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TOY MODEL OF NEUTRINO OSCILLATIONS IN A MATTER
N. V . MIKHEEV

Neutrino oscillations are rare examples of the macroscopic manifestation of an elementary particle having wave properties. The simple mechanical analogue of this microscopic phenomena is demonstrated. Amplitude beatings of coupled pendulums exactly reproduce all neutrino oscillations shapes, including the resonant neutrino transitions in the matter . . ­ , . , .
.., 2000

60- , : , ­ , , ­ [1]. , , , , , , . 1958 . , ( [2]): e = cos 1 + sin 2 , µ = - sin 1 + cos 2 . (1)

­ , 1 2 ­ m1 m2 . . , . (, e µ), µ- x e-
e µ 2 2 m c = sin ( 2 ) sin ----------------- x . 2 hp 22

(2)

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m2 ­ 1 2 , p ­ , c ­ , h ­ . , , e , 1 ­ e µ :

©

..

83



e e 2 2 m c = 1 ­ sin ( 2 ) sin ----------------- x . 2 hp 22

(3)

(2) (3) x ( ) 2 hp = -------------2 , 2 m c (4)

, , , = /4 ( ). . 1 . , . .
ee

. 2. . ­ () ,

1

­ ( ) . . , , d x1 12 2 --------- + 1 x 1 + -- x 2 = 0, 2 2 dt d x2 12 2 --------- + 2 x 2 + -- x 1 = 0, 2 2 dt
2 2

(5)

0
eµ

x

1

x1, 2 1, 2 ­ , ­ , . (5) , x1, 2 y1, 2 : x 1 = cos y 1 + sin y 2 , x 2 = ­ sin y 1 + cos y 2 , : (6)

0

x

. 1. . ee ­ e e , eµ ­ e µ

tg ( 2 ) = -------- , 2
2

= 2 ­ 1 .
2 2 2

(7)

(6) (5) (7),
2 d y2 d y1 2 --------- + 2 y 2 = 0, --------- + 1 y 1 = 0, 2 2 2 dt dt 12 2 2 22 4 1, 2 = -- ( 1 + 2 - ( ) + ) , + 2 2

" " (. 2), . ( , ), -

(8)

1, 2 , . , y1 y2 ,

84

, 6, 5, 2000



, x1 x2 . (6) x1, 2 y1, 2 (1), e, µ , 1, 2 ­ , e, µ x1, 2 , 1, 2 y1, 2 . x1, 2(t ), , : x1 ( 0 ) = a0 , dx 1 ( 0 ) dx 2 ( 0 ) x 2 ( 0 ) = --------------- = --------------- = 0. dt dt (9) ­ . ( ) , /2, , = /4, . (, e µ) [3, 4]. , , [5, 6]. , "" ­ ­ . , , , , 1, 2 , ( , 2) (7) (8). 2 2, : 0 1, (7). , (1 = 2 2 = 0), = = /4. . 3 . a 1, 2 ( t ) , ( ). (5) (8) (6) (t). (8), 1, 2 . ( ).
2

x 1 ( t ) = a 0 ( cos cos 1 t + sin cos 2 t ) ,
2 2

x 2 ( t ) = a 0 sin cos ( ­ cos 1 t + cos 2 t ) .

(10)

, ­ . , , 1, 2 , . 2 / 1, 2 2 /(2 - - 1). , x1, 2(t) , , . :
2 2 2 2 2 2 2 2 2 ( 2 ­ 1 ) t a 1 ( t ) = 2 x 1 ( t ) = a 0 1 ­ sin ( 2 ) sin ---------------------- , 2 2 2 2 2 2 ( 2 ­ 1 ) t a 2 ( t ) = 2 x 2 ( t ) = a 0 sin ( 2 ) sin ---------------------- . 2

(11)

, 2 = 0 ­ , = /4 (. (7)). (11) (2) (3), . , . , , , ­ m2, , -

/2 /4

0

tr

t

. 3. . tr ­

..

85



t

1, 2 ( t ) = 1, 2 ( ) d .
0



(12)

(9) x 1 ( t ) = a 0 ( cos cos 0 cos 1 + sin sin 0 cos 2 ) , x 2 ( t ) = a 0 ( ­ sin cos 0 ( cos 1 + cos sin 0 cos 2 ) ) , (13)

0 ­ t. , x1, 2(t) , . :
2 2 2 2 2 ­ 1 a 1 ( t ) = a 0 cos ( ­ 0 ) ­ sin ( 2 ) sin ( 2 0 ) sin ---------------- , 2 (14) 2 2 2 2 2 ­ 1 a 2 ( t ) = a 0 sin ( ­ 0 ) + sin ( 2 ) sin ( 2 0 ) sin ---------------- . 2

( ) , . . µ e (, ), ( ). , , , ­ . , (, ). "" .
1. .. // . . 1983. . 141, 4. . 675­709. 2. .., .. // . 1977. . 123, 2. . 181­215. 3. .., .. // . 1987. . 153, 1. . 3­58. 4. .. ­ "" ? // . 1998. 6. . 86­91.

a 1, 2 ( t ) . 4. ,
2
2 a1(t)

a2()

a

0 2 a2(t) a2()

t

5. .. // . 1997. 8. . 79­85. 6. .. // . 1996. 10. . 99­105.

.. ***
0 t

. 4. . e e () e µ ( )

, - , . ­ , . 60 .

86

, 6, 5, 2000