About Neutrino Oscillations
Vacuum Oscillations
Some Random Notes
Oscillations in Matter (Adiabatic Transitions)
Oscillation Formula
Since adiabatic, no transitions happen between two instantaneous mass eigenstates.
So to calculate the transition probability we only need to:
- Calculate the matrix that rotates the system from instataneous mass states to flavor states; $$ \begin{pmatrix} \ket{\nu_{\mathrm{e}}} \\ \ket{\nu_{\mathrm{x}}} \end{pmatrix} = \begin{pmatrix} \cos \theta_{\mathrm{m}} & \sin \theta_{\mathrm{m}} \\ -\sin \theta_{\mathrm{m}} & \cos \theta_{\mathrm{m}} \end{pmatrix} \begin{pmatrix} \ket{\nu_{\mathrm{L}}}\\ \ket{\nu_{\mathrm{H}}} \end{pmatrix}, $$ where $$ \tan 2\theta_m = \frac{\sin 2\theta_v}{\cos 2\theta_v - \lambda/\omega_v}. $$
- Calculate the initial distribution on the two instataneous mass states. Given initial condition as flavor states $$ \begin{pmatrix} \psi_{\mathrm{e}} \\ \psi_{\mathrm{x}} \end{pmatrix} = \begin{pmatrix} \psi_{\mathrm{e}}(0) \\ \psi_{\mathrm{x}}(0) \end{pmatrix} = \begin{pmatrix} \text{psie0} \\ \text{psix0} \end{pmatrix}. $$ The instantaneous mass states are $$ \begin{pmatrix} \text{psiL0}\\ \text{psiH0} \end{pmatrix} = \begin{pmatrix} \cos \theta_{\mathrm{m}}(0) & -\sin \theta_{\mathrm{m}}(0) \\ \sin \theta_{\mathrm{m}}(0) & \cos \theta_{\mathrm{m}}(0) \end{pmatrix}\begin{pmatrix} \text{psie0} \\ \text{psix0} \end{pmatrix} = \begin{pmatrix} \cos \theta_m(0) \text{psie0} - \sin \theta_m(0) \text{psix0} \\ \sin\theta_m(0) \text{psie0} + \cos \theta_m(0) \text{psix0} \end{pmatrix} $$
- Calculate the flavor states from the instatenous mass states. $$ \begin{pmatrix} \ket{\nu_{\mathrm{e}}}\\ \ket{\nu_{\mathrm{x}}} \end{pmatrix} = \begin{pmatrix} \cos\theta_m & \sin \theta_m \\ -\sin\theta_m & \cos \theta_m \end{pmatrix}\begin{pmatrix} \text{psiL0}\\ \text{psiH0} \end{pmatrix} = \begin{pmatrix} \cos\theta_{\mathrm{m}}\text{psiL0} + \sin \theta_{\mathrm{m}}\text{psiH0} \\ -\sin\theta_{\mathrm{m}} \text{psiL0} + \cos\theta_{\mathrm{m}} \text{psiH0} \end{pmatrix} $$
- What can also be calculated is the energy split between the two instantaneous mass states $$ \omega_{\mathrm m}(x) = \omega_{\mathrm v} \sqrt{ (\lambda(x)/\omega_{\mathrm v} - \cos \left(2\theta_{\mathrm v} \right) )^2 + \sin^2\left( 2\theta_v \right) } $$
Data Table
Data Table Generated From JSON File
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