Abstract
As is well known, the neutrino was originally postulated by Wolfgang Pauli in order to ‘save’ the conservation of energy in beta decay. As it turned out later after careful examination.
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Notes
- 1.
Wolfgang Pauli in 1930 initially chose the name ‘neutron.’ The term ‘neutrino’ was introduced later by Enrico Fermi. In 1956, the electron neutrino was detected experimentally for the first time, and in 1962, the muon neutrino. The tauon was observed in 1975, but the corresponding neutrino only in 2000. There may be still other types of neutrinos. These (as yet hypothetical) sterile neutrinos interact only via gravity and not—like the other neutrinos—through the weak interaction (hence the adjective ‘sterile’). See e.g. J. Kopp et al., Are there sterile neutrinos at the eV scale?, Phys. Rev. Lett. 107, 091801 (2011).
- 2.
We remind the reader: \(E^{2}=m_{0}^{2}c^{4}+p^{2}c^{2}\).
- 3.
Call the Hamiltonian for free neutrino motion \(H\). We have \(H\left| \nu _{1}\right\rangle =E_{1}\left| \nu _{1}\right\rangle \) and \(H\left| \nu _{2}\right\rangle =E_{2}\left| \nu _{2}\right\rangle \) with \(\Delta E=E_{1}-E_{2}>0\).
- 4.
Here, \(H\) denotes a (still) unknown operator and not the well-known operator \(-\frac{\hbar ^{2}}{2m}{\varvec{\nabla }}^{2}+V\). Double meanings of this type are quite common in quantum mechanics. We will learn the reason for this in later chapters.
- 5.
Numerical examples: the electron in this system of units has a rest mass of about 0.5 MeV. The accelerator LHC operates with protons of energies of up to 7 TeV.
- 6.
Of course, this is a key parameter—if it is 10\(^{-6}\) eV instead of 10\(^{-3}\) eV, then the length increases correspondingly by a factor of 1000.
- 7.
A recent review which also contains the values cited in Table 8.1 is given by G.L. Fogli et al., ‘Global analysis of neutrino masses, mixings and phases: entering the era of leptonic CP violation searches’, http://arxiv.org/abs/1205.5254v3 (2012).
- 8.
The first three matrices are (except for the phase shift \(\delta \)) the rotation matrices \(D_{x}\left( \theta _{23}\right) D_{y}\left( \theta _{13}\right) D_{z}\left( \theta _{12}\right) \). The first matrix describes e.g. a rotation by the angle \(\theta _{23}\) around the \(x\) axis.
- 9.
The precise determination of these angles is a current topic; see for instance Eugenie S. Reich, ‘Neutrino oscillations measured with record precision’, Nature 08 March 2012, where the measurement of the angle \(\theta _{13}\) is discussed.
- 10.
Since measured values are real, we can interpret them as eigenvalues of Hermitian operators.
- 11.
These properties are not mutually exclusive: A unitary operator or a projection operator can also be e.g. Hermitian.
- 12.
Since these operators exhibit only a few species and are fairly well-behaved, one could also speak of a ‘pet zoo’.
- 13.
These are essentially the three components of the orbital angular momentum operator for angular momentum \(1\); see Chap. 16, Vol. 2.
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Pade, J. (2014). Neutrino Oscillations. In: Quantum Mechanics for Pedestrians 1: Fundamentals. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-00798-4_8
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