$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
FullPage
overview
significance
derivation
related experiments
associated concepts
Neutrinos has three flavours, but we can simply this to a spin-1/2 system with two flavours to show that neutrinos oscillate between their states.

Significance

Coming soon

Derivation

The energy of a neutrino is $$\begin{align*} E&=\sqrt{m^2c^4+p^2c^2} \\ &\approx pc + \frac{m_ic^2}^2{2pm} \end{align*}$$.
Assume that $v_e$ and $v_{\mu}$ are quantum states of a spin-1/2 system. Then the state of a neutrino is a linear superposition of eigenstates of the Hamiltonian (energy) $|v_1\rangle$ and $|v_2\rangle$ with respective energies $pc+\frac{m_1^2c^4}{2pc}$ and $pc+\frac{m^2_2c^4}{2pc}$. Then we can write $|v_e\rangle$ and $|v_{\mu}\rangle$ in terms of the possible measurements of energy for $$\begin{align*} |v_e\rangle &= a|v_1\rangle + b|v_2\rangle \\ |v_{\mu}\rangle = c|v_1\rangle + d|v_2\rangle \end{align*}$$ We initialize the system in $|v_e\rangle$. Then the probability of a change in flavour is the same as the probability of a spin flip, which means $$\begin{align*} \text{Prob}_{v_2\to v_{\mu}} \porp \sin^2[\frac{E_1-E_2}{2}t] \end{align*}$$ which means that $$\begin{align*} E_1-E_2\prop(m_1^2-m_2^2), \end{align*}$$ which neutrino oscillations are proportional to the difference of the squared mass of neutrios mass eigenstates.

Associated Concepts

Coming soon
Neutrinos has three flavours, but we can simply this to a spin-1/2 system with two flavours to show that neutrinos oscillate between their states.

Significance

Coming soon

Derivation

The energy of a neutrino is $$\begin{align*} E&=\sqrt{m^2c^4+p^2c^2} \\ &\approx pc + \frac{m_ic^2}^2{2pm} \end{align*}$$.
Assume that $v_e$ and $v_{\mu}$ are quantum states of a spin-1/2 system. Then the state of a neutrino is a linear superposition of eigenstates of the Hamiltonian (energy) $|v_1\rangle$ and $|v_2\rangle$ with respective energies $pc+\frac{m_1^2c^4}{2pc}$ and $pc+\frac{m^2_2c^4}{2pc}$. Then we can write $|v_e\rangle$ and $|v_{\mu}\rangle$ in terms of the possible measurements of energy for $$\begin{align*} |v_e\rangle &= a|v_1\rangle + b|v_2\rangle \\ |v_{\mu}\rangle = c|v_1\rangle + d|v_2\rangle \end{align*}$$ We initialize the system in $|v_e\rangle$. Then the probability of a change in flavour is the same as the probability of a spin flip, which means $$\begin{align*} \text{Prob}_{v_2\to v_{\mu}} \porp \sin^2[\frac{E_1-E_2}{2}t] \end{align*}$$ which means that $$\begin{align*} E_1-E_2\prop(m_1^2-m_2^2), \end{align*}$$ which neutrino oscillations are proportional to the difference of the squared mass of neutrios mass eigenstates.

Associated Concepts

Coming soon
FullPage
overview
significance
derivation
related experiments
associated concepts
FullPage
overview
significance
derivation
related experiments
associated concepts
Neutrinos has three flavours, but we can simply this to a spin-1/2 system with two flavours to show that neutrinos oscillate between their states.

Significance

Coming soon

Derivation

The energy of a neutrino is $$\begin{align*} E&=\sqrt{m^2c^4+p^2c^2} \\ &\approx pc + \frac{m_ic^2}^2{2pm} \end{align*}$$.
Assume that $v_e$ and $v_{\mu}$ are quantum states of a spin-1/2 system. Then the state of a neutrino is a linear superposition of eigenstates of the Hamiltonian (energy) $|v_1\rangle$ and $|v_2\rangle$ with respective energies $pc+\frac{m_1^2c^4}{2pc}$ and $pc+\frac{m^2_2c^4}{2pc}$. Then we can write $|v_e\rangle$ and $|v_{\mu}\rangle$ in terms of the possible measurements of energy for $$\begin{align*} |v_e\rangle &= a|v_1\rangle + b|v_2\rangle \\ |v_{\mu}\rangle = c|v_1\rangle + d|v_2\rangle \end{align*}$$ We initialize the system in $|v_e\rangle$. Then the probability of a change in flavour is the same as the probability of a spin flip, which means $$\begin{align*} \text{Prob}_{v_2\to v_{\mu}} \porp \sin^2[\frac{E_1-E_2}{2}t] \end{align*}$$ which means that $$\begin{align*} E_1-E_2\prop(m_1^2-m_2^2), \end{align*}$$ which neutrino oscillations are proportional to the difference of the squared mass of neutrios mass eigenstates.

Associated Concepts

Coming soon
Neutrinos has three flavours, but we can simply this to a spin-1/2 system with two flavours to show that neutrinos oscillate between their states.

Significance

Coming soon

Derivation

The energy of a neutrino is $$\begin{align*} E&=\sqrt{m^2c^4+p^2c^2} \\ &\approx pc + \frac{m_ic^2}^2{2pm} \end{align*}$$.
Assume that $v_e$ and $v_{\mu}$ are quantum states of a spin-1/2 system. Then the state of a neutrino is a linear superposition of eigenstates of the Hamiltonian (energy) $|v_1\rangle$ and $|v_2\rangle$ with respective energies $pc+\frac{m_1^2c^4}{2pc}$ and $pc+\frac{m^2_2c^4}{2pc}$. Then we can write $|v_e\rangle$ and $|v_{\mu}\rangle$ in terms of the possible measurements of energy for $$\begin{align*} |v_e\rangle &= a|v_1\rangle + b|v_2\rangle \\ |v_{\mu}\rangle = c|v_1\rangle + d|v_2\rangle \end{align*}$$ We initialize the system in $|v_e\rangle$. Then the probability of a change in flavour is the same as the probability of a spin flip, which means $$\begin{align*} \text{Prob}_{v_2\to v_{\mu}} \porp \sin^2[\frac{E_1-E_2}{2}t] \end{align*}$$ which means that $$\begin{align*} E_1-E_2\prop(m_1^2-m_2^2), \end{align*}$$ which neutrino oscillations are proportional to the difference of the squared mass of neutrios mass eigenstates.

Associated Concepts

Coming soon
FullPage
overview
significance
derivation
related experiments
associated concepts