shallow wells. We compare the shallow-well expansions for both kind of functions and show that they do not exhibit the cubic term appearing in the exact series. It is also shown that the deep-well expansion for the harmonic functions agree with the first terms of perturbation theory.