Energy Eigenstates of the Hamiltonian
Not what you're looking for?
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?
Purchase this Solution
Solution Summary
The solution evaluates an expression for operator algebra.
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 ...
Purchase this Solution
Free BrainMass Quizzes
The Moon
Test your knowledge of moon phases and movement.
Variables in Science Experiments
How well do you understand variables? Test your knowledge of independent (manipulated), dependent (responding), and controlled variables with this 10 question quiz.
Introduction to Nanotechnology/Nanomaterials
This quiz is for any area of science. Test yourself to see what knowledge of nanotechnology you have. This content will also make you familiar with basic concepts of nanotechnology.
Intro to the Physics Waves
Some short-answer questions involving the basic vocabulary of string, sound, and water waves.
Basic Physics
This quiz will test your knowledge about basic Physics.