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Energy Eigenstates of the Hamiltonian

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?

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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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