3章 シュレディンガー方程式と波動関数

1 25. 3 .. 19 . Erwin Schr . odinger 1925 .. 2 2. u(x, t) = u(x, t) ( ). t2 x2. u(x, t) .. 2 u/ t2 .. m . p E . p2. E = ( ). 2m E .. E = h = h . ( ). k p . p k = . ( ). h 26 3 .. p2 h2 k2. h = E =. = ( ). 2m 2m (kx t) . h2 2 . h = ( ). t 2m x2. t .. (x, t) = A sin(kx t). ( ). ( ) . 2 2 . h h2 k2. h = . h A cos(kx t), = A sin(kx t) ( ). t 2m x2 2m ( ) . ( ) . sin cos . (x, t) = A ei(kx t) ( ). ( ) . 2 2 . h h2 k2. h h A ei(kx t). = i = A ei(kx t) ( ). t 2m x2 2m ( ) . h2 2. i . h (x, t) = (x, t) ( ). t 2m x2. ( ) . Schr . odinger equation . ( ) t x . h . ( ) ( ) E p .. i . h (x, t) = h (x, t) = E (x, t), ( ). t . i . h (x, t) = h k (x, t) = p (x, t). ( ). x p E x . k = p/ . h = E/ . h x . ( ) (x, t) . wave function . 27. x p . Hamiltonian H V (x, t) . m . p2. H = + V (x, t) ( ). 2m H .. i . h (x, t) = H (x, t) ( ). t .. x . x .. x u u x t u(x, t) . u t . x P u(x, t) Q' Q. P x x + x Q P'. u(x + x, t) Q P.

2 U + u x u . x P (x ) Q ( x + x ). P . Q x : . P Q . P Q equation of motion . P Q . x . 2 u/ t2 x . 2u x = sin Q sin P. t2. 28 3 . Q P .. u u u 2u sin Q sin P = tan Q tan P = = x = x x Q x P x x x2.. 2 2. u(x, t) = u(x, t) ( ). t2 x2. wave equation .. ( ) x . x t u(x, t) = A sin(kx t) = A sin 2 ( ). T. sine wave x t . A amplitude k wave number . angular frequency t = wave length T . x = period x = . frequency . 1 2 . = , k = , = 2 . ( ). T . ( ) = kx t phase . k x t = 0 . x . vph = = = ( ). t k phase velocity . x . ( ) sin cos ( ) . f . u = f (kx t). = kx t . df ( ) 2. f (kx t) = = k f (kx t), f (kx t) = k2 f (kx t). x d x x2. 29. t . df ( ) 2. f (kx t) = = f (kx t), f (kx t) = 2 f (kx t). t d t t2.. 1 2 2. 2 t2 f (kx t) =. vph x2. f (kx t) ( ). = vph k u(x, t) = f (kx t) .. g(kx + t) f (kx t) x . g(kx + t) x . u(x, t) = f (kx t) + g(kx + t) f g . f g f + g . superposition . sin(kx t) sin cos cos(kx t).

3 U(x, t) = ei(kx t) = cos(kx t) + i sin(kx t). u(x, t) . 30 3 .. probability density (x, t) . (x, t) = (x, t) (x, t) = | (x, t) |2 ( ). t x . t t .. d3 x (x, t) = 1 ( ). V.. ( ) . ( ) . conservation of probability .. V (x, t) m . 2. h 2 . i . h = +V . t 2m x2 ( ). h2 2 . i . h = + V . t 2m x2. x [ a, b ] .. b b . d .. dx (x, t) = dx + . ( ). dt a a t t ( ) .. b d dx (x, t). dt a b . 1. h 2 2 .. = dx . i 2m a x2 x2. b . h 1.. = . dx . i 2m a x x x . 1. h h 1 . = + ( ). i 2m x x x=b i 2m x x x=a 31.. 1. h . jx (x, t) = ( ). i 2m x x ( ) . b d dx (x, t) = jx (x = b, t) + jx (x = a, t) ( ). dt a [ a, b ] . ( ) x = b . x = a . [ a, b ] ( ) x . probability current density [ x, x + x ] . x 0 . x+ x d 1. lim dx (x, t). x 0 dt x x jx (x + x, t) + jx (x, t) ( x,t ). = lim x 0 x . x+ x 1. lim dx (x, t) = (x, t). x 0 x x jx jx (a,t ) jx (b,t ). x a b (x, t) jx (x, t). + = 0 ( ). t x : .. ( ) .. 1.

4 H . j(x, t) = ( ) ( ) ( ). i 2m ( ) . (x, t). + j(x, t) = 0 ( ). t . ( ). j ( ) . equation of continuity . 32 3 .. 1 (x, t) 2 (x, t) ( 1 , 2 ) V .. ( 1 , 2 ) = d3 x 1 (x, t) 2 (x, t). ( ). V.. 0 1 (x, t) 2 (x, t) . ( 1 , 2 ) = 0 ( ). (x, t) .. 3. ( , ) = d x (x, t) (x, t) = d3 x | (x, t) |2 > 0. ( ). V V. norm .. (x, t) = ( , ) ( ). V 1 . ( a, b ) = a b = ax bx +. ay by + az bz . 0 0 . a1 1 + a2 2 = 0 a1 = a2 = 0 ( ).. { i } 1 1 . orthonormal system .. ( i , j ) = d3 x i (x, t) j (x, t) = ij ( ). V. ij .. 1 i=j ij = ( ). 0 i = j 33.. (x, t) ak (t) { uk (x) } .. (x, t) = ak (t)uk (x), ( ). k complete system .. E . ( ) p ( ) . operator .. E i . h , ( ). t . h h . h ( px , py , pz ) , , ( ). i x i y i z .. O O . expectation value . O = ( , O ). ( ). x .. h . px = ( , px ) = d3 x (x, t) (x, t) ( ). V i x x .. x = ( , x ) = d3 x (x, t) x (x, t) ( ). V. x x . ( x, y, z ) ( x, y, z ) ( ).. O.

5 O = ( ). 34 3 . O eigenvalue eigen function ( ) eigenvalue equation . O . O k = k k , ( ). spectrum discrete spectrum continuous spectrum .. O .. ( , O ) = d3 x (x, t) O (x, t). V . = d3 x (x, t) (x, t) = ( , ). ( ). V. O conjugate operator . = O ( ).. ( , O ) = ( O , ). ( ).. ( O ) = O ( ).. Hermitian operator . self-conjugate operator . O = O. ( ).. O = O ( k ) = k . ( ). ( ) .. k d3 x k (x, t) k (x, t). V.. 3. = d x k (x, t) O k (x, t) = d3 x O k (x, t) k (x, t). V V . 3. = d x O k (x, t) k (x, t) = d3 x k (x, t) k (x, t). V V.. 3. = ( k ) d x k (x, t) k (x, t) ( ). V. 35.. ( ) . ( ) .. k = ( k , ) = d3 x k (x, t) (x, t) = 0. ( ). V. O k .. k d3 x (x, t) k (x, t). V.. 3. = d x (x, t) O k (x, t) = d3 x O (x, t) k (x, t). V . V . 3. = d x (x, t) k (x, t) = d3 x (x, t) k (x, t). ( ). V V.. ( k ) d3 x (x, t) k (x, t) = 0 ( ). V.. O . O . O k = k k , k = k k . ( ).. [ O, ] = O O = 0. ( ).

6 [ , ] commutator . commutation relation O .. = ak k . ( ). k O .. O = O ak k = ak O k = ak O k k = ak k k k ( ). k k k k.