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    One dimensional Quantum Mechanics problems

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    1,
    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.

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

    © BrainMass Inc. brainmass.com June 3, 2020, 11:22 pm ad1c9bdddf
    https://brainmass.com/physics/energy/one-dimensional-quantum-mechanics-problems-276248

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    Let the eigenbasis of the Hamiltonian H be denoted by such that:
    (1.1)
    Also we can order the energy levels in such a way that:
    (1.2)
    Now, we can write the normalized wave function as a linear combination of the eigenstates:
    (1.3)
    And since is normalized we have:
    (1.4)
    Now,

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

    (1.6)
    Using (1.4) ...

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