Calculating neutrino oscillations requires a lot of rotations between different bases.
There are three most used bases, vacuum basis which diagonalizes the vacuum Hamiltonian, flavor basis where the flavor states stay, and instataneous matter basis which diagonializes the Hamiltonian with constant matter interaction or the instataneous matter interaction.
The rotation from vacuum basis wavefunction
where the rotation is
The rotation from matter basis wavefunction
where the roation is
The matter mixing angle is determined by the matter potential
where
Rotate Twice
A rotation from vacuum basis to flavor basis then to matter basis is simply the addition of the two rotation, i.e.,
Rotate
Rotate
Given the vacuum basis Hamiltonian
Similarly, the flavor basis Hamiltonian
Vacuum Basis Hamiltonian
The Hamiltonian in this basis is composed of vacuum Hamiltonian which is diagonalized and the matter potential rotated from flavor basis,
Flavor Basis Hamiltonian
As a consistancy check, we now rotate Hamiltonian in vacuum basis to flavor basis.
Numerical Calculation of The Rotations
To write clean code, it is better to define and test there rotations first.
In vacuum basis, Hamiltonian with matter interaction is
where we have got a contribution of
where
Derivation of
Plug the transformation into Schrodinger equation,
Multiplying on both sides of the equation the Hermitian conjugate of the transformation matrix
the two sides becomes
We choose the condition that
which removes the second term in the Hamiltonian in vacuum basis. Finally we have the equation of motion in this new basis
We could even remove all the
The general solution of
where the constant can always be set to 0, which tells us that
Constant Matter Density
As a check, for constant
In this new basis, the Hamiltonian becomes
For a Hamiltonian with matter interection,
where
where
In general the Hamiltonian after the rotation is written in a form
in which
and
Another Form of Matter Potential
For the perturbation we could also make it traceless without changing the probabilities.
Position Dependent Rotation
If the rotation is position dependent, i.e., the matter profile
In the case of discussion here,
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