# Confined Neutrino Oscillation

Pralay Chakraborty<EMAIL_ADDRESS>Subhankar Roy
<EMAIL_ADDRESS>Department of Physics, Gauhati University, India

###### Abstract

A thought experiment is designed to speculate on the neutrino being trapped in
an impenetrable potential well. Considering both relativistic and non-
relativistic scenarios, we delve into several interesting facets connected to
the flavour oscillation.

###### keywords:

Neutrino Oscillation , Infinite Square Well Potential, Non-Relativistic Case,
Relativistic Case.

††journal: Physics Letters A

##  1 Introduction

The idea of neutrino was first introduced by Wolfgang Pauli in the year of
1930. Initially it was assumed that hardly it could be detected in the
experiments, but later in the year of 1956 it was detected experimentally. The
neutrinos belong to the lepton family and they exhibit peculiar
characteristics which cannot be understood with the help of present
understanding of Standard Model (SM) [1, 2, 3] of Particle physics. For
example, why the neutrinos posses mass is still unanswered in the framework of
SM. But from the quantum mechanics point of view, if the neutrino is massive,
then it must exhibit a peculiar phenomenon called neutrino oscillation [4, 5,
6] which says that the neutrino with certain flavour changes its identity from
one to another while it travels in space, and this prediction agrees well to
the experimental observation [7, 8, 9, 10, 11]. So far, we know that the
neutrinos participate only in weak interaction and thus the Universe is almost
transparent to the neutrinos. So, as per the present understanding of
neutrinos and their interaction is concerned, we understand that hardly, there
exists a possibility to trap a neutrino. However, extremely compact stars such
as neutron stars are characterized by the presence of trapped null geodesics,
which are specific paths followed by subatomic particles like photons and
neutrinos, in the curved space-time. When the neutron stars enter a phase in
their evolutionary cycle that permits geodetical motion of neutrinos, it may
so happen that inside the neutron star a certain portion of the neutrinos
produced within their dense interior becomes trapped or confined within the
star [12]. These trapped neutrinos are unable to escape the strong
gravitational field of these dense objects. In such extreme conditions, the
interactions between neutrinos become highly influential [13]. These
interactions can trigger flavour oscillations, leading to a collective
behaviour where neutrinos of different energies and flavours synchronize and
oscillate together as a group [14, 15]. In the present work, we set a thought
experiment to explore the consequences on neutrino oscillation if somehow the
neutrino could have been confined within a certain potential well. The
findings from this study though speculative and based on hypothetical
circumstances might be relevant in the context of above mentioned systems.

The theoretical formulation of the probability of conversion from one flavour
to another for neutrino was first proposed by Bruno Pontecorvo in the year of
1957 [16], based on the principles of quantum mechanics. However, the theory
holds good for high energetic relativistic free neutrinos [17, 18]. In the
present work, as we assume that the neutrino is bound within certain potential
well and hence we shall explore the possibilities concerning both: non-
relativistic and relativistic scenarios. However, for the latter case, the
situation is little difficult to deal with because of the appearance of the
Klein Paradox [19, 20, 21, 22, 23] in the bound system. What we expect is
that, being bound, the neutrino is available to stay in any of the energy
states allowed by the system. As a result, it is expected that the Probability
of conversion from one flavour to another will certainly be affected by the
quantised energies of the neutrino mass eigenstates.

The plan of the paper is given as follows: In section 2, we discuss the
flavour mixing and the parametrization of the PMNS matrix. We study the
neutrino oscillation probability inside an infinite square well potential in
section 3. The section 4 is devoted to some observations regarding the present
work. We write the summary and discussion of our work in section 5.

##  2 Parametrization of PMNS matrix

The quantum mechanics says that the neutrino mass eigenstates are not
equivalent to the neutrino flavour states, rather the mass eigenstates mix to
produce the neutrino flavour states. Experimentally, the flavour states are
observable and not the mass eigenstates. The neutrino mass eigenstates,
$\nu_{i=1,2,3}$ carry definite mass eigenvalues, $m_{i=1,2,3}$. The neutrino
flavour states, $\nu_{l=e,\mu,\tau}$ are expressed as a linear superposition
of the $\nu_{i}$ as shown below.

$\begin{bmatrix}\nu_{eL}\\\ \nu_{\mu L}\\\ \nu_{\tau L}\\\
\end{bmatrix}=U\begin{bmatrix}\nu_{1}\\\ \nu_{2}\\\ \nu_{3}\\\ \end{bmatrix},$
(1)

where,

$U\,=\,\begin{bmatrix}U_{e1}&U_{e2}&U_{e3}\\\ U_{\mu 1}&U_{\mu 2}&U_{\mu 3}\\\
U_{\tau 1}&U_{\tau 2}&U_{\tau 3}\\\ \end{bmatrix}.$ (2)

.

The $U$ is an unitary matrix and it physically signifies how the mass
eigenstates mix to produce the flavour states. The matrix, U is known as
Pontecorvo-Maki-Nakagawa-Sakata(PMNS) matrix and it is experimentally
observable. As per the standard parametrization scheme adopted by Particle
Data Group (PDG), the $U$ is represented as shown in the following,

$U=\begin{bmatrix}c_{12}c_{13}&c_{13}s_{12}&s_{13}e^{-i\delta}\\\
-c_{23}s_{12}-c_{12}s_{13}s_{s23}e^{i\delta}&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}&c_{13}s_{23}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}s_{13}c_{23}e^{i\delta}&c_{13}c_{23}\\\
\end{bmatrix}\begin{bmatrix}e^{i\alpha}&0&0\\\ 0&e^{i\beta}&0\\\ 0&0&1\\\
\end{bmatrix},$ (3)

where, $c_{ij}=\cos\theta_{ij}$, $s_{ij}=\sin\theta_{ij}$. The angles,
$\theta_{12}$, $\theta_{23}$ and $\theta_{13}$ are termed as solar,
atmospheric and reactor mixing angles. The phase angle, $\delta$ is called the
Dirac CP violating phase; and $\alpha$ and $\beta$ are called the Majorana
phases.

For two neutrino scenario, the neutrino mixing is depicted as shown in the
following

$\begin{bmatrix}\nu_{e}\\\ \nu_{\mu}\\\
\end{bmatrix}=\begin{bmatrix}\cos\theta&\sin\theta\\\
-\,\sin\theta&\cos\theta\\\ \end{bmatrix}\begin{bmatrix}\nu_{1}\\\ \nu_{2}\\\
\end{bmatrix},$ (4)

where, $\theta$ is the mixing angle.

##  3 Neutrino oscillation Probability Inside an Infinite Square Well
Potential

The conversion from one flavour to another in the process of neutrino
oscillation is studied in terms of the Probability
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ which from Pontecorvo’s original formulation
can be expressed as,

$P_{\nu_{e}\rightarrow\nu_{\mu}}\,=\,\sin^{2}(2\,\theta)\,\sin^{2}\Bigg{(}\frac{\Delta
m_{21}^{2}c^{3}l}{4E\hslash}\Bigg{)},$ (5)

where, $\Delta m_{21}^{2}\,=\,m^{2}_{2}-m^{2}_{1}$ and the angle $\theta$ is
the mixing angle. The $l$ is the distance between source and detector and $E$
stands for the energy of the neutrinos. This is to be noted that the above
relation is true for two neutrino flavour scenario, and while deriving the
same the neutrinos are considered as free ultra relativistic particles. But in
connection with the present work, we shall try to look into the possibilities
of neutrino oscillation inside an one dimensional infinite square-well
potential. In general, the form of a infinite square well potential is defined
in the following way,

$V(x)=\Bigg{\\{}\begin{matrix}0&\,\,\,\,0<x<L\\\ \\\
\infty&\,\,\,\,\text{otherwise}\end{matrix},$ (6)

where, L is the length of the box. Whether the particle in a box behaves as
relativistic or non-relativistic depends upon the parameter
$L_{c}\,=\,\frac{\lambda_{c}}{2\,\pi}$, where, $\lambda_{c}$ is the Compton
wavelength of the particle inside the well. The $\lambda_{c}$ can be expressed
as shown below,

$\lambda_{c}=\frac{h}{mc},$ (7)

where, $m$ is the mass of the particle. Depending upon the condition whether
$L>>L_{c}$ or $L\lesssim L_{c}$, we identify the particle inside the box as
non-relativistic and relativistic respectively [24]. Keeping aside the spin
$1/2$ nature of neutrino, we employ both Schr$\ddot{o}$dinger equation (for
non-relativistic case) and Klein-Gordon equation (for relativistic case)
respectively to study the probability of flavour conversion. The same is
studied in the framework of both Dirac and Majorana equation which respects
the spin $1/2$ nature of neutrino.

In the present work, we stick to the normal ordering of the neutrino masses
($m_{3}>m_{2}>m_{1}$). The value of $m_{3}$ is fixed at $0.06\,eV/c^{2}$ and
$m_{1}$ and $m_{2}$ are calculated from two mass squared differences
($\Delta\,m_{21}^{2}$ and $\Delta\,m_{31}^{2}$) as $0.0318\,eV/c^{2}$ and
$0.0329\,eV/c^{2}$. We highlight that the sum of three neutrino masses
($m_{1}\,+\,m_{2}\,+\,m_{3}\,=\,0.124\,eV/c^{2}$) considered in our work is
consistent with the cosmological data [25]. In SI unit system, the mass
eigenvalues are of the order of $10^{-38}kg$. From Eq. (7), the Compton
wavelength associated with the mass eigenvalues comes out to be $\approx
10^{-5}\,m$. For the non-relativistic case, we set the well length $L$ at $1$
meter which is $>>L_{c}$ while for relativistic scenario, the well length is
set at $10^{-5}$ meter which is of the order of the Compton wavelength of
neutrino.

We solve the Schr$\ddot{o}$dinger equation for the well and obtain the mass
eigenstates as shown in the following,

$\displaystyle\nu_{1}\,$ $\displaystyle=$
$\displaystyle\,\sqrt{\frac{2}{L}}\,\sin\Big{(}\frac{n_{1}\pi
x}{L}\Big{)}e^{-\frac{i}{\hslash}E_{n_{1}}t},$ (8) $\displaystyle\nu_{2}\,$
$\displaystyle=$ $\displaystyle\,\sqrt{\frac{2}{L}}\,\sin\Big{(}\frac{n_{2}\pi
x}{L}\Big{)}e^{-\frac{i}{\hslash}E_{n_{2}}t},$ (9)

where, $n_{i}$ is the quantum number associated with the allowed energy
states. In the present work, the quantum numbers $n_{1}$ and $n_{2}$ are
coupled to the neutrino mass eigenstates $\nu_{1}$ and $\nu_{2}$ respectively.
The quantised energy of the particle in $n^{th}_{i}$ state inside the
potential is given by,

$E_{n_{i}}=\frac{n^{2}_{i}\,\pi^{2}\,\hslash^{2}}{2\,m_{i}\,L^{2}},$ (10)

where, $m_{i}$ is the mass of the particle.

We find that the particle may lie in any one of the quantized energy states
depicted by the quantum number $n$. In this regard, we expect that the
oscillation probability will depend on the quantum number of the corresponding
mass eigenstates.

In our case, an electron type neutrino is trapped inside a hypothetical
infinite square well potential. We first consider two flavour oscillation and
derive the expression for $P_{\nu_{e}\rightarrow\nu_{\mu}}$ (see A) as shown
in the following,

$P_{\nu_{e}\rightarrow\nu_{\mu}}\,=\,\sin^{2}(2\,\theta)\,\sin^{2}\Big{[}\frac{\pi^{2}\,\hslash\,t}{4\,L^{2}}\Big{(}\frac{n_{1}^{2}}{m_{1}}-\frac{n_{2}^{2}}{m_{2}}\Big{)}\Big{]}.$
(11)

It is seen that not only the masses of corresponding mass eigenstates of
$\nu_{1}$ and $\nu_{2}$, but also the energy states where the mass eigenstates
lie will contribute to the oscillation probability.

Similarly, for three neutrino case, the expression of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ can be derived as shown below,

$\displaystyle P_{\nu_{e}\rightarrow\nu_{\mu}}$ $\displaystyle=$
$\displaystyle-4\,Re\,(U_{e1}U^{*}_{\mu 1}U^{*}_{e2}U_{\mu
2})\,\sin^{2}\Big{[}\frac{\pi^{2}\hslash
t}{4L^{2}}\Big{(}\frac{n^{2}_{2}}{m_{2}}-\frac{n^{2}_{1}}{m_{1}}\Big{)}\Big{]}+2\,Im\,(U_{e1}U^{*}_{\mu
1}U^{*}_{e2}U_{\mu 2})\,\sin\Big{[}\frac{\pi^{2}\hslash
t}{2L^{2}}\Big{(}\frac{n^{2}_{2}}{m_{2}}-\,$ (12)
$\displaystyle\frac{n^{2}_{1}}{m_{1}}\Big{)}\Big{]}\,-4\,Re\,(U_{e1}U^{*}_{\mu
1}U^{*}_{e3}U_{\mu 3})\,\sin^{2}\Big{[}\frac{\pi^{2}\hslash
t}{4L^{2}}\Big{(}\frac{n^{2}_{3}}{m_{3}}-\frac{n^{2}_{1}}{m_{1}}\Big{)}\Big{]}+2\,Im\,(U_{e1}U^{*}_{\mu
1}U^{*}_{e3}U_{\mu 3})\,\sin\Big{[}\frac{\pi^{2}\hslash t}{2L^{2}}$
$\displaystyle\,\Big{(}\frac{n^{2}_{3}}{m_{3}}-\frac{n^{2}_{1}}{m_{1}}\Big{)}\Big{]}\,-\,4\,Re\,(U_{e2}U^{*}_{\mu
2}U^{*}_{e3}U_{\mu 3})\,\sin^{2}\Big{[}\frac{\pi^{2}\hslash
t}{4L^{2}}\Big{(}\frac{n^{2}_{3}}{m_{3}}-\frac{n^{2}_{2}}{m_{2}}\Big{)}\Big{]}+2\,Im\,(U_{e2}U^{*}_{\mu
2}U^{*}_{e3}U_{\mu 3})\,$ $\displaystyle\sin\Big{[}\frac{\pi^{2}\hslash
t}{2L^{2}}\Big{(}\frac{n^{2}_{3}}{m_{3}}-\frac{n^{2}_{2}}{m_{2}}\Big{)}\Big{]}.$

For relativistic case, dealing with the problems related to trapping potential
is challenging because one may encounter a conjecture termed as Klein Paradox.
Precisely, the reflected flux from the wall is greater than that of the
incident flux. However, we may avoid this problem and rescue the one-particle
picture by adopting the concept of position-dependent mass [26, 27],

$m(x)=\Bigg{\\{}\begin{matrix}m&\,\,\,\,0<x<L\\\ \\\
\infty&\,\,\,\,\text{otherwise}\end{matrix}.$ (13)

We solve Klein-Gordon equation to obtain the mass eigenstates. We see that the
states are identical to those appearing in Eqs. (8-9)provided the expression
of quantised energy, $E_{n_{i}}$ of the mass eigenstate $\nu_{i}$ is given by,

$E_{n_{i}}\,=\,\sqrt{\frac{n_{i}^{2}\pi^{2}\hslash^{2}c^{2}}{L^{2}}\,+\,m_{i}^{2}c^{4}},$
(14)

where, $i\,=\,1,2$. Now, if we consider a relativistic electron type neutrino
trapped inside an infinite square-well potential, the expression for
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ (in case of two flavour oscillation) can be
derived in the following manner,

$\displaystyle P_{\nu_{e}\rightarrow\nu_{\mu}}\,$ $\displaystyle=$
$\displaystyle\,\sin^{2}({2\theta})\,\sin^{2}\Big{[}\frac{t}{2\hslash}\Big{\\{}\Big{(}\frac{n^{2}_{1}\pi^{2}\hslash^{2}c^{2}}{L^{2}}\,+\,m^{2}_{1}c^{4}\Big{)}^{\frac{1}{2}}-\Big{(}\frac{n^{2}_{2}\pi^{2}\hslash^{2}c^{2}}{L^{2}}\,+\,m^{2}_{2}c^{4}\Big{)}^{\frac{1}{2}}\Big{\\}}\,\,\Big{]}.$
(15)

If we take the non-relativistic limit on Eq. (15), the expression of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ is obtained as shown below,

$\displaystyle P_{\nu_{e}\rightarrow\nu_{\mu}}^{NRL(KG)}\,$ $\displaystyle=$
$\displaystyle\,\sin^{2}(2\,\theta)\,\sin^{2}\Big{[}\frac{\pi^{2}\,\hslash\,t}{4\,L^{2}}\Big{(}\frac{n_{1}^{2}}{m_{1}}-\frac{n_{2}^{2}}{m_{2}}\Big{)}\Big{]}.$
(16)

It is evident from Eq. (16), under non-relativistic limit, the
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ becomes identical to the
Schr$\ddot{o}$dinger case. On the other hand, under the ultra-relativistic
limit, the Eq. (15) take the following form,

$\displaystyle P_{\nu_{e}\rightarrow\nu_{\mu}}^{URL(KG)}\,$ $\displaystyle=$
$\displaystyle\,\sin^{2}({2\theta})\,\sin^{2}\Big{[}\frac{t}{2\hslash}\Big{\\{}\frac{h\pi
c}{L}(n_{1}-n_{2})+\frac{c^{3}L}{2\pi\hslash}\Big{(}\frac{m_{1}^{2}}{n_{1}}-\frac{m_{2}^{2}}{n_{2}}\Big{)}\Big{\\}}\,\,\Big{]}.$
(17)

Now, if we consider the spin half nature of the neutrino, the neutrino
oscillation can be studied in the light of Dirac equation. We solve the Dirac
equation inside the well and obtain the following form of the mass
eigenstates[24],

$\displaystyle\nu_{i}$ $\displaystyle=$ $\displaystyle
Be^{i\frac{k_{i}L}{2}}\begin{pmatrix}2\cos\big{(}k_{i}x-\frac{k_{i}L}{2}\big{)}\chi\\\
2i\,r\sin\big{(}k_{i}x-\frac{k_{i}L}{2}\big{)}\sigma_{x}\chi\end{pmatrix}e^{-\frac{i}{\hslash}E_{i}t},\quad\quad
i=1,2.$ (18)

where, $\chi$ is either $(1,0)^{T}$ or $(0,1)^{T}$ and
$r=\frac{ck_{i}\hslash}{E+mc^{2}}$. The Pauli spin matrix, $\sigma_{x}$ is
chosen in a basis where it is diagonal(This is in agreement with the fact that
the movement of neutrino is constrained in the $x$ direction). Here, $k_{i}$
stands for the wave number associated with $\nu_{i}$. However, $k_{i}$ takes
discrete values which can be found out by solving the following transcendental
equation,

$\tan(k_{i}L)\,=\,-\,\frac{\hslash k_{i}}{m_{i}c},$ (19)

and further, the discrete energy eigenvalues $E_{i}$ of the state $\nu_{i}$
are worked out using the following relation,

$E_{i}\,=\,\sqrt{\hslash^{2}c^{2}k^{2}_{i}\,+\,m_{i}^{2}c^{4}},\,\,\,k_{i}\,\neq\,0.$
(20)

Now, if we consider two neutrino case, the expression for
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ can be obtained as shown below,

$\displaystyle P_{\nu_{e}\rightarrow\nu_{\mu}}\,$ $\displaystyle=$
$\displaystyle\,\sin^{2}({2\theta})\,\sin^{2}\Big{[}\frac{t}{2\hslash}\Big{\\{}\Big{(}\hslash^{2}c^{2}k^{2}_{1}\,+\,m^{2}_{1}c^{4}\Big{)}^{\frac{1}{2}}\,-\,\Big{(}\hslash^{2}c^{2}k^{2}_{2}\,+\,m^{2}_{2}c^{4}\Big{)}^{\frac{1}{2}}\Big{\\}}\,\,\Big{]}.$
(21)

Under the non-relativistic limit, Eq. (21) can be expressed as,

$\displaystyle P_{\nu_{e}\rightarrow\nu_{\mu}}^{NRL(Dirac)}\,$
$\displaystyle=$
$\displaystyle\,\sin^{2}({2\theta})\,\sin^{2}\Big{[}\frac{\hslash
t}{4}\Big{(}\frac{k_{1}^{2}}{m_{1}}-\frac{k_{2}^{2}}{m_{2}}\Big{)}\,\Big{]},$
(22)

whereas, for the ultra-relativistic limit, the Eq. (21)appears as shown below,

$\displaystyle P_{\nu_{e}\rightarrow\nu_{\mu}}^{URL(Dirac)}\,$
$\displaystyle=$
$\displaystyle\,\sin^{2}({2\theta})\,\sin^{2}\Big{[}\frac{t}{2\hslash}\Big{\\{}\hslash
c\,(k_{1}-k_{2})+\frac{c^{3}}{2\hslash}\Big{(}\frac{m_{1}^{2}}{k_{1}}-\frac{m_{2}^{2}}{k_{2}}\Big{)}\Big{\\}}\,\Big{]},$
(23)

Similarly, for three neutrino case, we can derive the expression of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ as shown below,

$\displaystyle P_{\nu_{e}\rightarrow\nu_{\mu}}$ $\displaystyle=$
$\displaystyle\,-\,4\,Re\,(U_{e1}U^{*}_{\mu 1}U^{*}_{e2}U_{\mu
2})\,\sin^{2}\,\Big{[}\frac{t}{2\hslash}\Big{\\{}\big{(}k^{2}_{2}\hslash^{2}c^{2}\,+\,m^{2}_{2}c^{4}\big{)}^{\frac{1}{2}}-\big{(}k^{2}_{1}\hslash^{2}c^{2}\,+\,m^{2}_{1}c^{4}\big{)}^{\frac{1}{2}}\Big{\\}}\Big{]}$
$\displaystyle+\,2\,Im\,(U_{e1}\,U^{*}_{\mu 1}U^{*}_{e2}U_{\mu
2})\,\sin\,\Big{[}\frac{t}{\hslash}\Big{\\{}\big{(}k^{2}_{2}\hslash^{2}c^{2}\,+\,m^{2}_{2}c^{4}\big{)}^{\frac{1}{2}}-\big{(}k^{2}_{1}\hslash^{2}c^{2}\,+\,m^{2}_{1}c^{4}\big{)}^{\frac{1}{2}}\Big{\\}}\Big{]}\,$
$\displaystyle\,-\,4\,Re\,(U_{e1}U^{*}_{\mu 1}U^{*}_{e3}U_{\mu
3})\,\sin^{2}\,\Big{[}\frac{t}{2\hslash}\Big{\\{}\big{(}k^{2}_{3}\hslash^{2}c^{2}\,+\,m^{2}_{3}c^{4}\big{)}^{\frac{1}{2}}-\big{(}k^{2}_{1}\hslash^{2}c^{2}\,+\,m^{2}_{1}c^{4}\big{)}^{\frac{1}{2}}\Big{\\}}\Big{]}\,$
$\displaystyle+\,2\,Im\,(U_{e1}\,U^{*}_{\mu 1}U^{*}_{e2}U_{\mu
2})\,\sin\,\Big{[}\frac{t}{\hslash}\Big{\\{}\big{(}k^{2}_{3}\hslash^{2}c^{2}\,+\,m^{2}_{3}c^{4}\big{)}^{\frac{1}{2}}-\big{(}k^{2}_{1}\hslash^{2}c^{2}\,+\,m^{2}_{1}c^{4}\big{)}^{\frac{1}{2}}\Big{\\}}\Big{]}\,$
$\displaystyle\,-\,4\,Re\,(U_{e2}U^{*}_{\mu 2}U^{*}_{e3}U_{\mu
3})\,\sin^{2}\,\Big{[}\frac{t}{2\hslash}\Big{\\{}\big{(}k^{2}_{3}\hslash^{2}c^{2}\,+\,m^{2}_{3}c^{4}\big{)}^{\frac{1}{2}}-\big{(}k^{2}_{2}\hslash^{2}c^{2}\,+\,m^{2}_{2}c^{4}\big{)}^{\frac{1}{2}}\Big{\\}}\Big{]}$
$\displaystyle+\,2\,Im\,(U_{e2}U^{*}_{\mu 2}U^{*}_{e3}U_{\mu
3})\,\sin\,\Big{[}\frac{t}{\hslash}\Big{\\{}\big{(}k^{2}_{3}\hslash^{2}c^{2}\,+\,m^{2}_{3}c^{4}\big{)}^{\frac{1}{2}}-\big{(}k^{2}_{2}\hslash^{2}c^{2}\,+\,m^{2}_{2}c^{4}\big{)}^{\frac{1}{2}}\Big{\\}}\Big{]}.\,$

Here, we wish to highlight that the above equation is true for the Klein
Gordon scenario as well. But in contrast, for Klein Gordon case, $k_{i=1,2,3}$
appearing in above equation is equivalent to $k_{n_{i=1,2,3}}$, where,
$k_{n_{i}}=\frac{n_{i}\pi}{L}$.

##  4 Important Observations

As it is discussed in section 3, in addition to the mass eigenvalues, the
quantum number $n$ also contribute to the neutrino oscillation probability. In
this connection, we can expect that the frequency of oscillation ($f$) will
also vary with the energy (i.e., quantum number $n$) of the mass eigenstates.
In this regard, we concentrate on two cases:

  1. (i)

case-I: The neutrino mass eigenstates carry same quantum number (For two
neutrino scenario, $n=n_{1}=n_{2}$ and the same for three neutrino case refers
to $n=n_{1}=n_{2}=n_{3}$).

  2. (ii)

case-II: The neutrino mass eigenstates bear different quantum numbers.

Needless to mention that the situation $n_{1}=n_{2}$ does not imply that the
two neutrinos lie in the same energy eigenstate because they differ in masses.
The variation of the oscillation frequency with respect to the quantum number
$n$ can be visualised from the parameter $f^{\prime}$, 111In the present work,
though the quantum number $n$ is a desecrate variable, the $f^{\prime}$ is
obtained to understand the variation of the oscillation frequency with respect
to the quantum number $n$. where, $f^{\prime}=\frac{df}{dn}$.

For the non-relativistic scenario, the expression of oscillation frequency
$f_{NR}$ can be derived in the following manner,

$f_{NR}\,=\,\frac{\pi\,\hslash}{8\,L^{2}}\Big{(}\frac{n_{1}^{2}}{m_{1}}-\frac{n_{2}^{2}}{m_{2}}\Big{)}.$
(25)

For the case-I, it is seen that the oscillation frequency increases if the
mass eigenstates take higher values $n$ (see Figs. 1-2). The variation of the
frequency with respect to the quantum number $n$ is given by,

$f^{\prime}=\frac{n\pi\hslash}{4L^{2}}\Big{(}\frac{1}{m_{1}}-\frac{1}{m_{2}}\Big{)}.$
(26)

From Eq. (26), we see that the change in oscillation frequency is directly
proportional to the quantum number $n$. So it is obvious that the frequency
will increase with the increase in the quantum number. For the case-II, we
notice that oscillation frequency as well as the amplitude of oscillation
varies when the energy of the mass eigenstates varies(see Figs. 3-4). This is
because altering the energy states will certainly shift the frequency as the
masses corresponding to the mass eigenstates are different.

Similar to the non-relativistic case, we emphasize over two possible
scenarios. The oscillation frequency for Klein-Gordon case $f_{KG}$ is given
by,

$\displaystyle f_{KG}\,$ $\displaystyle=$
$\displaystyle\,\frac{1}{4\pi\hslash}\Bigg{[}\Big{(}\frac{n^{2}_{1}\pi^{2}\hslash^{2}c^{2}}{L^{2}}\,+\,m^{2}_{1}c^{4}\Big{)}^{\frac{1}{2}}-\Big{(}\frac{n^{2}_{2}\pi^{2}\hslash^{2}c^{2}}{L^{2}}\,+\,m^{2}_{2}c^{4}\Big{)}^{\frac{1}{2}}\Bigg{]}.$
(27)

For the case-I, it is found that the frequency of oscillation decreases as the
energy of the mass eigenstates increases (see Figs. 5-6). In the relativistic
limit, the variation of the frequency with respect to the quantum number $n$
is given by,

$f^{\prime}=\frac{Lc^{3}}{8\pi^{2}\hslash^{2}n^{2}}\,(m_{2}^{2}-m_{1}^{2}).$
(28)

It is clearly seen from Eq. (28) that the quantity $f^{\prime}$ is inversely
proportional to square of the quantum number $n$. Under non-relativistic limit
of the probability corresponding to the Klein-Gordon case, the expression of
frequency is given by,

$f_{NRL}=\frac{1}{4\pi\hslash}\Big{[}\frac{\pi^{2}\hslash^{2}}{2L^{2}}\Big{(}\frac{n_{1}^{2}}{m_{1}}-\frac{n_{2}^{2}}{m_{2}}\Big{)}\Big{]}$
(29)

The variation of frequency with respect to the quantum number $n$ can be
visualised as shown in the following,

$f_{NRL}^{\prime}=\frac{n\pi\hslash}{4L^{2}}\Big{(}\frac{1}{m_{1}}-\frac{1}{m_{2}}\Big{)}.$
(30)

It is to be noted the $f_{NRL}^{\prime}$ is identical to Eq. (26). Similarly,
for the Dirac case, the oscillation frequency $f_{D}$ can be calculated from
the expression as shown in the following,

$\displaystyle f_{D}\,$ $\displaystyle=$
$\displaystyle\,\frac{1}{4\pi\hslash}\Bigg{[}\Big{(}\hslash^{2}c^{2}k^{2}_{1}\,+\,m^{2}_{1}c^{4}\Big{)}^{\frac{1}{2}}\,-\,\Big{(}\hslash^{2}c^{2}k^{2}_{2}\,+\,m^{2}_{2}c^{4}\Big{)}^{\frac{1}{2}}\Bigg{]}.$
(31)

In the above expression, the discrete values of $k_{i=1,2}$ are estimated
using Eq.(19). For this we first plot the left hand side and right hand side
of the said equation against $k_{i}$ 7) and estimate the points where the
plots intersect and then using the Newton-Raphson method, we determine the
possible values of $k_{i}$ for different mass eigenstates (see Table 2).

Similar to Klein-Gordon scenario, it is observed that the oscillation
frequency decreases as the mass eigenstates with higher $k$ values (see Figs.
8-9). This cannot be explained with the help of the parameter $f^{\prime}$ as
two mass eigenstates have different masses. In this way, the two mass
eigenstates cannot posses same wave number.

For the case-II, we observe similar circumstances for Klein-Gordon and Dirac
scenarios. It is seen that the amplitude and the oscillation frequency vary as
the quantum numbers associated with the mass eigenstates are not same (see
Figs. 10-13).

Further, if we look into the fact that neutrinos are electromagnetically
neutral particles, they might resemble their own anti-particle state i.e., a
Majorana Particle [28, 29]. However, the scenarios remain akin to the Dirac
case as we perceive the same transcendental equation for the allowed wave
number $k_{i}$ [30]. Apart from that, we obtain plots of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ analogous to Dirac case.

We highlight that we choose $\theta\,=\,45^{\circ}$ for two neutrino case
while for three neutrino case, we choose the best fit values of the mixing
angles as per the oscillation data (see Table 1) [31, 32].

###  4.1 Conditions for No Flavour Oscillation

In the theory of standard neutrino oscillation, where the neutrino travels in
free space, the oscillation stops when $\Delta m^{2}_{21}=0$. This signifies
that the neutrino mass eigenvalues cannot be degenerate. If the two mass
eigenstates $\nu_{1}$ and $\nu_{2}$ have got same masses then its impossible
to distinguish between the two states and under this scenario, there arises no
question of mixing and hence the neutrino oscillation disappears. However, we
see that if neutrino is confined within a well, then $\nu_{1}$ and $\nu_{2}$
can be discriminated by virtue of the quantum numbers $n_{1}$ and $n_{2}$, in
addition to their distinct masses $m_{1}$ and $m_{2}$ respectively. So, for
the sake of discussion, even if we may speculate a situation when
$m_{1}=m_{2}$, the two states will still remain distinguishable unless
$n_{1}=n_{2}$ and thus, mixing as well as flavour oscillation do occur. Unlike
the confined case, for free space, the said speculation will neither lead to
mixing nor to oscillation. However, for a confined neutrino, the oscillation
will not occur under some specific conditions which are discussed below.

In non-relativistic scenario, we exploit Eq. (11) and Eq. (12) to draw a
important consequence. If two neutrino mass eigenstates posses different
quantum number ($n_{1}\neq n_{2}$), we witness non zero value of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ even if the neutrino mass eigenstates are
degenerate i.e., $m_{1}=m_{2}$ (see Figs. 14 and 15).

From Eq. (11), it is seen that the oscillation will not occur (two neutrino
case) for the following condition,

$m_{1}\,=\,\Bigg{(}\frac{n^{2}_{1}}{n^{2}_{2}}\Bigg{)}\,m_{2}.$ (32)

For three neutrino framework, the condition for zero oscillation probability
is shown below,

$m_{3}\,=\,\Bigg{(}\frac{n^{2}_{3}}{n^{2}_{1}}\Bigg{)}\,m_{1}\,=\,\Bigg{(}\frac{n^{2}_{3}}{n^{2}_{2}}\Bigg{)}\,m_{2}.$
(33)

For the relativistic scenario, the obtained results are similar results to the
non-relativistic case (see Figs. 14-14 and 15-15). However, the oscillation
will not occur for some specific conditions.

For the Klein-Gordon scenario, there will be no oscillation if,

$\Delta\,m^{2}_{21}\,=\,\frac{\pi^{2}\hslash^{2}}{L^{2}c^{2}}\,(n^{2}_{1}-n^{2}_{2}),$
(34)

in view of the two neutrino framework, whereas for three neutrino case, the
neutrino oscillation vanishes if,

$\frac{\Delta m^{2}_{21}}{(n^{2}_{1}-n^{2}_{2})}=\frac{\Delta
m^{2}_{31}}{(n^{2}_{1}-n^{2}_{3})}=\frac{\Delta
m^{2}_{32}}{(n^{2}_{2}-n^{2}_{3})}=\Bigg{(}\frac{\pi^{2}\hslash^{2}}{L^{2}c^{2}}\Bigg{)}.$
(35)

Similarly, for the Dirac case, we exploit Eq. (21) and perceive the condition
for no oscillation as shown below,

$\Delta\,m^{2}_{21}\,=\,\frac{\hslash^{2}}{c^{2}}\,(k^{2}_{1}-k^{2}_{2}),$
(36)

for two neutrino case and the same for three neutrino framework, is obtained
as in the following,

$\frac{\Delta m^{2}_{21}}{(k^{2}_{1}-k^{2}_{2})}=\frac{\Delta
m^{2}_{31}}{(k^{2}_{1}-k^{2}_{3})}=\frac{\Delta
m^{2}_{32}}{(k^{2}_{2}-k^{2}_{3})}=\Bigg{(}\frac{\hslash^{2}}{c^{2}}\Bigg{)}.$
(37)

It is observed that the quantum numbers associated with the mass eigenstates
play a crucial role in the neutrino flavour conversion when the neutrino is
trapped inside a infinite square well potential.

##  5 Summary and Discussion

In our work, we follow the Pontecorvo’s approach and set a thought experiment
to study the neutrino oscillation inside a confined region. We formulate our
work by considering both relativistic and non-relativistic frameworks. The
oscillation is expressed with respect to time and a few important consequences
are discussed. We have studied the probability of flavour conversion
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ considering two cases where the mass
eigenstates $\nu_{1}$ and $\nu_{2}$ are attributed to the same quantum number
and the case when the quantum numbers are different. For the first case, we
encounter a significant difference between the oscillation frequency for
relativistic and non-relativistic scenarios. For the relativistic scenario, it
is seen that the oscillation frequency decreases as the mass eigenstates
corresponds to excited energy states while for the non-relativistic scenario,
the results are completely opposite. For the second case, we observe a change
in the oscillation frequency when the energies of the mass eigenstates change.
However, for the three neutrino case, a change in both frequency and amplitude
of oscillation is observed when the mass eigenstates are placed in higher
energy states. Apart from this, we discuss some conditions for which the
neutrino oscillation will not occur for both the relativistic and non-
relativistic cases. A similar attempt is made by applying the wave packet
approach in the non-relativistic framework [33]. But in the present work, we
deal with different equations of motions and explore the relativistic scenario
as well.

It is needless to mention that the present work is based on a speculation. The
neutrinos have got very low interaction cross-section with matter and the
world is almost transparent to them. So, confining the neutrinos inside a
potential well is not at all straight-forward. However, there may exist some
scenarios where a very dense object like the core of a neutron star may act as
an obstacle to neutrinos and thus, the latter may get trapped there for
several seconds [34, 12]. In this dense environment, the neutrino-neutrino
interactions may trigger the self induced flavour conversion of neutrinos
[13]. So, in this context, we hope that our study may find some relevance.

##  6 Acknowledgement

The research work of Pralay Chakraborty is supported by Innovation in Science
Pursuit for Inspired Research (INSPIRE), Department of Science and Technology,
Government of India, New Delhi vide grant No. IF190651.

## References

  * [1] S. L. Glashow, Partial Symmetries of Weak Interactions, Nucl. Phys. 22 (1961) 579–588.  doi:10.1016/0029-5582(61)90469-2.
  * [2] S. Weinberg, A Model of Leptons, Phys. Rev. Lett. 19 (1967) 1264–1266.  doi:10.1103/PhysRevLett.19.1264.
  * [3] M. Herrero, The Standard model, NATO Sci. Ser. C 534 (1999) 1–59.  arXiv:hep-ph/9812242, doi:10.1007/978-94-011-4689-0_1.
  * [4] Y. Farzan, M. Tortola, Neutrino oscillations and Non-Standard Interactions, Front. in Phys. 6 (2018) 10.  arXiv:1710.09360, doi:10.3389/fphy.2018.00010.
  * [5] S. A. R. Ellis, K. J. Kelly, S. W. Li, Current and Future Neutrino Oscillation Constraints on Leptonic Unitarity, JHEP 12 (2020) 068.  arXiv:2008.01088, doi:10.1007/JHEP12(2020)068.
  * [6] G. Fantini, A. Gallo Rosso, F. Vissani, V. Zema, Introduction to the Formalism of Neutrino Oscillations, Adv. Ser. Direct. High Energy Phys. 28 (2018) 37–119.  arXiv:1802.05781, doi:10.1142/9789813226098_0002.
  * [7] Y. Fukuda, et al., Measurements of the solar neutrino flux from Super-Kamiokande’s first 300 days, Phys. Rev. Lett. 81 (1998) 1158–1162, [Erratum: Phys.Rev.Lett. 81, 4279 (1998)].  arXiv:hep-ex/9805021, doi:10.1103/PhysRevLett.81.1158.
  * [8] Y. Fukuda, et al., Evidence for oscillation of atmospheric neutrinos, Phys. Rev. Lett. 81 (1998) 1562–1567.  arXiv:hep-ex/9807003, doi:10.1103/PhysRevLett.81.1562.
  * [9] J. Boger, et al., The Sudbury neutrino observatory, Nucl. Instrum. Meth. A 449 (2000) 172–207.  arXiv:nucl-ex/9910016, doi:10.1016/S0168-9002(99)01469-2.
  * [10] Q. R. Ahmad, et al., Direct evidence for neutrino flavor transformation from neutral current interactions in the Sudbury Neutrino Observatory, Phys. Rev. Lett. 89 (2002) 011301.  arXiv:nucl-ex/0204008, doi:10.1103/PhysRevLett.89.011301.
  * [11] Q. R. Ahmad, et al., Measurement of day and night neutrino energy spectra at SNO and constraints on neutrino mixing parameters, Phys. Rev. Lett. 89 (2002) 011302.  arXiv:nucl-ex/0204009, doi:10.1103/PhysRevLett.89.011302.
  * [12] Z. Stuchlik, J. Hladik, M. Urbanec, Neutrino trapping in braneworld extremely compact stars, Gen. Rel. Grav. 43 (2011) 3163–3190.  arXiv:1108.5767, doi:10.1007/s10714-011-1229-z.
  * [13] A. Mirizzi, I. Tamborra, H.-T. Janka, N. Saviano, K. Scholberg, R. Bollig, L. Hudepohl, S. Chakraborty, Supernova Neutrinos: Production, Oscillations and Detection, Riv. Nuovo Cim. 39 (1-2) (2016) 1–112.  arXiv:1508.00785, doi:10.1393/ncr/i2016-10120-8.
  * [14] H. Duan, G. M. Fuller, J. Carlson, Y.-Z. Qian, Coherent Development of Neutrino Flavor in the Supernova Environment, Phys. Rev. Lett. 97 (2006) 241101\.  arXiv:astro-ph/0608050, doi:10.1103/PhysRevLett.97.241101.
  * [15] H. Duan, G. M. Fuller, J. Carlson, Y.-Z. Qian, Simulation of Coherent Non-Linear Neutrino Flavor Transformation in the Supernova Environment. 1. Correlated Neutrino Trajectories, Phys. Rev. D 74 (2006) 105014.  arXiv:astro-ph/0606616, doi:10.1103/PhysRevD.74.105014.
  * [16] B. Pontecorvo, Mesonium and anti-mesonium, Sov. Phys. JETP 6 (1957) 429.
  * [17] S. M. Bilenky, B. Pontecorvo, Again on Neutrino Oscillations, Lett. Nuovo Cim. 17 (1976) 569.  doi:10.1007/BF02746567.
  * [18] S. M. Bilenky, B. Pontecorvo, Lepton Mixing and Neutrino Oscillations, Phys. Rept. 41 (1978) 225–261.  doi:10.1016/0370-1573(78)90095-9.
  * [19] A. Calogeracos, N. Dombey, Klein tunneling and the Klein paradox, Int. J. Mod. Phys. A 14 (1999) 631–644.  arXiv:quant-ph/9806052, doi:10.1142/S0217751X99000312.
  * [20] N. Dombey, A. Calogeracos, Seventy years of the Klein paradox, Phys. Rept. 315 (1999) 41–58.  doi:10.1016/S0370-1573(99)00023-X.
  * [21] A. Hansen, F. Ravndal, Klein’s Paradox and Its Resolution, Phys. Scripta 23 (1981) 1036.  doi:10.1088/0031-8949/23/6/002.
  * [22] R. K. Su, G. G. Siu, X. Chou, Barrier penetration and Klein Paradox, J. Phys. A 26 (1993) 1001–1005.  doi:10.1088/0305-4470/26/4/026.
  * [23] A. Calogeracos, N. Dombey, History and physics of the Klein paradox, Contemp. Phys. 40 (1999) 313–321.  arXiv:quant-ph/9905076, doi:10.1080/001075199181387.
  * [24] P. Alberto, C. Fiolhais, V. M. S. Gil, Relativistic particle in a box, European Journal of Physics 17 (1) (1996) 19–24.  doi:10.1088/0143-0807/17/1/004.
URL https://doi.org/10.1088/0143-0807/17/1/004

  * [25] N. Aghanim, et al., Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6, [Erratum: Astron.Astrophys. 652, C4 (2021)].  arXiv:1807.06209, doi:10.1051/0004-6361/201833910.
  * [26] P. Alberto, S. Das, E. C. Vagenas, Relativistic particle in a three-dimensional box, Phys. Lett. A 375 (2011) 1436–1440.  arXiv:1102.3192, doi:10.1016/j.physleta.2011.02.030.
  * [27] P. Alberto, S. Das, E. C. Vagenas, Relativistic particle in a box: Klein–Gordon versus Dirac equations, Eur. J. Phys. 39 (2) (2018) 025401\.  arXiv:1711.06313, doi:10.1088/1361-6404/aa9b43.
  * [28] H. Arodź, Relativistic Quantum Mechanics of the Majorana Particle, Acta Phys. Polon. B 50 (2019) 2165–2187.  arXiv:2002.07482, doi:10.5506/APhysPolB.50.2165.
  * [29] L. Borsten, M. J. Duff, Majorana Fermions in Particle Physics, Solid State and Quantum Information, Subnucl. Ser. 53 (2017) 77–121.  arXiv:1612.01371, doi:10.1142/9789813208292_0002.
  * [30] S. De Vincenzo, On real solutions of the Dirac equation for a one-dimensional Majorana particle, Results Phys. 15 (2019) 102598.  arXiv:2007.03072, doi:10.1016/j.rinp.2019.102598.
  * [31] I. Esteban, M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz, A. Zhou, The fate of hints: updated global analysis of three-flavor neutrino oscillations, JHEP 09 (2020) 178.  arXiv:2007.14792, doi:10.1007/JHEP09(2020)178.
  * [32] M. C. Gonzalez-Garcia, M. Maltoni, T. Schwetz, NuFIT: Three-Flavour Global Analyses of Neutrino Oscillation Experiments, Universe 7 (12) (2021) 459.  arXiv:2111.03086, doi:10.3390/universe7120459.
  * [33] L. Johns, Neutrino oscillations in a trapping potential, Int. J. Mod. Phys. A 34 (28) (2019) 1950160.  arXiv:1906.01673, doi:10.1142/S0217751X19501604.
  * [34] S. Reddy, M. Prakash, Neutrino scattering in a newly born neutron star, Astrophys. J. 478 (1997) 689–700.  arXiv:astro-ph/9610115, doi:10.1086/303804.

##  Appendix A Important Derivations

To derive the probability expression Eq. (11), we would like to start from the
Pontecorvo’s approach of finding the oscillation probability.

Inside the well, the wave functions at $t=0$,

$\displaystyle\mid\nu_{1}(t=0)\rangle\,$ $\displaystyle=$
$\displaystyle\,\sqrt{\frac{2}{L}}\,\sin\Big{(}\frac{n_{1}\pi x}{L}\Big{)},$
(38) $\displaystyle\mid\nu_{2}(t=0)\rangle\,$ $\displaystyle=$
$\displaystyle\,\sqrt{\frac{2}{L}}\,\sin\Big{(}\frac{n_{2}\pi x}{L}\Big{)},$
(39)

For two neutrino scenario, the neutrino mixing is depicted as shown in the
following

$\begin{bmatrix}\mid\nu_{e}\rangle\\\ \mid\nu_{\mu}\rangle\\\
\end{bmatrix}=\begin{bmatrix}\cos\theta&\sin\theta\\\
-\,\sin\theta&\cos\theta\\\ \end{bmatrix}\begin{bmatrix}\mid\nu_{1}\rangle\\\
\mid\nu_{2}\rangle\\\ \end{bmatrix},$ (40)

where, $\theta$ is the mixing angle.

At $t=0$, we consider the neutrino is purely electron type. In this regard,
the wave function corresponds to $\nu_{e}$ is given by,

$\mid\psi(t=0)\rangle=\cos\theta\mid\nu_{1}(t=0)\rangle+\sin\theta\mid\nu_{2}(t=0)\rangle.$
(41)

The mass eigenstates $\mid\nu_{1}(t=0)\rangle$ and $\mid\nu_{2}(t=0)\rangle$
will evolve in time $t$ as shown below:

$\displaystyle\mid\nu_{k}(t)\rangle=e^{-\frac{i}{\hslash}\,\hat{H_{k}}t}\mid\nu_{k}(t=0)\rangle=e^{-\frac{i}{\hslash}\,E_{n_{k}}t}\mid\nu_{k}(t=0)\rangle,\quad\quad
k=1,2.$ (42)

At time $t$,

$\mid\psi(t)\rangle=\cos\theta
e^{-\frac{i}{\hslash}E_{n_{1}}t}\mid\nu_{1}(t=0)\rangle+\sin\theta
e^{-\frac{i}{\hslash}E_{n_{2}}t}\mid\nu_{2}(t=0)\rangle.$ (43)

From Eq. (40), we can write,

$\displaystyle\mid\nu_{1}(t=0)\rangle$ $\displaystyle=$
$\displaystyle\cos\theta\mid\nu_{e}\rangle-\sin\theta\mid\nu_{\mu}\rangle,$
(44) $\displaystyle\mid\nu_{2}(t=0)\rangle$ $\displaystyle=$
$\displaystyle\sin\theta\mid\nu_{e}\rangle+\cos\theta\mid\nu_{\mu}\rangle,$
(45)

and we rewrite Eq. (43) as shown in the following,

$\displaystyle\mid\psi(t)\rangle$ $\displaystyle=$
$\displaystyle\cos\theta(\cos\theta\mid\nu_{e}\rangle-\sin\theta\mid\nu_{\mu}\rangle)e^{-\frac{i}{\hslash}E_{n_{1}}t}+\sin\theta(\sin\theta\mid\nu_{e}\rangle+\cos\theta\mid\nu_{\mu}\rangle)e^{-\frac{i}{\hslash}E_{n_{2}}t},$
(46) $\displaystyle=$ $\displaystyle
e^{-\frac{i}{\hslash}E_{n_{1}}t}(\cos^{2}\theta+\sin^{2}\theta
e^{\frac{i}{\hslash}(E_{n_{1}}-E_{n_{2}})t})\mid\nu_{e}\rangle-e^{-\frac{i}{\hslash}E_{n_{1}}t}(1-e^{\frac{i}{\hslash}(E_{n_{1}}-E_{n_{2}})t})\cos\theta\sin\theta\mid\nu_{\mu}\rangle.$
$\displaystyle=$ $\displaystyle
c_{e}\mid\nu_{e}\rangle+c_{\mu}\mid\nu_{\mu}\rangle.$

where, $c_{e}=\langle\nu_{e}\mid\psi(t)\rangle$ and
$c_{\mu}=\langle\nu_{\mu}\mid\psi(t)\rangle$.

Now, the probability that the electron neutrino $\nu_{e}$ oscillate into muon
neutrino $\nu_{\mu}$ is given by,

$\displaystyle P(\nu_{e}-\nu_{\mu})$ $\displaystyle=$ $\displaystyle
c_{\mu}c^{*}_{\mu}$ (47) $\displaystyle=$
$\displaystyle(1-e^{\frac{i}{\hslash}(E_{n_{1}}-E_{n_{2}})t})(1-e^{-\frac{i}{\hslash}(E_{n_{1}}-E_{n_{2}})t})\cos^{2}\theta\sin^{2}\theta$
$\displaystyle=$
$\displaystyle\frac{1}{4}(2-2\cos\frac{(E_{n_{1}}-E_{n_{2}})t}{\hslash})\sin^{2}2\theta$
$\displaystyle=$
$\displaystyle\sin^{2}2\theta\,\sin^{2}\frac{(E_{n_{1}}-E_{n_{2}})t}{2\hslash}$

From Eq. (10), we put the expressions of $E_{n_{1}}$ and $E_{n_{2}}$ in Eq.
(47) and obtain the expression for $P(\nu_{e}-\nu_{\mu})$ as shown in the
following,

$P_{\nu_{e}\rightarrow\nu_{\mu}}\,=\,\sin^{2}(2\,\theta)\,\sin^{2}\Big{[}\frac{\pi^{2}\,\hslash\,t}{4\,L^{2}}\Big{(}\frac{n_{1}^{2}}{m_{1}}-\frac{n_{2}^{2}}{m_{2}}\Big{)}\Big{]}.$
(48)

We would like to highlight an important discussion here. In principle, the
mass eigenstate $\nu_{1}$ may correspond to any of the states like
$\nu_{1}^{1}$, $\nu_{1}^{2}$, $\nu_{1}^{3}\cdots$ for $n_{1}=1,2,3,\cdots$
respectively and the same holds good for $\nu_{2}$ as well. Under this
scenario, the mixing of neutrinos can be visualised as shown below.

$\displaystyle\nu_{e}$ $\displaystyle=$ $\displaystyle
c_{1}^{1}\nu_{1}^{1}+c_{1}^{2}\nu_{1}^{2}+c_{1}^{3}\nu_{1}^{3}+\cdots+c_{2}^{1}\nu_{2}^{1}+c_{2}^{2}\nu_{2}^{2}+c_{2}^{3}\nu_{2}^{3}+\cdots$
(49) $\displaystyle\nu_{\mu}$ $\displaystyle=$ $\displaystyle
d_{1}^{1}\nu_{1}^{1}+d_{1}^{2}\nu_{1}^{2}+d_{1}^{3}\nu_{1}^{3}+\cdots+d_{2}^{1}\nu_{2}^{1}+d_{2}^{2}\nu_{2}^{2}+d_{2}^{3}\nu_{2}^{3}+\cdots$
(50)

Hence,

$\begin{pmatrix}\nu_{e}\\\
\nu_{\mu}\end{pmatrix}=\tilde{U}\begin{pmatrix}\nu^{1}_{1}\\\ \nu^{2}_{1}\\\
\nu^{3}_{1}\\\ \vdots\\\ \nu^{1}_{2}\\\ \nu^{2}_{2}\\\ \nu^{3}_{2}\\\
\vdots\end{pmatrix}.$ (51)

We understand that $\tilde{U}$ is a rectangular matrix. If we believe that the
neutrino mixing matrix is unitary, then certainly at a time, only two states
can contribute towards the mixing. Now this mixing can happen in three ways:
(i) mixing between two states having same mass but different quantum numbers
($m_{1}=m_{2},\,n_{1}\neq n_{2}$) (ii) mixing between two states having same
quantum number but different masses ($n_{1}=n_{2},\,m_{1}\neq m_{2}$) and
(iii) that between the states having different quantum numbers and different
masses ($m_{1}\neq m_{2},\,n_{1}\neq n_{2}$).

Parameters | btf $\pm$ $1\,\sigma$ |  $3\,\sigma$ range
---|---|---
$\theta_{12}/^{\circ}$ | $33.44^{+0.77}_{-0.74}$ | 31.27 - 35.86
$\theta_{23}/^{\circ}$ | $49.2^{+1.0}_{-1.3}$ | 39.5 - 52
$\theta_{13}/^{\circ}$ | $8.57^{+0.13}_{-0.12}$ | 8.20 - 8.97
$\delta_{CP}/^{\circ}$ | $194^{+52}_{-25}$ | 105 - 405
$\frac{\Delta\,m_{21}^{2}}{10^{-5}\,eV^{2}}$ | $7.42^{+0.21}_{-0.20}$ | 6.82 - 8.04
$\frac{|\Delta\,m_{31}^{2}|}{10^{-3}\,eV^{2}}$ | $2.515^{+0.028}_{-0.027}$ | 2.431 - 2.599
Table 1: $3\,\nu$ oscillation parameters obtained from global analysis of neutrino data in normal ordering [31]. Mass Eigenstates | $k_{1}(\,m^{-1})$ | $k_{2}(\,m^{-1})$ | $k_{3}(\,m^{-1})$
---|---|---|---
$m_{1}$ | $220185$ | $502241$ | $805122$
$m_{2}$ | $221522$ | $503187$ | $805770$
$m_{3}$ | $246048$ | $523767$ | $820822$
Table 2: Numerical values of $k_{1}$, $k_{2}$ and $k_{3}$ with respect to
$m_{1}$, $m_{2}$ and $m_{3}$ respectively.

Figure 1: (a) Shows the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$
for $n_{1}=1$ and $n_{2}=1$ in case of two flavor oscillation in non-
relativistic case. (b) Plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$
is shown when $n_{1}=2$ and $n_{2}=2$. (c) Presents the plot of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ for $n_{1}=3$ and $n_{2}=3$.
From the plots, we notice that the oscillation frequency increases as the
energy of the mass eigenstates increases.

Figure 2: (a) Displays the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against
$t$ when $n_{1}=1$, $n_{2}=1$ and $n_{3}=1$ for three flavor oscillation in
non-relativistic scenario . (b) Plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$
against $t$ is shown where $n_{1}=2$, $n_{2}=2$ and $n_{3}=2$. (c) Gives the
plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ for $n_{1}=3$, $n_{2}=3$
and $n_{3}=3$. From the plots, we see a increase in oscillation frequency with
the increase in energy of the mass eigenstates.

Figure 3: (a) Presents the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against
$t$ for $n_{1}=1$ and $n_{2}=2$ and vice-versa for two flavor oscillation in
non-relativistic case. (b) Plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against
$t$ is shown for $n_{1}=1$ and $n_{2}=3$ and vice-versa. (c) Manifests the
plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ for $n_{1}=2$ and
$n_{2}=3$ and vice-versa. From the plots, we find that the oscillation
frequency changes with the change in energy of the mass eigenstates.

Figure 4: (a) Shows the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$
for three flavor oscillation in non-relativistic case, where, $n_{1}$, $n_{2}$
and $n_{3}$ can have values $(1,1,2)$, $(1,2,1)$ and $(2,1,1)$ respectively.
(b) Gives the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ for the
combinations of quantum numbers $(1,1,3)$, $(1,3,1)$ and $(3,1,1)$
respectively. (c) Plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ is
shown where $n_{1}$, $n_{2}$ and $n_{3}$ are chosen to be $(1,2,3)$, $(2,3,1)$
and $(3,1,2)$ respectively. From the plots, it is clear that the frequency as
well the amplitude of the oscillation changes as the energy of the mass
eigenstates changes.

Figure 5: (a) Shows the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$
where, $n_{1}=1$ and $n_{2}=1$ for two flavor oscillation in relativistic case
(Klein-Gordon). (b) The plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$
is manifested for $n_{1}=2$ and $n_{2}=2$. (c) Displays the plot of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ when $n_{1}=3$ and $n_{2}=3$.
From the plots, we notice a decrease in the oscillation frequency as the
energy of the mass eigenstates increases.

Figure 6: (a) Shows the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$,
where, $n_{1}=1$, $n_{2}=1$ and $n_{3}=1$ for three flavor oscillation in
relativistic scenario (Klein-Gordon). (b) The plot of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ is manifested for $n_{1}=2$,
$n_{2}=2$ and $n_{3}=2$. (c) Presents the plot of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ for $n_{1}=3$, $n_{2}=3$ and
$n_{3}=3$ respectively. From the plots, we observe that the oscillation
frequency decreases with the increase in energy of the mass eigenstates.

Figure 7: (a) Presents the plot of $\tan(kL)$ and $\frac{-\hslash k}{m_{1}c}$
against the wave number $k$. (b) The plot of $\tan(kL)$ and $\frac{-\hslash
k}{m_{2}c}$ against the wave number $k$ is displayed. (c) Manifests the plot
of $\tan(kL)$ and $\frac{-\hslash k}{m_{3}c}$ against the wave number $k$.
From figure, we encounter three intersection points of the above said
expressions. From these points, the values of $k_{1}$, $k_{2}$ and $k_{3}$ are
calculated for three mass eigenstates.

Figure 8: (a) Displays the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against
$t$ where, $k_{1}=220185\,m^{-1}$ and $k_{2}=221522\,m^{-1}$ for two flavor
oscillation in relativistic case (Dirac). (b) Presents the plot of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ for $k_{1}=502241\,m^{-1}$ and
$k_{2}=503187\,m^{-1}$. (c) Manifests the plot of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ when $k_{1}=805122\,m^{-1}$ and
$k_{2}=805770\,m^{-1}$. From the plots, we find that oscillation frequency
decreases with the increase in the energy of the mass eigenstates.

Figure 9: (a) The plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ is
shown, where, $k_{1}=220185\,m^{-1}$, $k_{2}=221522\,m^{-1}$ and
$k_{3}=246048\,m^{-1}$ for three flavor oscillation in relativistic scenario
(Dirac). (b) Presents the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against
$t$ for $k_{1}=502241\,m^{-1}$, $k_{2}=503187\,m^{-1}$ and
$k_{3}=523767\,m^{-1}$. (c) Manifests the plot of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ when $k_{1}=805122\,m^{-1}$,
$k_{2}=805770\,m^{-1}$ and $n_{3}=820822\,m^{-1}$ respectively. From the
plots, we perceive that the oscillation frequency decreases when the energy of
the mass eigenstates increases.

Figure 10: (a) Gives the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$
for $n_{1}=1$ and $n_{2}=2$ and vice-versa when two flavor oscillation is
considered in relativistic case (Klein-Gordon). (b) Presents the plot of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ for $n_{1}=1$ and $n_{2}=3$ and
vice-versa. (c) Displays the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against
$t$, where, $n_{1}=2$ and $n_{2}=3$ and vice-versa. From the figure, we notice
rarely any change in oscillation frequency for above mentioned energies.

Figure 11: (a) Presents the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against
$t$ when three flavor oscillation in relativistic case is considered (Klein-
Gordon), where, $n_{1}$, $n_{2}$ and $n_{3}$ can have values $(1,1,2)$,
$(1,2,1)$ and $(2,1,1)$ respectively. (b) Shows the plot of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ for the combinations of quantum
numbers $(1,1,3)$, $(1,3,1)$ and $(3,1,1)$ respectively. (c) Plot of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ is manifested where the values
of $n_{1}$, $n_{2}$ and $n_{3}$ are chosen to be $(1,2,3)$, $(2,3,1)$ and
$(3,1,2)$ respectively. From the plots, we observe that the frequency and the
amplitude of oscillation changes with the change in energy of the mass
eigenstates.

Figure 12: (a) Manifests the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against
$t$ for the combinations $k_{1}=220185\,m^{-1}$, $k_{2}=503187\,m^{-1}$ and
$k_{1}=502241\,m^{-1}$, $k_{2}=221522\,m^{-1}$ when two flavor oscillation is
considered in relativistic case (Dirac). (b) Displays the plot of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ for the combinations of the
values of the wave number; $k_{1}=220185\,m^{-1}$, $k_{2}=805770\,m^{-1}$ and
$k_{1}=805122\,m^{-1}$, $k_{2}=221522\,m^{-1}$. (c) Presents the plot of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$, where, the wave number $k$ can
have values $k_{1}=502241\,m^{-1}$, $k_{2}=805770\,m^{-1}$ and
$k_{1}=805122\,m^{-1}$, $k_{2}=503187\,m^{-1}$. From the figure, we observe
very small change in oscillation frequency for above mentioned energies.

Figure 13: (a) The plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ is
shown for three flavor oscillation in relativistic case (Dirac) for
combinations of the wave number ($k_{1}=220185\,m^{-1}$,
$k_{2}=503187\,m^{-1}$, $k_{3}=246048\,m^{-1}$), ($k_{1}=220185\,m^{-1}$,
$n_{2}=221522\,m^{-1}$, $k_{3}=523767\,m^{-1}$) and ($k_{1}=502241\,m^{-1}$,
$k_{2}=221522\,m^{-1}$, $k_{3}=246048\,m^{-1}$). (b) Gives the plot of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ where the combinations of
quantum numbers are ($k_{1}=220185\,m^{-1}$, $n_{2}=805770\,m^{-1}$,
$k_{3}=246048\,m^{-1}$), ($k_{1}=220185\,m^{-1}$, $k_{2}=221522\,m^{-1}$,
$k_{3}=820822\,m^{-1}$) and ($k_{1}=805122\,m^{-1}$, $k_{2}=221522\,m^{-1}$,
$k_{3}=246048\,m^{-1}$). (c) Displays the plot of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$, where, the values of $k_{1}$,
$k_{2}$ and $k_{3}$ are chosen in the following combinations,
($k_{1}=805122\,m^{-1}$, $n_{2}=221522\,m^{-1}$, $k_{3}=523767\,m^{-1}$),
($k_{1}=220185\,m^{-1}$, $k_{2}=503187\,m^{-1}$, $k_{3}=820822\,m^{-1}$) and
($k_{1}=502241\,m^{-1}$, $k_{2}=805770\,m^{-1}$, $k_{3}=246048\,m^{-1}$). From
the plots, we observe that the frequency and the amplitude of oscillation
changes when the energy of the mass eigenstates changes.

Figure 14: (a) Presents the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against
$t$ for $m_{1}=m_{2}$ in case of two flavor oscillation in non-relativistic
case. (b) Displays the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$
for two flavor oscillation in relativistic case (Klein-Gordon), where,
$m_{1}=m_{2}$. (c) Shows the plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against
$t$ for $m_{1}=m_{2}$ for two flavor oscillation in relativistic case (Dirac).
From the plots, we notice a non-zero oscillation probability even if the mass
eigenvalues are degenerate.

Figure 15: (a) The plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ is
manifested for $m_{1}=m_{2}=m_{3}$, when three flavor oscillation in non-
relativistic case is considered. (b) Presents the plot of
$P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ where, $m_{1}=m_{2}=m_{3}$ for
three flavor oscillation in relativistic case (Klein-Gordon). (c) Displays the
plot of $P_{\nu_{e}\rightarrow\nu_{\mu}}$ against $t$ for three flavor
oscillation in relativistic case (Dirac) for $m_{1}=m_{2}=m_{3}$. From the
plots, we see that the non-zero oscillation probability exists for degenerate
neutrino mass eigenvalues.