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Schrodinger 1 2 3 4 Schrodinger 5 6 Schrodinger
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1
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3 deBroglie (1) (2) (3)
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1. O
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2. 1
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1.;2. .
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r r Born
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(r) Born (r) |(r)|2 | (r)|2 r | (r)|2 x y z r xyz
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trd=dxdydz(r,t)d W( r, t) = C| (r,t)|2 dC1tr w( r, t ) = {dW(r,
t )/ d} = C | (r,t)|2 V t W(t) = V dW = Vw( r, t ) d= CV | (r,t)|2
d
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2 C | (r , t)|2 d= 1, C C = 1/ | (r , t)|2 d | (r , t)|2 d , C
0,
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3 2 4 (r , t ) C (r , t ) C t r1 r2 (r, t) C (r, t) (r , t ) C
(r , t )
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(r , t ) | (r , t )|2 d= A A |(A)-1/2 (r , t )|2 d= 1 (A)-1/2 (r
, t ) (r , t )(A)-1/2 (r , t ) exp{i} (r , t )
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4I Dirac x=x0 fx Fourier k=px/, dk= dpx/,
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II t=0 A12 2 = 1 A1= [2]-1/2,
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2
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= C11 + C22 ||2 = |C11+ C22|2 = (C1*1*+ C2*2*) (C11+ C22) = |C1
1|2+ |C22|2 + [C1*C21*2 + C1C2*12*] 1 2
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1 2 ,..., n ,... = C11 + C22 + ...+ Cnn + ... ( C1 , C2 ,...,Cn
,...) 12...n...12 = C11 + C22 .C1 C2
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p deBroglie p p
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(r,t) r C(p, t) p (r,t) p
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(r,t) C(p, t)
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3 1 2 1 2 3 4Hamilton
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:
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1 (x) | (x)|2 x(r) |(r)|2 r x2(x)
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(x) c(px) (x)(x)px(x) px x px1
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(r) *(r)(r),F
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23
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4Hamilton
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4 Schrodinger V(r) Schrodinger
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1926Schrodinger (1): (2)
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t r p 1
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2 3 p, E1t = t0 ( r, t0) 21( r, t ) 2( r, t ) ( r, t)= C11( r, t
) + C22( r, t ) ,
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E t
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E = p2/2 43(1)(2)
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V(r) Schrodinger V(r) 4
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() Schrodinger N i (i = 1, 2,..., N) ( r1, r2, ..., rN ; t) i
Ui(ri) V(r1, r2, ..., rN) Schrodinger
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Hamilton Z Coulomb i Coulomb
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5
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t r
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Schrodinger
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J(7) Eq.7 Eq.7 Gauss S
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1
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1. Born d W(r, t) = |(r, t)|2 d 2. (r, t) 3.Schrodinger1
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S 2. :1. Born w(r, t) = *(r, t) (r, t)t r(r, t) r, t2
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3 I II I II Schrodinger
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6 Schrodinger
Schrodinger Hamilton
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Schrodinger Schrodinger V(r)t t, r
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Schrodinger (r)t=0(r,0) t=2E/h de Broglie E (r,t)(r,t)
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Hamilton1Hamilton (r, t)E(r, t) Schrodinger
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2 1 + 2 E H H
3
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( r, t) E1 Schrodinger2 E 3 n En 4 Cn
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21
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1. 2. Schrodinger 3. ||2 t3tt
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2.2 2.12.3
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Schrodinger 1 2 3 47 8 9
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7
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V(x,y,z) = V1(x) + V2(y) + V3(z) S- (x,y,z) = X(x) Y(y) Z(z) E =
Ex + Ey + EzS- V(x,y,z) Schrodinger
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S 1S 2 3 4
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1 S V(x) I II III I(x),II(x) III (x)22
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3 (-a) = (a) = 02S-1 2x - C2=0
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2 x = -a I(-a) = II(-a) 0 = A cos(-a + ) A sin(-a + )= 0 1
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(1)+(2)(2)-(1)
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n
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I n = m
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I II m = 2 n m = 2n+1
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= 0 4 A
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S S [] S
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1
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2n = 0 , E = 0, = 0 n = k, k=1,2,...n
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4n*(x) = n(x)5
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2.3 2.4 2.8
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1 2 1 2 3 4 5 6 8
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x = Asin( t + )1 F = - kxV0 = 0 V = 0
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2 Vx x = a V V0 x = a
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(a, V0)
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1 2 3 4 5 6
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1 HamiltonSchrodinger x
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2 > 1
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H() H() H()() 0() H() 2. H()
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3. k k
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b0 k b1 k b0 0, b1=0. Heven(); b1 0, b0=0. Hodd().
bk+2(k+2)(k+1)- bk 2k + bk(-1) = 0 bk H = co Hodd + ce Heven = (co
Hodd + ce Heven e) exp[-2/2]
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3(I)=0 exp[-2/2]|=0 = 1 Heven()|=0 = b0 Hodd()|=0 = 0 (II) H()
exp[2] H() exp[2] H() x=0, x =0,
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H() H() n bn 0, bn+2 = 0.
)
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4 H() Hn() Hn() n 2nHn() = 2n+1
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H0 = 1, H1=2 H2 = 2H1-2nH0 = 42-2 H0=1 H2=42-2 H4 = 164-482+12
H1=2 H3=83-12 H5=325-1603+120(x)
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5 ( ) exp[-2] =Hn n2n dnHn /dn = 2n n! (I)=x d= dx (II)Hn()
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63. E0={1/2} 01.Hn()(2)n n= n=2.nn exp[-2/2]n Hn() n
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4. w0() = |0()|2 = = N02 exp[-2] = 0 ||1 | x|< 1(|x| =
1)V(x)=(1/ 2)2 x2 = {1/2} = E0
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n n [-a, a] 5.
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1 Hamilton 1.
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2
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3 N N n1, n2, n3 n1 , n2 n3 = N - n1 - n2N{n1 , n2, n3 }
N= n1 + n2 + n3
n1
n2
0
0,
1,
...,
N
N+1
1
0,
1,
...,
N-1
N
2
0,
1,
...,
N-2
N-1
...,
...,
...,
...,
...
...
N
0,
1
N ( N= n1 + n2 + n3 ), {n1 , n2, n3 }
(1/2)(N+1)(N+2)
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Schrodinger1 V(x)-q xx
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2 V(x)
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3Hamilton Hamilton
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4Schrodinger Schrodinger Schrodinger
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2.5
3.83.93.12
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9
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( E x
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1E > V0 E > 0, E > V0, k1 > 0, k2 > 0.
Schrodinger
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1,2,3 exp[-iEt/] xx x > a III C'=0 1. 2.
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: I D = JD/JIII R = JR/JI x > a III x x < 0 I x D R3.
4.
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J = Aexp[ik1x] * = A* exp[-ik1x]= Cexp[ik1x] = Aexp[-ik1x]
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: D+R=1 x > a III
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k2=ik3k3=[2(V0-E)/ ]1/2 k2 ik3 sin ik3a = i sinh k3a E < V0 D
+ k2=[2(E-V0)/ ]1/2 E < V0 k2 tunnel effect .2E < V0
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1k3a >> 14 k1 k3 E V0/2, D0 = 4
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1: E=1eV, V0 = 2eV, a = 2 10-8 cm = 2 D 0.51a=5 10-8cm = 5 D
0.024 p/e 1840 a = 2 D 2 10-38 Gamow 2:
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2 x1 x2V(x)
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1 2
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1( N H3 ) (NH3)N H NN N 1. R-ST-U E N2. NH32.3786 1010 Hz
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2 (b)
