Explore BrainMass

Explore BrainMass

    Energy Eigenstates of the Hamiltonian

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    See the attached files.

    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?

    © BrainMass Inc. brainmass.com March 4, 2021, 9:57 pm ad1c9bdddf


    Solution Preview

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

    But we know that in the coordinate space we have:
    By the same token:
    When we apply the operators and consecutively we get:

    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.