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    One dimensional Quantum Mechanics problems

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    Show that for any normalized |u> , the inequality E0 <= <u|H|u> holds, where E0 is teh lowest energy eigenvalue.

    Prove the following theorem: Every attractive potential in one dimension has at least one bound state.

    Consider the potential V = -aV0 * delta(x)
    Show that it admits a bound state of energy E = -ma^2 * V0^2 / 2h^2

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    Solution Preview

    Please see the attachment.

    Let the eigenbasis of the Hamiltonian H be denoted by such that:
    Also we can order the energy levels in such a way that:
    Now, we can write the normalized wave function as a linear combination of the eigenstates:
    And since is normalized we have:

    Due to (1.2) and the fact that we can write equation (1.5) as an inequality:

    Using (1.4) ...

    Solution Summary

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