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    Energy Eigenstates of the Hamiltonian

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    1. Operator Algebra. Evaluate the following expressions:

    See attached for equations

    Neutrino Oscillation made oversample. Neutrinos come in three varieties that we know of: the electron neutrino (V_e) the tau neutrino (V_T) and the muon neutrino (which is irrelevant to this problem). Nuclear fusion in the sun's interior produces electron neutrino. A major problem describes (in a super simplified way) one possible explanation.

    Suppose the Hamiltonian of the sun (which describes the interaction of neutrinos with the rest of the sun) can be described as follows:

    (See attached)

    Find the energy eigenstates of this Hamiltonian in terms of the "basis vectors" (V_e) and (V_T), and the corresponding energy eigenvalues.

    Suppose that t = 0, an electron neutrino (V_e) is produced in the center of the sun. What is the state at time t > 0, in terms of the parameters a and b?

    What is the probability that the original electron neutrino will be observed as a tau neutrino at time t?

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    https://brainmass.com/physics/angular-momentum/energy-eigenstates-hamiltonian-286171

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    Solution Preview

    See the attached files.
    First, let's recall that in the eigenbasis of we have:
    (1.1)
    Now let's consider the translation operator:
    (1.2)
    For small a we can express the right hand side as a Taylor series:
    (1.3)
    But this can be also written as:
    (1.4)
    Hence:

    (1.5)
    But we know that in the coordinate space we have:
    (1.6)
    Thus:
    (1.7)
    By the same token:
    (1.8)
    And:
    (1.9)
    When we apply the operators and consecutively we get:

    Hence:
    (1.10)
    So the basis is also an eigenbasis for and applying it on the eigenbasis yields the ...

    Solution Summary

    The solution evaluates an expression for operator algebra.

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