相対論的量子力学 - lab.twcu.ac.jp

1 1 1. ( ) .. 1.. 2. 4 .. 2. 3 .. 3.. 4.. 5.. 6.. 8. 2 8.. 8.. 9.. 9.. 10.. 11. 3 12.. 12.. 12.. 13.. 14.. 15. 4 16.. 16. ( ) .. 17. 5 18.. 18. : .. 19.. 22. Coulomb .. 22.. 24. 6 25. 7 27. Maxwell .. 27.. 30.. 1. 1. , 13, , 1973. 2. , , , 2002. 3. , 10, , 1993 ( ). 4. , , , , 1 . , , 1969. 5. Bjorken, and Drell, Relativistic Quantum Mechanics , McGraw-Hill, 1965. 6. Bjorken, and Drell, Relativistic Quantum Fields , McGraw-Hill, 1965. 1 . ( ) . 1.. 2.. 3. n .. un = n un ( ).. = ( ). 4.. 3.. E p x .. n un .. un .. (t, x) = an (t)un (x) ( ). n an . 2. 5.. n n un an |an (t)|2.. 1 .. |an |2 = 1. n < > .. < > (t) = |an (t)| n = dx (x, t) (x, t). 2. ( ). n . 6.. h2 2. i . h (t, x) = H (t, x) = (t, x) + V (t, x) (t, x). ( ). t 2m t x (t, x) . (t, x) = (t, x) (t, x) ( ). t (t, x) . 1 . 1 .. (t, x). + div j(t, x) = 0, div j j ( ).

2 T j 3 .. h j ( ) ( ). 2im ( ) ( ) .. 4 . c . c = 1010 cm/s ( ). E p c .. E 2 (pc)2 = m2 c4 ( ). 4 4 . E. p = (p0 , p) ( , p) ( ). c 4 . c 4 0 . x = (x0 , x) (ct, x) ( ). 3. 4 d4 x 4 . 3 d3 x . d4 x dx0 dx1 dx2 dx3 = cdtd3 x. ( ). 0 i = 1, 2, 3 4 . , , , ( , , = 0, 1, 2, 3) 4 A = (A0 , Ai ) B = (B 0 , B i ).. A B = A0 B 0 A1 B 1 A2 B 2 A3 B 3 ( ). 4 . A = (A0 , Ai ) = (A0 , Ai ) ( ).. 3. A B = A B = A0 B0 + A1 B1 + A2 B2 + A3 B3 ( ). =0. 4 4 .. 1 0 0 0.. 0 1 0. 0 = . = ( ). 0 0 1 0 . 0 0 0 1. A B = A B = A B ( ).. 3 . i, j, k, = 1, 2, 3 . 2 dxi dx i . dx j = aj k dxk ( ).. (dx j )2 = (dxk )2 aj l aj k = lk ( ).. at a = 1 deta = 1 ( ). deta = +1 SO(3) . SO(3) . j k . aj k = j k + j k ( ). aj k aj l = kl l k = k l ( ). 4. 3 3 1 2 1 2.. 1 2 = 2 1 d ( ).. 0 1 0.. a(d )j k = 1 + d 1 0 0 ( ). 0 0 0.. 0 1 0.

3 Da( ) . = 1 0 0 a( ) ( ). d . 0 0 0.. cos sin 0.. a( )j k = sin cos 0 ( ). 0 0 1. 4 ( ) .. 1 0 0 0.. 0 cos . sin 0 .. a( ) = ( ). 0 sin cos 0 . 0 0 0 1.. 4 . 2 dx . dx . dx = a dx ( ).. ds2 = dx dx ( ). ds2 = ds 2 a a = ( ). at a = deta = 1 ( ). +1 SO(1, 3) .. 1 . 1. d 0 1 . 2.. ( ), ( ) . 5. 3.. cosh sinh 0 0.. sinh cosh 0 0 . a( ) = ( ).. 0 0 1 0 .. 0 0 0 1. v . v 1 1. tanh = , cosh = = ( )2 ( ). c 1 2 1 vc 4 p 4 A . x v A 4 .. A 0 0 0 A0. 1 1 . A 0 0 . = A , v, . 1. ( ). A 2 0 2 . 0 1 0 A c 1 2. 3 A3. A 0 0 0 1. 4 ( ) .. V . p2. E= +V ( ). 2m (V = 0) . E p . ( )2. E. = p2 + m2 c2 ( ). c .. h i h E=h , kj . pj = h ( ). t i xj . E 1 . p0 = p0 = = i h = i h 0 , c c t x h . pj = p = . j = i h j ( ). i xj x .. i h . p = i h ( ). x . V . x .. (x V ) = V ( ). x . 6. ( ) (t, x) . ( ) ( ). (t, x) . ( )2 ( )2.

4 I . h . h (t, x) = (t, x) + m2 c2 (t, x) ( ). c t i xj (Klein) (Gordon).. ( ) , .. ( , ) = i d3 x [ (t, x) 0 (t, x) 0 (t, x) (t, x)] ( ). 0 ( ) 4 j . j i( ( ) ( ) ) ( ). ( ) . j = 0 ( ). ( ) . d ( , ) = 0 ( ). dt ( ) .. (Dirac) .. N .. 1 (t, x).. 2 (t, x) . (t, x) = ( ).. N (t, x). 7.. 1 . 4 N N j , j = 1, 2, 3 . m . i . h (t, x) h . = j j (t, x) + mc (t, x) ( ). c t i x .. ( )2 ( )( ). i . h j . h h k . = + mc + mc . c t i xj i xk ( )2. h . = + m2 c2 . ( ). i xj j 1 .. j k + k j = 2 jk , ( ). j + j = 0, 2 = 1. ( ).. j , . j = j , = ( ). , i . 1. +1 1 ( ( ) j = k , ( ) 2 ). 2.. Tr i = Tr = 0 ( ). ( ( ) 1 cyclic . j ). , j (N = ) . N = 2 1/2 . ( ) ( ) ( ). 1 0 1 2 0 i 3 1 0. = , = , = . ( ). 1 0 i 0 0 1. 2 1 , j 2 2 .. N = 2 N 4 . N = 4 . ( ) ( ). j 0 j 1 0. = , = , ( ). j 0 0 1.. 4 . (t, x) ( ). 8.

5 ( ) ( ). k + c =0 ( ). t xk 4 . j = 0 ( ). 0 j 0 . j k . j k c k , j 0 c , j = (j 0 , j 1 , j 2 , j 3 ) ( ).. m = 0 2 2 . 2 . (t, x) hc j . i . h = (t, x) = j pj c (t, x) ( ). t i xj 1.. j k = jk + i jkl l ( j ) = j ( ). jkl . 123 = +1 ( ). 2. ( ) ( ) ( ) . 2 .. j , . 0 = , i = i . ( ). 0 j . 0 = 0 , j = j ( ). 2 k . 1 ( ) . + = 2 ( ).. ( ). i h mc (t, x) = 0 ( ). 9. 5 . 5 = i 0 1 2 3 ( ).. 5 + 5 = 0, = 0, 1, 2, 3, ( 5 )2 = 1 ( ).. = U DP.. U ( ).. ( ) ( ) ( ). 0 1 0 j 0 j 5 0 1. DP = , DP = , j = 1, 2, 3, = ( ). 0 1 j 0 1 0. ( ) 2 2 .. ( ) ( ) ( ). 1 1 0 2 2 0 3 3 0. = , = , = ( ). 0 1 0 2 0 3. ( ) ( ) ( ). 1 0 I 2 0 iI 3 I 0. = , = , = ( ). I 0 iI 0 0 I. 0 j j DP = 3 , DP = i 2 j , DP = 3 , DP = 1 j ( ).. (x) (x ) = S(a) (x), x = ax ( ).. (x) = S(a 1 ) (x ), x = a 1 x ( ). S . S(a 1 ) = [S(a)] 1 S 1 (a) ( ).

6 X . = = a ( ). x x x x ( ) ( ). i . h . mc (x) = 0 i h mc (x ) = 0 ( ). x x S 1 (a) . a = S 1 (a) S(a) ( ). a = + .. i S(a) = 1 , = ( ). 4. 10. ( ) . 2i( ) = [ , ] ( ).. i = [ , ] ( ). 2. 1. ( ) ( ) . 2.. ( ) ( ). l 0 0 i j jk jk = = jkl 0j = =. 0j ( ). 0 l i j 0.. 12 a( ) . d = 1 2 = 12 . a( + d ) = a(d ) a( ) ( ). S(a( + d )) = S(a(d )) S(a( )) ( ). dS(a( )) i = 12 S(a( )) ( ). d 2. ( ). i . ( 12 )2 = +1 S(a( )) = exp 12 = cos + i 12 sin ( ). 2 2 2. 12 ( ) 3 S3 . 12. S3 = h ( ). 2. 12 2 . S( = 2 ) = cos = 1 ( ). 4 .. (r, , + 2 ) = S( = 2 ) (r, , ) = (r, , ) ( ). 1 d = 0 1 = 01 . dS(a( )) i = 01 S(a( )) ( ). d 2. ( ). i . ( 01 )2 = 1 S(a( )) = exp 01 = cosh i 01 sinh ( ). 2 2 2. 01 /2 S(a( )).. ( ) .. 11.. 4 4 .. 0 4 .. 0 = ( 1 , 2 , 3 , 4 ) 0 = ( 1 , 2 , 3 , 4 ) ( ).. jk = jk , 0j = 0j , 0 0 = ( ).

7 ( ) S . 0 S 0 = S 1 ( ).. (x ) = (x ) 0 = (x)S 0 = (x) 0 S 1 = (x)S. 1. ( ). 4 4 .. 4 . 1 , 2 2 2 1, 2, . 2 . 4 A , B , . 1 2 A B , 1 2 A , . 1 2 A B . ( ).. ( ).. SDirac = d4 x (x) i h mc (x) ( ).. = ei , = e i ( ).. ( ). SDirac SDirac = d4 x (x) i h mc (x) = SDirac ( ).. ( ). (t, x) (t, x) = 0 ( ). t 12. 3 .. = mc2 i h ( ). t 4 E p j = 1, , 4 .. E 2 = p2 c2 + m2 c4 , ( ). Et p x (t, x) = w(p) exp i +i . ( ). h h w(p) .. 1 0.. 0 i mc2 t 1 i mc2 t = . 1 e h , = . 2 e h ( ).. 0 0 . 0 0.. 0 0.. 0 i mc2 t 0 i mc2 t . = 3 e h , = h 4. 0 e ( ). 1 . 0 1. {. mc2 +1 r = 1, 2. r (x) = wr (p = 0)e i r h t , r = ( ). 1 r = 3, 4. ( ) 3 Sz . ( ) . ( ). l 0. jk = jkl ( ). 0 l 4 3 . h . h w1 , w3 , Sz = + , w2 , w4 , Sz = ( ). 2 2.. ( x ) v ( x ) v . (x ) . (x ) = S( ( v)) (x) ( ). mc2 p x . exp( i r t) = exp( i r ) ( ).}

8 H h v S( ( v)) = e 2 01 , i tanh = ( ). c p x . r (x ) = wr (p)e i r h , 01 = i 1 = i 1 1 ( ). 13. wr (p) = e 2 01 wr (p = 0). i ( ).. = cosh 1 sinh wr (p = 0). 2 2.. 1 0 0 tanh 2.. 0 1 tanh . 0 . = cosh 2 . 2 0 tanh 2 1 0 .. tanh 2 0 0 1. w (p = 0). r ( ). mvc 2. tanh 1 (v/c) pc tanh = = = ( ). 2 1 + 1 tanh . 2 mc 2. + mc2 E + mc2. 1 tanh2 . mv mc2 . p= , E= = (mc2 )2 + (pc)2 ( ). 1 (v/c)2 1 (v/c)2.. cosh + 1 E + mc2. cosh = = ( ). 2 2 2mc2.. r (x) = wr (p)e i r (p x . / . h). ( ).. (p r mc)wr (p) = 0, r (p)(p r mc) = 0. w ( ).. r (p)wr (p) = rr r w ( ).. E. (wr ( r p)) wr ( r p) = rr ( ). mc2. p 4 .. 4. r w r (p)w r (p) = ( ). r=1. p = 0 . ( )-( ) S( (p)) S( (p)) 1 p = 0 . S S 1 = 1 .. Pr . Pr Pr = Pr rr ( ).. (p mc)(p + mc) = (p mc)(p + mc) = p2 (mc)2 = 0 ( ). 14. ( p + mc)( p + mc) = (p )2 2p mc + (mc)2.

9 = p2 2p mc + (mc)2 = 2mc ( p + mc) ( ). 4 p + (p) ( + (p)). ( ) . p + mc (p) = ( ). 2mc 1+ z z 2 4 s =. (0, 0, 0, 1) p = (E/c, 0, 0, 0) . 4 . s p = 0 ( ). s 4 (s) . 1 + z 1 + i 1 2 1 + 5 3 0. = =. 2 2 2. 1 + 5 s 0. = (s), s p = 0 ( ). 2.. 1. 2 e e .. me (15)MeV/c2 ( ). 1928.. me E p . E 2 = p2 c2 + m2 c4 ( ). p E .. E= p2 c2 + m2 c4 , E = p2 c2 + m2 c4 ( ).. ( ) ( ) ( ) .. 1930 hole .. 15.. 1/2 .. p E = p2 c2 + m2 c4 e .. p E = + p2 c2 + m2 c4 +e .. m = E 2 p2 c2 /c2 . q Sz .. 1932 .. C .. Minimal interaction e > 0 . e Q Q . Q = 1 Minimal interaction .. p = i . h p eA = i h eA ( ). x x Minimal interaction . [ ] [ ]. ( ). h mc = 0 i h eA mc = 0. i ( ). x x A Q . [ ]. ( eQ ) . h i i A mc (x) = 0 ( ). h .. C . c C T = C 0T ( ).. 16. = 0 (T ) .. [ ]. ( eQ ) . i . h + i A mc c (x) = 0 ( ). h ( ) ( ) . C.

10 C 1 C = T ( ). C . C C = 1 ( ). C .. 0 0 0 1.. 0 0 1 0 . 2 0 . C = i = ( ).. 0 1 0 0 . 1 0 0 0.. 1.. c = T C 1 ( ). 2.. T. 1c 2c = 1T 2 ( ). T. 1c 2c = + 1T T 2 ( ). T. 1c 2c = 1T T T 2 ( ). T. 1c 5 2c = 1T 5T 2 ( ). 4 .. p = i . h . p + eA = i h + eA ( ). x x . [ ]. ( )( ).. g i h + eA i h + eA (mc) = 0. 2. ( ). x x 0 . 17. 0 . [ ]. ( ). i h + eA mc = 0.. ( ). x . [ ][ ]. ( ) ( ). h + eA + mc i h + eA mc = 0. i ( ). x x . = i ( ).. [. ( )( ). h + eA i h + eA (mc)2. g i . x x ]. ( )( ). i i h + eA i h + eA = 0 ( ). x x . [ ]. ( )( ) . i i h + eA i h + eA = i i h + eA , i h + eA . x x 2 x x . ( ). A A 1. = i (ie . h) e h F ( ). 2 x x 2. 4 2 F . A A . F ( ). x x . 1/2 . [ ]. ( )( ) 1.. g i h + eA i h + eA (mc) + e h F = 0. 2 . ( ). x x 2. 0 . ( ). ( ). i Et . (t, x) = u(x)e h , u(x) = ( ).. [( )2 ]. E + e 2 1.