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A simple formula from which a geometrical picture of neutrino oscillations with two flavors may be constructed is derived from the equation of motion for the neutrinos. Applications of the picture to the nonadiabatic as well as adiabatic Mikheyev-Smirnov-Wolfenstein effects in the solar-neutrino problem are given. A generalization of the picture to the three-generation case is also briefly discussed.The neutrino oscillation can be described as the rotation of a unit vector v J t ) representing a neutrino of flavor a around the mass eigenstate v i ( t ) . This is analogous to the precession of a magnetic dipole in a (steady and varying) external magnetic field. The projection of v,( t ) on v,( t ) is related to the amplitude of finding vi ( t ) in the state v,(t). Although the geometrical representation of neutrino oscillation is a convenient way to understand and analyze the oscillation problems, the representation itself is not unique. For example, Mikheyev and ~m i r n o v ' (MS) have used an orthogonal basis with the axes v,, Rev,, and Imv2 to describe the oscillation of two generations of neutrinos. In their picture, the flavor vector, e.g., v, will rotate around v , with an angle 0 starting from the initial position v, ( 0 ) =Rev, = (cos0, sine, 0). For oscillations in matter, known as the ~i k h e~e v -s m i r n o v 2 -~o l f e n s t e i n~ (MSW) effects, the mass eigenstates v , and v2 in vacuum are replaced by the effective (in matter) mass eigenstates v:"' and vkm'. Although this representation correctly describes the oscillations both in vacuum and matter, the choice of the orthogonal basis is somewhat arbitrary.On the other hand, ~e s s i a h ,~ and Kim, Nussinov and sze5 have recently discussed the use of a geometrical picture in three-dimensional Euclidean flavor space deduced from the two-valued representation in flavor space of the rotation group. They have independently applied this picture to the case of adiabatic approximation of the MSW effects in the solar-neutrino problem.In this paper we present a clear and simple derivation of the geometric picture discussed in Refs. 4 and 5 starting from the original equation of motion for two generations of neutrinos. Applications of the picture to nonadiabatic as well as adiabatic MSW effects in the solarneutrino problem are given. A generalization to the three-generation case is also discussed.We will start with the case of two neutrino flavors. Let $=(v, v,lT be a neutrino state expressed in the weak basis and normalized such that $+I)= 1. The equation of motion for neutrinos in matter is given by where A =~V '~G~N , E with GF the Fermi constant, N, the electron number density in matter, and E the neutrino energy. Also in Eq. ( I ) , ~= m ; -m : , the masssquared difference. In a vacuum, we have A=O. In this case Eq. (1) is the equation of motion for the weakeigenstate neutrinos in a vacuum.We will rewrite Eq. ( I ) in the form where ~= ( 1 / 2 E ) [ -? A s i n 2 0 +~( A c o s 2 8 -when expressed in terms of the orthonormal unit vectors (^x,^y,^z) whic...

The Mikheyev-Smirnov-Wolfenstein model of resonant solar-neutrino oscillations is discussed for three generations of leptons. Assuming adiabatic transitions, bounds for the p-and e-neutrinos mass-squared difference are obtained as a function of the e-p mixing angle 0,. The allowed region in the A21,0-01 plot that would solve the solar-neutrino problem is shown to be substantially larger than that of the two-generation case. In particular, the difference between the two-and three-generation cases becomes significant for 0 , larger than -20".