ΦΑΡΑΚΟΣ- ΚΒΑΝΤΟΜΗΧΑΝΙΚΗ 2ος ΤΟΜΟΣ (ΣΗΜΕΙΩΣΕΙΣ) -
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Κβαντομηχανική ΙΙ Πρόχειρες σημειώσεις του μαθήματος Κωνσταντίνος Φαράκος, Αν. Καθηγητής Τομέας Φυσικής Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών Εθνικό Μετσόβιο Πολυτεχνείο 26 Ιανουαρίου 2011
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Transcript of ΦΑΡΑΚΟΣ- ΚΒΑΝΤΟΜΗΧΑΝΙΚΗ 2ος ΤΟΜΟΣ (ΣΗΜΕΙΩΣΕΙΣ) -
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,.
pi
26 2011
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2
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1 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 , pi . . . . 11.1.2 . . . . . . 2
1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 pi . . . . . . . . . . . . . . . 6
1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2 , . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.3 pi (A)2 pi pi A . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.4 pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Schrodinger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.1 Schrodinger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.4 Schrodinger pi pi . . . . . . . . . . . . . . . . 16
1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.6 - . . . . . . . . . . . . . . . 221.7 - Dirac . . . . . . . . . . . . . . . . . . . . . . . . 26
1.7.1 pi -Dirac . . . . . . . . . . . . . . . . . . . . . . . . . . 271.7.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.7.3 pi . . . . . . . . . . . 28
1.8 Dirac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.8.1 pi pi . . . . 301.8.2 Schwartz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.8.3 pi Schmidt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.8.4 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.8.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.8.6 pi . . . . . . . . . . . . . . . . . . . . . . 331.8.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 Schrodinger- pi 372.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.1.2 pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.2.1 pi . . . . . . . . . . . . . . . . . . . . . 502.2.2 pi pipi . . . . . . . . . . . . . . . . . . . 52
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ii
2.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.3.1 , pi Hermite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 673.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.1.1 pi pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.1.2 pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.3 pi . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.3.1 pi pi pi pi, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4 Coulomb- Schrodinger 794.1 - pi . . . . . . . . . . . . . . . . . . . . . . . 794.2 - pi . . . . . . . . . . . . . . . . . . . . . . . 814.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.5 . . . . . . . . . . . . . . . . . . . . . . . 86
5 - spin- 915.1 . . . . . . . 915.2 pi . . . . . . . . . . . . . . . 945.3 pi pi . . . . . . . . . . . . . . . . . . . . . 97
5.3.1 pi j = l . . . . . . . . 985.4 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.5.1 spin1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
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1
1.1
pi pi , - pi pi . pi . pi pi . , - 1, 2 , . pipi pi , -. . pi .
1.1.1 , pi
:
() a b S, a+ b S. c1a+ c2b S, pi c1, c2 pi .
() a - e1, e2 e3, pi pi :
a = a1e1 + a2e2 + a3e3
pi .
() a pi :
|a| = (a21 + a22 + a23)1/2() pi pi
a b = |a| |b| cos
pi . = pi/2 a b = 0
-
2
() ekeiej = ij
pi ij Kronecker
ij =
{0, = i 6= j1, = i = j
ek . a = a1e1+a2e2+a3e3, b = b1e1+b2e2+b3e3 :
a b = a1b1 + a2b2 + a3b3pi . pi pi .
1.1.2 -
pi pi . pi n(x) , Hilbert. pi pi .
:
() ( pi pi pi pi). 1(x) 2(x) S,
(x) = c11(x) + c22(x)
pi S, c1, c2, .
()
1,2 =
?1(x)2(x) dx
pi ?1(x) 1(x), dx = dx1dx2dx3, R3. :
1,1 0, 1, a2 = a1,2 1,2 + 3 = 1,2+ 1,3
, , < . 1,2 = 0 .
() pi S S pi, A:
A , S S
A = A
(x,
d
dx, . . .
)()
A(c11 + c22 + c33) = c1(A1) + c2(A2) + c3(A3)
-
1.2 3
1.2
(1) pi Hilbert
C = A+ B, : C = (A+ B) = A + B
(2) pi C = A B C = A(B), S
(3) D = [A, B] D = A(B) B(A)
(, D 6= 0) [A, B] = A B B A
, .
A = x, B =
x,
[A, B] = x
(
x
) x
(x) =
[A, B
]= 1
.
(4) A1
A A1 = A1 A = 1(A A1) = A(A1) = = A1(A)
(5)
A = a A. A a A pi - . , an, pi
n : An = ann
A . a A pi , -, . 1,2, . . . ,k ,
11 + 22 + . . .+ kk = 0
1 = 2 = . . . = k = 0
(6) A ()?(x)
(A(x)
)dx =
(A(x)
)?(x) dx, , S
pi :
, A = A,
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4
(7) A A ?(A)
dx =
(A
)? dx
, A = A,
: pi (AB).?(AB
)dx =
(A
)?B dx =
(BA
)? dx
pi : ?(AB
)dx =
((AB)
)? dx
pi(AB) = BA
: (A) = A?(A)dx =
(A)?dx =
((A)?dx
)?=
(?(A)dx
)?=
( ((A)
)?dx
)?=
?(A)dx, ,
A = (A)
(8) pi
A = A
.
(9) g(A) A :
g(x) pi Taylor pi :
g(x) =
+n=0
1
n!g(n)(0)xn
g(A) =+n=0
1
n!g(n)(0)An
: pi T () pi pi
T ()(x) = (x+ )
:
(x+ ) = (x) + (x)+1
2(x)2 + . . . =
n=0
an
n!
dn(x)
dxn
= eddx(x) = T ()(x)
T () = e ddx .
-
1.2 5
1.2.1
() .
()
(AB) = BA = BA = AB
() pi .?nAn dx = an
?nn dx
,
?nAn dx =
(An)
?n dx =
(nn)
?n dx
= ?n
?nn dx
n = ?n() .
?mAn dx = an
?mn dx
?mAn dx =
(Am)
?n dx = ?m
?mn dx
?m = mpi
(n m)
?mn dx = 0
an 6= am : ?mn dx = 0
A = , pi pipi
?nn dx = 1
?mn dx = mn
() pi, pi .
: A . pipi A . pi pi pi , S :
=n
cnn,
pi An = nn
?nm dx = nm.
pi cn pi :?n dx =
m
cm
?nm dx =
m
cmmn
pim cmnm = cn
cn =
?n dx
-
6
() () pi pi .
=
?(A) dx =
(A)? dx
? =
[?(A) dx
]?=
(A)? dx =
pi : A pi S pi , .
pi. S, 1 2 = 1 + 2, .
A pi pi , :?A dx =
(?A
)?=
(A)? dx
(1 + 2)?A(1 + 2) dx =
(1 + 2)
[A(1 + 2)
]?dx.
?1A1 dx+
?1A2 dx+
?
?2A1 dx+ ||2
?2A2 dx
=
1(A1)
? dx+
2(A1)
? dx+ ?
1(A2)? dx+ ||2
2(A2)
? dx
?1(A2) dx =
(A2)
?1 dx 1,2
pi A .
1.2.2 pi
1. A(r,p) pi (r) (p) A(r, p) pi pi r = r p = i~.2. A(r,p) A(r, p), pi pi pi
A(r,i~)n(r) = nn(r) 1. p = i~ -pi .
:
pi pipi x
+
?1(px2) dx = i~ +
?1d2dx
dx
= (i~)[ +
d
dx(?12) dx
+
(d?1dx
2 dx
)]= (i~) [?12]
+
+ (i~) +
(d1dx
)?2 dx
= + +
(i~d1
dx
)?2 dx
=
+
(px1)?2 dx
pi pi px, py, pz pi pi.
-
1.2 7
2. ( ) V ? = V , , H pi .
H =p2
2m+ V (r), p2 = p2x + p
2y + p
2z
pi px = i~ ddx
V = V (x).
+
?1H2 dx =1
2m
+
?1p2x2 dx+
+
?1V (x)2 dx
=1
2m
+
(px1)?px2 dx+
+
(V (x)1)?2 dx
=1
2m
+
(p2x1)?2 dx+
+
(V (x)1)?2 dx =
+
(H1)?2 dx
p2y p2z. , p2x2(x) px(px2). 2 0 x d2/dx 0, x .
-
8
1.3
1.3.1
A pi pi a1, a2, . . . , a . N N1 a1, . . . , N a . A pi
A = N1a1 + . . .+NaN
= a1N1N
+ . . .+ aNN
= a1f1 + . . .+ af =
k=1
akfk
pi fk k . N fk Pk, pi pi .
A = a1P1 + . . .+ aP =k=1
akPk
k=1
Pk = 1
G(A) A, g = G(a)
G(A) =k=1
gkPk =
k=1
G(ak)Pk
1.3.2 ,
pi pi A , pi A pi pi pi a, (
a da2, a+
da
2
) P (a)da
pi P (a) = pi pi
A = +
aP (a)da
P (a1 < a < a2) =
a2a1
P (a) da
+
P (a) da = 1
G(A) A:
G(A) = +
G(a)P (a) da
1.3.3 pi (A)2 pi pi A
(A)2 = (A A)2, A = (A A)2(A)2 =
+
(a A)2P (a) da
=
+
a2P (a) da 2A +
aP (a) da A
+A2 +
P (a) da =1
= A2 2A2 + A2 = A2 A2
-
1.4 Schrodinger 9
:
(A)2 =
k=1
(ak A)2P (ak)
=k
a2kPk 2Ak
akPk + A2 = A2 2A2 + A2
= A2 A2
pi , pi pi pi 1. pi .
(A)2 = 0 a = Api .
=1
(ak A)2 P = 0
P 0 (a A)2 0 piA = a0 P0 = 1, P = 0 k 6= k0,
P = 1.
1.3.4 pi
pi n (n- ) .
In = An = +
anP (a) da
pi In (n = 1, 2, . . . ,) pi pi G(A) pi pi pi Taylor. F (A, ) = eiA,
f() = eiA = n
(i)n
n!An
=n
(i)n
n!An =
n
(i)n
n!In
f() .
f() = eiA = +
eiaP (a) da
P (a) =
1
2pi
+
f()eiad
In f() P (a)
1.4 Schrodinger
pi pi (r, t) , pi pi . pi pi . pi (r, t) pi pi pi Schrodinger:
-
10
[ ~
2
2m2 + V (r, t)
](r, t) = i~
(r, t)
t
pi pi . . , pi ,:
E =p2
2m+ V
,p = i~
H(r, t) =p2
2m+ V (r, t) = ~
2
2m2 + V (r, t)
. H -. pi Schrodinger :
H = i~
t
pi , V (r, t) = 0:
i~t
= ~2
2m
2
x2=
p2
2m = E
E =p2
2m
= Aei(kxt) pipi ,
t= i = iE
~
= E~ = E
h
= 2pi ~ = h2pi
x= ik
2
x2= (ik)2 = k2
pi pi Schrodinger :
E = ~2
2m(k2)
pi E = p2/2m , :
p = ~k p = h2pi
2pi
= hp
De Broglie
(x, t) = Aei(pxEt)/~
h Planck 6, 63 1034 Joule sec.
~ =h
2pi= 1, 05 1034 Joule sec.
-
1.4 Schrodinger 11
pi pi pi:De Broglie
= h/p, =E
h E = p
2
2m
pi ; pi Schrodinger
p = = i~ x
1.4.1 Schrodinger[ ~
2
2m2 + V (r, t)
](r, t) = i~
(r, t)
t(1.1)
Schrodinger pi . pi :
(r, t) = (r)(t)
pi V = V (r). (1.1), pi , :[
~2
2m2 + V (r)
] = i~
t
(r)(t) pi pi,
H(r)
(r)
x,y,z
= i~1
t t
x, y, z, t. , W . pi pi :
[ ~
2
2m2 + V
] = W
i~(t)
t= W(t) (t) = eiWt/~
(r, t) = (r)eiWt/~ W ; W E (r),pi pi pi V = 0. pi,
H(r) = E(r) (1.2)(r, t) = (r)eiEt/~ (1.3)
E (r) . pi pi (1.2) (1.3) .
1.4.2
pi x pi . pipi pi , pi . , pi.
pi pi ( Born), pi pi pi pi pi , pi
P (r, t) = (r, t)(r, t)
-
12
pi t pi pi
C =
(r, t)(r, t)d3x
pi pipi . pipi t. pi pi pi . .
C ( ).
dC
dt=
d
dt
+
(x, t)(x, t)
=
+
t dx+
+
tdx
i~
t= H
i~
t= (H)
t
=i~H,
t=i
~(H)
dCdt
=i
~
+
(H) dx i~
+
(H) dx
H
(H) dx =
(H) dx
dCdt
= 0
pi +
dx . pi - t = t0 t.
1.4.3
) (r, t) pi pi - (r) H , pi pi pipi , pipi . ( H = E),
V
(r, t)(r, t)d3x
pipi. pi pi . pi
d3x = r2 sin dr d d
|(r)| r r
,||2 =
r r2
d3x r
r2+2 dr
2+ 2 < 1 2+ 3 < 0
< 32
-
1.4 Schrodinger 13
2+ 2 = 1 dr
r' ln r,
pi pi. 2+ 2 > 1 r pi pi.
) pi: n(r, t) En, n(r, t) pi pi (r, t) pi pi pi
(r, t) =n
nn(r, t)
n (0 n 1) n . pi |n|2 = nn pi En E. ,
n =
n(r, t)(r, t) d3x
n
|n|2 =n
nn = 1
||2 pi pi.
(r, t) =n
nn(r, t) =n
nn(r)eiEnt/~
n(r)m(r) d3x = nm
n(r, t)(r, t) d3x =
m
m
n(r)m(r) d
3xei(EmEn)/~
=m
mnmei(EmEn)t/~ = n
(r, t) , pi
(r, t)(r, t) d3x = 1
1 =V
d3x =n
m
nm
n(r)m(r)ei(EmEn)t d3x
=n
m
nmnmei(Enm)t
=n
nn
(r, t), pi pi n, pi Schrodinger. :
i~
t=n
n(i~)nt
=n
nHn = H
n pi pi (r, t) t0, t0 = 0
n =
n(r)(r, 0) d3x
-
14
(r, 0), pi S(t)
(r, t) = S(t)(r, 0)
pi Schrodinger:
i~St
(r, 0) = HS(t)(r, 0)
[i~S
t HS
](r, 0) = 0 t
i~St
= HS, Ht
= 0
S(t) = eiHt/~,(r, 0) =
n
nn(r)
(r, t) = eiHt/~(r, 0) =n
neiHt/~n(r)
=n
nn(r)eiEnt/~
eiHtn(r) = eiEnt/~n(r)
Hn = Enn.
)
n(r, t) En. pipi
n(r, t)n(r, t) = n(r)n(r)e
iEnt/~eiEnt/~ = n(r)n(r)
. , . pi n(r, t),
(r, t) =n
ann(r, t)
(r, t)(r, t) =
n
al anl (r)n(r) +
n,ln 6=l
analnle
i(ElEn)t/~
pi pi , pi..
nl =En El~
n,l .
)
P (x, t) = (x, t)(x, t) pi x :
x = +
xP (x, t) dx =
+
(x, t)(x(x, t)) dx
-
1.4 Schrodinger 15
pi x(x, t) = x(x, t).
pi xn:
xk =
(x, t)xk(x, t) dx
pi A pi pi A, ,
A =
(r, t)A(r, t) d3x
,Ak =
Ak d3x
, :
(r, t) =n
ann(r, t), Hn(r) = Enn(r)
E =
H d3x
E =n
|an|2En
(r, t) A pi ,
A =
A :A =
A d3x =
d3x =
pi (A)2 = A2 A2 = 2 2 = 0
pi pi pi A .
n(r, t) A H , (r, t) pi n :
=n
ann, an
A =
A d3x =n
anann
. , |an|2 = nan = Pn = n A .
A .
H A, (r, t) (r, t) =
n ann(r, t):
Hn = Enn
-
16
A :
A =
A d3x =n,m
anam
nAm d3x
=n,m
anamei(EmEn)t/~
n(r)Am(r) d3x
mn = (Em En)/~.
Anm =
n(r)Am(r) d3x
(n,m) pi A. pi A pi Anm = Amn.
A =n,m
anameimntAnm
.
Anm =
nAm d
3x, Amn =
mAn d
3x
(Amn) =(
m(An) d3x
)=
(An)
m d3x
=
nAm d
3x = Anm
A pi Ak .
1.4.4 Schrodinger pi pi
pi piN rk = (xk, yk, zk) pk = (pxk, pyk, pzk), :
H =
Nk=1
p2k2mk
+ V (r1, . . . , rN )
:
pk = i~(
xk,
yk,
zk
)
H =
Nk=1
~2
2mk2k + V (r1, . . . , rN )
H(r1, . . . , rN , t) = i~
t(r1, . . . , rN , t)
pi 1 pi r1, . . ., N pi rN :
P (r1, . . . , rN , t)dV1 dVN = (r1, . . . , rN , t)(r1, . . . , rN , t)dVdV = dV1 dVN = d3x1 d3xN
(r1, . . . , rN , t) = (r1, . . . , rN )(t)
H = E (t) = eiEt/~
-
1.5 17
1.5
A, B
[A, B] = AB BA
[A, B] = 0, .
A,B pi pi . A,B , [A, B] 6= 0 .
(1) , pi ( pi ) .
pi. A , k A:
Ak = kk
A, k B: Bk = kk. A B pi. ,
(AB BA)k = kkk kkk = 0
pi k: =k ckk
(AB BA) = AB BA = A(B) B(A)= A(
k
ckkk) B(k
ckkk)
=k
ckkkk k
ckkkk = 0
(2) A, B , A , , B. pi . Ak = kk ,
B(Ak) = A(Bk)
B(Ak) = kBk
A(Bk) = k(Bk) = Bk A k . pi k = kk
Bk = kk
(3) A, B pi , - pi . pi .
[A, B] + [B, A] = 0
[A, A] = 0
[A, B + C] = [A, B] + [A, C]
-
18
[A+ B, C] = [A, C] + [B, C]
[A, BC] = ABC BCA = ABC BCA BAC= [A, B]C + B[A, C]
[AB, CD] =? = [A, C]BD + C[A, D]B + A[B, C]D + CA[B, D]
. :
[A, C], [A, D], [B, C], [B, D]
pipi pipi : pi pi pi , pi:
[A1A2 An, B1B2 Bk] =ij
(A1 Ai1)(B1 Bj1)[Ai, Bj ](Bj+1 Bk)(Ai+1 An)
=ij
(B1 Bj1)(A1Ai1)[AiBj ](Ai+1 An)(Bj+1 Bk)
[AB, CDE] = [A, C]BDE + C[A, D]BE + CD[A, E]B
+ A[B, C]DE + CA[B, D]E + CDA[B, E]
(1) [x, p] = i~, [p, x] = i~
[x, p](x) = xp(x) px(x) = i~xddx
+ i~d
dx(x)
=
i~xd
dx+
i~xd
dx+ i~
= i~(x), (x).
(2)
[x, p2] = p[x, p] + [x, p]p
= i~p+ i~p = 2i~p
= i~dp2
dp
[x, pk] = i~dpk
dp= i~kpk1
[x, A(x, p)] = i~A
p,
pi pi A(x, p) Taylor pi p pi .
-
1.5 19
(3)
[p, x2] = x[p, x] + [p, x]x = 2i~x
= i~x2
x
[p, A(x, p)] = i~Ax
(4) : [x, px] = [y, py] = [z, pz] = i~
[x, py] = [x, pz] = 0
.
(5) L = r p,
Lx = ypz zpyLy = zpx xpzLz = xpy ypx
[Lx, Ly] = [(ypz zpy), (zpx xpz)]= [ypz, zpx] [ypz, xpz] [zpy, zpx] + [zpy, xpz]= y[pz, z]px + 0 + 0 + x[z, pz]py
= i~ypx + i~xpy = i~Lz,
[Ly, Lz] = i~Lx [Lz, Lx] = i~Ly ( ) Levi-Civita ijk
123 = 1
= 0
{
[Li, Lj ] = i~ijkLki, j, k = 1, 2, 3
} . -, pi . :
L2 = L2x + L2y + L
2z
[L2, Lx] = [L2x, Lx] + [L2y, Lx] + [L2z, Lx]= 0 + Ly[Ly, Lx] + [Ly, Lx]Ly + Lz[Lz, Lx] + [Lz, Lx]Lz
= i~LyLz i~LzLy + i~LzLy + i~LyLz = 0 [L2, Lk] = 0, k
pi pi .
A :
[Li, Aj ] = i~ijkAk
[Li,A
2] = 0.
-
20
(1) Schwartz
:((x)(x) dx
)(
(x)(x) dx) (x)(x) dx2
, , |,|2
pi.
1 = ,,,
1,1 0
1,1 = (
? ,?
, ?
)( ,,
)dx
=
?dx ,?
?,, 0
,, ,?,
(2) C pi pi : C = C1 + iC2C = C1 iC2
pi C1, C2 .
pi. pi C1, C2:
C1 =C + C
2, C2 =
C C2i
C1 =1
2(C + C) = C1
C2 = 1
2i(C C) = C2
C1 + iC2 = C, C1 iC2 = C
(3) A B pipi - :
(A)(B) 12|[A,B]|
pi.(A)2 = A2 A2
(B)2 = B2 B2
pi A = 0, B = 0. A = A A B = B B.
(A)2 = A2 =
A2 d3x =
(A)(A) d3x
-
1.5 21
(B)2 = B2 =
B2 d3x =
(B)(B) d3x
A , pi pi B.
(A)2 = A, A, (B)2 = B, Bpi Schwartz, pi
(A)2(B)2 |A, B|2
(A) (B) |A, B|
A, B =
(A)(B) d3x =
(AB) d3x = AB
(A)(B) |AB| AB pi .
C = AB
C = BA = BA AB +BA
2= C1
AB BA2i
=[A,B]
2i= C2
C = C1 + iC2
C1 C2 pi ,
AB = C = C1+ iC2
|AB| =C12 + C22 |C2| =
12i [A,B] = 12 |[A,B]|
pi :(A)(B) 1
2|[A,B]|
(4) (i)
(x)(p) 12|[x, p]|
(x)(p) 12~
(ii)
(x)(E) 12
[x, H][x, H] = i~
H
px= i~px
m
pi px, x ,
(x)(E) ~2m|px|
, :
E = 0 (x)(E) = 0 px = 0. pi
(pk)(E) . . .(Lk)(E) . . .
-
22
(iii)
(Lx)(Lz) 12|[Lx, Lz]|
[Lx, Lz] = i~Ly (Lx)(Lz) ~
2|Ly|
(5) x, px = i~/x pi ((x) (p) = ~/2) :
(x x) = i(px px) (x) = Ae(xx)2/2~eipx/~
pi. A,B, Schwartz :
A = B
C1 = 0 AB = BA
AB dx =
BA dx(A)B dx =
(B)A dx
(B)(B) dx =
(B)(B) dx
= = i pi pi :
A = iaB,
pi a pi .
1.6 -
() A H , :
dAdt
=1
i~[A, H]+ A
t
pi.
i~
t= H
i~?
t= (H)
dAdt
=d
dt
(A) d3x =
?
tA d3x+
A
t d3x+
A
td3x
= 1i~
(H)A d3x+
1
i~
AH d3x+ A
t
=1
i~
(HA+ AH) d3x+ A
t
=1
i~[A, H]+ A
t
-
1.6 - 23
pi pi , pi pi-
At
= 0.
dAdt
=1
i~[A, H]
pi pi , .
[A, H] = 0 [An, H] = 0 npi pi A
[f(A), H] = 0
() pipi pipi .
pi pi .
( ) pi pi .
() d
dtH = H
t, [H, H] = 0
Ht = V (r, t)
t
V (r), ,
d
dtH = 0
, pi .
()
pi ,
d
dtp = 1
i~[p, H]
[pk, H] = i~ Hxk
= i~ Vxk
= i~Fk
ddtpk = i~
i~Fk = Fk
pi . xk . .
pi pi pi , pi pi pi , x, pi pi pi , ( ) .
() L
t= 0
-
24
: Li = ijkxjpk
[Li, H] = ijkxj [pk, H] + ijk[xj , H]pk
= ijkxj(i~) Hxk
+ ijk(i~)H
pjpk
= i~ijkxj Vxk
+ i~ijkpjmpk
= i~ijkxjFk +
[Li, H] = i~(r F)i
d
dtL = r F = N
(r F)k = 0 [Lk, H] = 0
pi pi k .() - Ehrenfest
:v = p
m
dxdt
=1
i~[x, H] = i~
i~Hp = p
m
dxdt
= vdpdt
=1
i~[p, H] = i~
i~Vx = F (x)
dpdt
= F , Newton
F (x) = 0 dpdt
= 0
p = = p0 dx
dt= p
m = p
m=p0m
= v0 x = v0t+ x0
F (x) = F ,
F =
(x, t)F(x, t)dx = F
dx = F
dpdt
= F pt = Ft+ p0
dxdt
=F
mt+ v0
x = 12
F
mt2 + v0t+ x0
F
m= a = pi
-
1.6 - 25
() Parity () pi : , r r ( ) pi pi . pi (r) (r). pi :
P(r) = (r) P
(P(r)
)= P
((r)
)= (r)
P 2 = 1 P .
P(r) = (r) = (r)(r) = P
(P(r)
)= P(r) = 2(r)
2 = 1 = 1P+(r) = +(r), P(r) = (r)
+ (+) pi ()
pi .
pi Parity :
pi
(r)P(r) d3x =
(r)(r) d3x =
(r)(r) d3x
=
(P(r))(r) d3x
+
(x)(x) dx =
()() d
=
+
()() d
V (r) = V (r) pi (r r) H pi ,2
x2
2
(x)2 =2
x2
x x. ,P (H) = H(r)(r) = H(r)
H(P) = H
((r)
) (P H HP ) = 0 [P , H] = 0
Parity . pi . pi .
-
26
1.7 - Dirac
pi .
Aa = aa
pi a pi pi pi . pi -pi, pi. : +
a(x)a(x) dx = (a, a
)
pi
(a, a) =
0, a 6= a
pi, a = a
pi pi pi . pi pi () pi pi pi . pi pi pi, pi pi pi - Dirac (a a). :
+
f(x)(x) dx = f(0)
f(x) 1
+
(x) dx = 1
pipi (, ) pi pi . pi : +
f(x)(x ) dx = f()
f(x)(x ) dx = f(), < < .
A a, a(x) pipi (x) :
(x) =
[pi a]
c(a)a(x) dx
-Dirac a pi c(a):
+
a(x)(x) dx =a
c(a)
+
a(x)a(x) dxda
=
a
c(a)(a a) da = c(a)
c(a) =
+
a(x)(x) dx
-
1.7 - Dirac 27
1.7.1 pi -Dirac
(x) = limL
sinxL
pix +
sinxL
pixdx =
2
pi
+0
sinxL
xdx =
2
pi
+0
sin y
ydy =
2
pi
pi
2= 1
limx0
sinxL
pix=L
pi
pi L 0 pi 2pi/L 0 L . pi x.
(x) =1
2pi
+
eikx dk
1
2pi
+
eikx dx = limL
1
2pi
LL
eikx dk
= limL
1
2pi
1
x
xLxL
eiy dy = limL
1
2ipix(eixL eixL)
= limL
sin(xL)
pix= (x)
1.7.2 -
() (x) =1
||(x)
() x(x) = 0
() (x) = (x)
() (x2 2) = 12|| [(x ) + (x+ )]
() (f(x)
)=k
(x xk)( dfdx)x=xk
, f(xk) = 0 pi .
pi.
() +
[x(x)]f(x) dx =
+
(x)[xf(x)] dx = 0f(0) = 0 f
() > 0: +
(x)f(x) dx =1
+
(y)f(ya
)dy =
1
f(0) =
1
+
(x)f(x)dx
< 0: = ||, y = x +
(x)f(x) dx =1
+
(y)f( y
)dy =
1||
+(y)f
( y
)dy
=1
|| +
(y)f( y
)dy =
f(0)
|| =1
|| +
(x)f(x) dx
() pi pi () = 1 .
-
28
(+) f(x) pi .
f(x) = f(xk) + f(xk)(x xk)
f(xk) = 0 f(x) = f (xk)(x xk) +
(f(x)
)(x) dx =
k
xk+xk
(f (xk)(x xk)
)(x) dx
=k
1
|f (xk)| xk+xk
(x xk)(x) dx
=k
(xk)
|f (xk)| =k
1
|f (xk)| +
(x xk)(x) dx
(f(x)) =k
(x xk)|f (xk)|
f(x) = x2 2 f (x) = 2x, x1 = , x2 = :
xa(x) = xa(x)
xa(x) = aa(x) (x a)a(x) = 0
a(x) = (x a)
1.7.3 pi
:
i~dp(x)dx
= pp(x) dp(x)dx
= ip
~p(x)
p(x) = Ne(ip/~)x
pi N . +
p(x)p(x) dx = (p p)
NN +
eix~ (p
p) dx = NN2pi(
1
~(p p)
)= |N |22pi~(p p) = (p p)
|N |2 = 12pi~
N = 12pi~
(x) =
+
c(p)p(x) dx
c(p) = +
p(x)(x) dx =12pi~
+
eip~x(x) dx
p pi p :
P (p) = |c(p)|2
-
1.8 Dirac 29
c(p) Fourier (x) +
(x)(x)dx =
+
c(p)c(p)dp
c(p) c(p) 0 p . :
p = +
(x)p(x) dx = +
p|c(p)|2 dp
pk = +
pk|c(p)|2 dp
(x) c(p) . p = p x = i~d/dpx.
x = +
c(p)xc(p)dp = +
(x)x(x)dx
pi.
x = +
(x)x(x)dx =1
2pi~
+
( +
eipx/~c(p)dp)x
( +
eiqx/~c(q)dq)
dx
=1
2pi~
+
( +
eipx/~c(p)dp)
(i~)( +
c(q)d
dq
(eiqx/~
)dq
)dx
+
c(q)d
dq
(eiqx/~
)dq =
+
d
dq
(c(q)eiqx/~
)dq
+
eiqx/~dc(q)
dqdq
= 0 +
eiqx/~dc
dqdq
c(q ) = 0
x = 12pi~
(i~) +
dpdqc(p)dc(q)
dq
+
ei(qp)x/~dx
= i~ +
dpdqc?(p)dc(q)
dq(q p) = i~
+
dpc(p)dc(p)
dp
=
+
c(p)xc(p)dp
pix = i~
d
dp
1.8 Dirac
(r, t) | ket . pi | bra . :
dx = |
| =(
dx)
=
dx = |
-
30
|3 = |1+ |2
4|3 = 4|1+ 4|2
3|4 = 1|4+ 2|4
A:
|A = |A| =
Adx
1.8.1 pi pi -
(r) =n
ann(r) an =
nd3x = n|
| =n
n||n =n
|nn|
|n pi | |n. - | |
| =k
|kk|, | =k
|nn|
| =k
|kk| =k
k|k|
| =k
|kk|
| =
d3x =k
kk, k = k|, k = k,
1.8.2 Schwartz
pi
1|1 2|2 |1|2|2
| = |2 1|21|1 |1
| 0, . pi
| =[2| 1|2
1|1 1|] [|2 1|21|1 |1
]= 2|2 1|2
1|21|1 0
2|2 1|1 1|21|2
-
1.8 Dirac 31
1.8.3 pi Schmidt
|k k = 1, 2, . . . , N pi |i i = 1, 2, . . . ,M N .
pi, |1 = N1|1 pi 1|1 = N1N11|1 = 1
N1 = 11|1 = 1|11/2 |1 = |1
[1|1]1/2
pi pi |2
|2 = N2(|2+ 12|1)
1|2 = 0 2|2 = 1 1|2+ 121|1 = 0
12 = 1|22|2 = N22
(2|2 |12|2) = 1 N2 =
(2|2 |12|2)1/2 :
|3 = N3(|3+ 13|1+ 23|2
) .
1.8.4 ,
.
Q|n = qn|n
q Q n , pi n ,pi Q q.
Q
1|Q2 = Q1|2 Q ket Q|, bra, |Q.
(Q) = Q, (AB) = BA, (A) = A
(A+B) = A +B pipi .
Q = Q,
. pi .
, Q1
QQ1 = Q1Q = 1
-
32
(Unitary) , U
:U = U1
| = U |, | = U |
| = U|U = |UU = |
UU = 1, UU = 1
[Q,Q] = 0
.
1.8.1. :
Q| = q|
Q| = q|
pi.
Q = q, Q = =
(Q)dx =
Qdx = q
( )
QQ = 1 QQ| = qQ| = qq| = |
qq = 1 q = ei
Pk = |kk|
Pk| = |kk| P 2k = Pk,
P 2k |k = Pk|kk| = |kk|kk|= |kk| = Pk|
pi
| =k
|kk| =k
Pk|
k
Pk =k
|kk| = 1
-
1.8 Dirac 33
1.8.5
pi
U = eiA
A .U = eiA, pi pi .pi pi eiAeiA = 1.
Q = |Q|
| | = U |pi U , Q pi Q:
Q Q = UQU
pi.Q = |Q|
Q = |Q| = U|UQU|U= |UU
1
QUU1
| = |Q|
1.8.6 pi
, |n, pi | .| =
n
|nn| =n
n|n
pi an = n|. n n = 1, 2, . . . , N, . . . n, pi
|
12...N...
= |
Q | | = Q| =
k
Q|kk|
n = n| = n|Q| =k
n|Q|kk|
12...N...
= Q
ij
12...N...
Qij = i|Q|j =
iQjdx
i =
j
Qijj
-
34
(i) pi (Q)ij = Qji
(ii) pi (Q)ij = Qji
(Q)ij = i|Q|j = Qi|j = j |Qi = Qji
(iii) Q = Q
(Q)ij = Qji = Qij
(iv) U = U1
(U)ij = (U1)ij
pi -
|k =l
ukl|l, i|j = ij i|j = ij
m|k =n,l
umnukln|l =l
umlukl = mk
A pi, :
AA = I l
Akl(A)lm =
l
AklAml = mk
AA = I l
(A)ml(A)lk = mk =l
AlmAlk
U .
pi Q, Q pi pi :
Q = UQU
1.8.7
Q| = |(Q I)| = 0
pi I .pi Q pi | an, - an pipi
det(Q I) = 0 pi pi , |. pi .
() pi ().
pi pi Q; |l pi Q
Qkl = k|Q|l
-
1.8 Dirac 35
|n Q pi ( pi )
Q|n = n|n
I =l
|ll|
Qnm = n|Q|m =k,l
n|kk|Q|ll|m =k,l
UknQklUlm
pi U Uij = i|j
Q = UQU
j = i |j |i.
|m =l
|ll|m =l
l|m|l
U .
-
36
-
2 Schrodinger - pi
2.1
2.1.1
pipi F = 0, V (x, t) = . Schrodinger :
~2
2m
2(x, t)
x2= i~
t
, x. (x, t) = (x)(t).
i~
t= E i~
t= E
(t) = eiE
~t
~2
2m
2
x2= E ~
2
2m
2
x2= E
k2 =2mE
~2, k =
2mE
~
2
x2= k2 (x) = Aeikx +Beikx
(x, t) = [Aeikx +Beikx]eiE~ t
E = ~.
(x, t) = Aei(kxt) +Bei(kx+t)
pi A,B pi pi pi pi.
pi pi (+x), (x). () , B = 0. , A = 0.
pipi pi pi pi pi .
=E
~=~
2mk2
-
38 Schrodinger - pi
pi pi .
pi, pi.. B = 0, :
P (x, t) = +(x, t)+(x, t) = |A|2
pi x t, pi pi x = , P = 0 (x)(P ) ~/2.
P (x, t) x, pipi x = pipi, P = pipi, pi pi :
(x, t) =12pi
A(k)ei(kxt) dk +
12pi
B(k)ei(kx+t) dk
= (k) = ~2m
k2.
peikx = i~ xeikx = (i~)(ik)eikx = ~keikx
p = ~k
E =~2k2
2m=
p2
2m
3
~2
2m
(2
x2+2
y2+2
z2
)= i~
t
(r, t) = (r)(t)
(t) = eiE~ t, E > 0.
~2
2m(2x +
2y +
2z) = E
2x + 2y +
2z =
2mE
~2 = k2 (2.1)
k2 =2mE
~2
(r) = 1(x)2(y)3(z)
(2.1) :
11
21x2
+1
2
22y2
+1
3
23z2
= k2
1
1
21x2
=
1
2
22y2
=
1
3
23z2
=
+ + = k2
pipi pipi , .
= k2x, = k2y, = k2z
-
2.1 39
k2x + k2y + k
2z = k
2
(r) = eikxxeikyyeikzz = eikr
k = e1kx + e2ky + e3kz, k2 = k2x + k
2y + k
2z
k2 =2mE
~2,
kx, ky, kz pi.
(r, t) = Aei(krt) +Bek(kr+t)
=E
~ : E pi kx, ky, kz.
pi
pi pi P (x, t) = pi x t, pi pi +
(x, t)(x, t) dx = 1
(pi pi ) pi , . pi .
pi pi pi pi pi .
J =n
s t ,
pi v = x/t, pi x t. :
J =n
s t x
x=
n
Vv
J = vpi pi ( x, t) v pi ( x, t). pi pi pi s V pi .,
J = v
()
t= J
d
dt
V
d3x = S
J da
pi (x, t) , pi .
pi pi - , Schrodinger . pi pi , pi pi pi pi r, t.
-
40 Schrodinger - pi
P = (r, t)(r, t)
(r, t)(r, t) d3x = N,
pi N , pi pi pi, . pi :
J = i ~2m{}
pi: ,
i~
t= H, H = ~
2
2m2 + V (r)
P = , V = V (r)
P
t=
t() =
?
t +
t
= [
1
i~H
]+
[ 1i~H
]=
1
i~
{ ~
2
2m2 + ~
2
2m2
}= ~
i2m
{22}
= ~i2m[]
= J
J = i ~2m{}
P
t= J , P =
2.1.2 pi
pi
V (x) =
0, x < 0V0 x > 0
Schrodinger:
~2
2m
2
x2+ V (x) = E
. E < V0 pi < x < 0 x = 0 pi. :
E = E + V, E > 0.
Schrodinger < x < + pi pipi :
(i) : < x < 0 1(ii) : 0 < x < + 2
-
2.1 41
E > 0 ( ;)E = H = T + V = T + V , V = 0
E = T T 0 pi.
T =
p2
2m dx =
1
2m
(p)(p) dx
=1
2m
dx =
1
2m, 0
pi = p. pi pi pi 1, 2 pi :
. pi pi pipi . pipi pi pipi x .
pi :
~2
2m
d21dx2
= E1 d21dx2
= 2mE~2
1
k21 =2mE
~2> 0
1(x) = Aeik1x +Beik1x
pi pipi pipipi pi pi . pi pi .
pi :
~2
2m
22x2
+ V02 = E2
22x2
= 2m~2
(E V0)2 = 2m~2 (V0 E)2
k22 =2m
~2(V0 E) > 0
2(x) = Cek2x +Dek2x
pi pi D = 0, pi pi x = pi pi.
pi pi , x = 0:
1(x = 0) = 2(x = 0)
A+B = Cpi :
d1dx
x=0
=d2dx
x=0
ik1A ik1B = k2C :
A+B = C
AB = ik2k1C
2A = C[1 +
ik2k1
]
-
42 Schrodinger - pi
C = 2k1k1 + ik2
A, B =k1 ik2k1 + ik2
A
pi E ( E < V0) .
1(x) = Ae
ik1x +k1 ik2k1 + ik2
Aeik1x
2(x) =2k1
k1 + ik2Aek2x
pi C = 0 V0 pipi ! x > 0 . V0
k2 0 x > 0 2(x > 0) 0,pi V0 = C = 0. 1(x) = A sin k1x 1(x = 0) = 0, .
pi :
x =1
k2=
~2m(V0 E)
pi = Aeik1x
= Ak1 ik2k1 + ik2
eik1x
= A2k1
k1 + ik2ek2x
pi pi pipipi Jpi, J , J.
Jpi = i~2m
(pixpi pixpi)
=i~2m{AA(ik1)AA(ik1)} = 2~k1
2mAA =
~k1m
AA
p1 = ~k1, v1 = p1m =~mk1
Jpi = AAv1
J = i~2m
(axa axa)
=i~2m
{AA
(k1 ik2)(k1 + ik2)(k1 + ik2)(k1 ik2) (2ik1)
}= ~k1
mAA
J = i~2m
(x x)
=i~1m
{2k1
k1 + ik2
2k1k1 ik2
[ek2x(k2)ek2x ek2x(k2)ek2x
]}=i~2m
4k21k21 + k
22
[] =
pi pi pi x > 0, pi, pi pi !!!
R T :
-
2.1 43
R =|J|Jpi
, T =JJpi
, T +R = 1
R T pi pi , . :
Jpi =~k12m|A|2, J = ~k1
m|A|2, J = 0
T = 0, R = 1 pi pi, pi , -pi pi .
. E > V0
pi :
1(x) = Aeik1x +Beik1x
k21 =2mE
~2> 0
pi :
d22dx2
= 2m~2
(E V0)2
k22 =2m
~2(E V0) > 0
2(x) = Ceik2x +:0
Deik2x
D = 0: pi x > 0 pi , pipipi pi .pi (x):{
A+B = C
ik1A ik1B = ik2C
C = 2k1k1 + k2
A, B =k1 k2k1 + k2
A
V0, E k2 k1 R 0, T 11(x) = Ae
ik1x +Ak1 k2k1 + k2
eik1x, x < 0
2(x) = A2k1
k1 + k2eik2x, x > 0
pi x < 0 pi pi x > 0. pi B = 0, .pi
R =|J|Jpi
=|B|2v1|A|2v1 =
|B|2|A|2 =
(k1 k2k1 + k2
)2T =
JJpi
=|C|2v2|A|2v1 =
4k21(k1 + k2)2
k2k1
=4k1k2
(k1 + k2)2
R+ T = 1
k1 k2, pi pi , R T . pi x = 0.
-
44 Schrodinger - pi
2.1.3 -
pi
V (x) =
0, x < 0V0, 0 < x < a0, x > a
pi pi x x < 0 pi pi .
. E < V0
(i) x < 0
~2
2m
d2
dx2= E
d2
dx2= 2mE
~2
k21 =2mE
~2> 0 1(x) = A1eik1x +B1eik1x
(ii) 0 < x < a
~2
2m
d2
dx2+ V0 = E
d2
dx2=
2m
~2(V0 E)
k22 =2m
~2(V0 E) > 0 2(x) = A2ek2x +B2ek2x
(iii) x > a
~2
2m
d2
dx2= E
d2
dx2= 2m
~2E
k21 =2m
~2E > 0 3(x) = A3eik1x +B3eik1x
pi : B3 = 0. pi (x > a) pi ()pi pi A3, pi pipipi pi .
x = 0 x = a: pi 0 < x < a , (x) 6= 0
1(x = 0) = 2(x = 0)
d1dx
(x = 0) =d2dx
(x = 0)
2(x = a) = 3(x = a)
d2dx
(x = a) =d2dx
(x = a)
J = i~2m
{
x
x
}
A1 +B1 = A2 +B2 (2.2)ik1A1 ik1B1 = k2A2 k2B2 (2.3)
ek2aA2 +B2ek2a = A3eik1a (2.4)
k2A2ek2a k2B2ek2a = ik1A3eik1a (2.5)
-
2.1 45
A,B. pi , pi pi -pi, A1, A1.
A3, E < V0
T =JJpi
=|A3|2~k1/m|A1|2~k1/m =
|A3|2|A1|2
pi A1 pi pipipi , B1 pi A3 pi .
(2.3) ik1 pi (2.2):
A1 +B1 = A2 +B2
A1 B1 = k2ik1
A2 k2ik1
B2
2A1 = A2
(1 +
k2ik1
)+B2
(1 k2
ik1
)
A1 = A2 ik1 + k22ik1
+B2ik1 k2
2ik1(2.6)
(2.5) k2 pi (2.4), (2.5) :
A2ek2a +B2e
k2a = A3eik1a
A2ek2a B2ek2a = A3eik1a ik1
k2
2A2ek2a = A3e
ik1a
(1 +
ik1k2
)
2B2ek2a = A3eik1a
(1 ik1
k2
)
A2 = A3e
(ik1k2)a k2 + ik12k2
B2 = A3e(ik1+k2)a
k2 ik12k2
(2.7)
(2.8)
(2.7) (2.8) (2.6) :
A1 = A3(k2 + ik1)
2
4ik1k2e(ik1k2)a A3 (k2 ik1)
2
4ik1k2e(ik1+k2)a
A1 = A3 eik1a
4ik1k2
[(k2 + ik1)
2ek2a (k2 ik1)2ek2a]
pi , A1 6= 0 A3 6= 0 . pi pi pi E < V0.
. "tunneling". pipi - pi pi pi.. pi .
: k2a 1 , ek2a pi :
-
46 Schrodinger - pi
(1)
A3 ' A1eik1a 4ik1k2(k2 ik1)2 e
k2a
T = |A3|2
|A1|2 16k21k
22
(k22 + k21)
2e2k2a (2.9)
(2)
2(0) = A2 +B2 = A3eik1a[ek2a + ek2a]
2(a) = A2ek2a +B2e
k2a = A3eik1a[ + ]
2(0)2(a)
' ek2a 2(a)2(0)
' ek2a
|2(a)|
2
|2(0)|2 ' T
pi =
k2 + ik12k2
+ = 1
pi
A1A1 =A3A3
16k21k22
{[(k2 + ik1)
2ek2a (k2 ik1)2ek2a]
[(k2 ik1)2ek2a (k2 + ik1)2ek2a]} (2.10) :
= (k2 + ik1)2(k2 ik1)2e2k2a (k2 + ik1)4 (k2 ik1)4+
+ (k2 + ik1)2(k2 ik1)2e2k2a
= (k22 + k21)
2(e2k2a + e2k2a) [(k2 + ik1)4 + (k2 ik1)4]= (k22 + k
21)
2(e2k2a + e2k2a) 4(k22 k21)2 + 2(k22 + k21)2= (k22 + k
21)
2[e2k2a + e2k2a + 2] 4(k22 k21)2= (k22 + k
21)
2(ek2a + ek2a)2 4(k22 k21)2= 4(k22 + k
21)
2 cosh2(k2a) 4(k22 k21)2
pi pi
(k2 + ik1)4 + (k2 ik1)4 = 4(k22 k21)2 2(k22 + k21)2
pi (2.10) :
A1A1 =A3A34k21k
22
{(k22 + k
21)
2 cosh2(k2a) (k22 k21)2}
T = A3A3
A1A1=
4k21k22
(k22 + k21)
2 cosh2(k2a) (k22 k21)2T +R = 1 R = 1 T
pi : cosh2 x sinh2 x = 1,
k21 =2m
~2E, k22 =
2m
~2(V0 E)
T =
1 + sinh2 k2a4E
V0
(1 E
V0
)1
-
2.1 47
k2a , :
sinh(k2a) =1
2
(ek2a ek2a) ek2a
2
T =
1 + e2k2a16E
V0
(1 E
V0
)1
' 16 EV0
(1 E
V0
)e2k2a,
pi (2.9)
T =1
1 + ' 1, 1, 1 + '
a, E V0, - k2a T , pi pi pi .
. E > V0
(i) : x < 0 ~2
2m1 = E1
k21 =2mE
~2> 0 1(x) = A1eik1x +B1eik1x
(ii) : 0 < x < a ~2
2m2 + V02 = E2
k22 =2m
~2(E V0) > 0
2(x) = A2eik2x +B2eik2x
(iii) : x > a ~2
2m3 = E3
3(x) = A3eik1x
pi . pi Ak, Bk.
A1 +B1 = A2 +B2
ik1A1 ik1B1 = ik2A2 ik2B2A2e
ik2a +B2eik2a = A3eik1a
ik2A2eik2a ik2B2eik2a = ik1A3eik1a
pi pi pipi (E < V0), pi k2 ik2 pi A3 A1 pi.. :
T =
1 + sin2(k2a)4E
V0
(E
V0 1)1
(i) E/V0 > 1 pi pi .(ii) E/V0 1, T 1, R 0.
-
48 Schrodinger - pi
(iii) E/V0 > 1, pi k2a,
k2a = npi T = 1 n = 1, 2, 3, . . .
a
2m
~2(E V0) = npi a2 2m~2 (E V0) = n
2pi2
En = ~2pi2
2ma2n2 + V0
(iv) pi pi pi pi 2.1.
R
-V0
E
r
V(r)
2.1
pi T (E) = f(E)e2k2a. f(E) E, e2k2a pi E a, pi pi a , . MeV pi pi pi 107 s 1010 .WKB approximation (Wentzel, Kramers, Brillouin)
(v) pi T (E) ' e2k2a
k2 =
2m
~2(V0 E)
T (E) ' |(a)|2
|(0)|2 ' e2k2a
pi k2 pi.
pi pi pi.. 2.2. pi pi pi x k2(x), pi
k2(x) =
2m
~2(V (x) E)
pi T (E) k2 pi pi pi pi .
k2 1a
x2x1
2m
~2(V (x) E) dx
-
2.1 49
V(x)
x1 x2 x
E
2.2
T (E) = e2k2a = exp{2 x2x1
2m
~2(V (x) E) dx
}
pi pi pi Gamow .
a = x2 x1
A =2
~
x2x1
2(V (x) E) dx
T (E) ' emA
pi :
T1(E)
T2(E)' e
m1
em2' e(
m1m2)
-
50 Schrodinger - pi
2.2
2.2.1 pi
. pipi pi pipi, pi:
V (x) =
0, 0 < x < a, x < 0 x > a pi 0 x a. pi pi 2, V0 (x) , x > a x < 0 pipi.
Schrodinger 0 < x < a :
~2
2m
d2
dx2= E d
2
dx2= 2mE
~2
k2x =2mE
~2
(x) = Aeikxx +Beikxx
(x = 0) = 0 A+B = 0 B = A(x) = 2Ai sin(kxx) = C sin(kxx)
(x = a) = 0 sin(kxa) = 0 kxa = npi
kx = npia, n = 1, 2, 3 . . .
:
E =~2
2mk2x
En = n2 pi2~2
2ma2, n = 1, 2, 3, . . .
E1 , . -, pi .
pi : +
dx = 1
C2 a
0
sin2(kx) dx =C2
k
ka=npi0
sin2 d =C2
npi/a
npi
2= C2
a
2= 1
C =
2
a
n(x) =
2
asin(npix
a
)
En = n2 pi
2~2
2ma2, n = 1, 2, 3, . . .
-
2.2 51
(i) : () pi pi (, )
pi , 0 < x < a, .(ii) pi , pi .(iii) pi pi pi:
E1 E2 piE3
(iv) ~ 0, m , .
(v) n EnEn
=2n+ 1
n2 0,
pi pi .(vi) -
x p ' ~ pi x ' a.,
(p)2 = p2 p2 p = 0
[x,H] = i~mp [x,H] = 0 pipi
(p)2 = p2 '(~
x
)2' ~
2
a2
:
E = p2
2m' ~
2
2ma2' E1
(vii) pi n pipi
E1 ' ~2
2ma2' E
: m = me = 0.5MeV, a = R ' 0.5 108 cm
E ' eV
: m = mp ' 2000 me, a = Rpi ' 1013 cm
EpiE
' meR2
mpR2pi.' 106 107
-
52 Schrodinger - pi
. pi
V (x, y, z) =
0 < x < a0, 0 < y < b
0 < z < cx < 0, x > a
, y < 0, y > bz < 0, z > c
~2
2m2 = E
( )
pi pi. 2.1.1 (x, y, z) = 1(x)2(y)3(z)
k pipi pi pipi () :
1(x) =
2
asin(npix
a
), n = 1, 2, 3, . . .
2(x) =
2
bsin(mpiy
b
), m = 1, 2, 3, . . .
3(z) =
2
csin
(lpiz
c
), l = 1, 2, 3, . . .
:
E = Ex + Ey + Ez = (k2x + k
2y + k
2z)~2
2m
E = ~2
2m
(pi2n2
a2+pi2m2
b2+pi2l2
c2
),
pi .
E = Enlm
n, l,m .
pi pi (n, l,m), pi.
. a = b = c
E = ~2pi2
2ma2(n2 + l2 +m2) = (n2 + l2 +m2)
n = 1, l = 1,m = 1 E111 = 3 n = 2, l = 1,m = 1 E = E211 = E121 = E112 = 6
...
2.2.2 pi pipi
V (x) =
{0, a < x < aV0, x < a, x > a
pi E < V0. pi :
(i) , x < a: ~2
2m
d21dx2
+ V01 = E1
-
2.2 53
(ii) , a < x < a: ~2
2m
d22dx2
= E2
(iii) , x > a: ~2
2m
d23dx2
+ V03 = E3
Schrodinger:
(), x < ad21dx2
=2m
~2(V0 E)1
k21 =2m
~2(V0 E) 1(x) = Aek1x +Bek1x
(), a < x < ad22dx2
= 2mE~2
2
k22 =2m
~2E 2(x) = Ceik2x +Deik2x
(), x > a3(x) = Fe
k1x +Gek1x
: x , (x) pipi , x 1(x ) 0 3(x) 0
B = 0, F = 0
1(x) = Aek1x, 3(x) = Gek1x
,2(x) = C sin k2x+ D cos k2x
C, D .
V (x) pi pi . - pi, parity .
. , (x) = (x)
A = G, C = 0 1(x) = Aek1x, 2(x) = D cos(k2x), 3(x) = Aek1x
:1(a) = 2(a)
d1dx
x=a
=d2dx
x=a
2(a) = 3(a)
d2dx
(a) =d3dx
(a)
pi pi .
D cos(k2a) = Aek1a
Dk2 sin(k2a) = Ak1ek1a
}
pi pi:
tan(k2a) =k1k2
-
54 Schrodinger - pi
. , (x) = (x)
1(x) = Aek1x, 2(x) = C sin(k2x), 3(x) = Aek1x
:C sin(k2a) = Aek1a
Ck2 cos(k2a) = Ak1ek1a
tan(k2a) = k2
k1
:
tan(k2a) =k1k2
=k1a
k2a=
k21a
2
k22a2
k21 =2m
~2(V0 E), k22 =
2m
~2E
z = k2a, z2 = k22a
2 =2m
~2Ea2
z20 =2m
~2V0a
2
tan(z) =z20 z2z2
=
(z0z
)2 1
0 /2 3/2 2 5/2 zz1
z2 z3 z0
2.3: pi z0 = 8 pi..
zk =
2m
~2Eka
2 Ek = ~2z2k
2ma2
pi :
tan(k2a) = k2k1
= k2ak1a
= k22a
2
k21a2
k2a = z, pitan z = z
z20 z2
-
2.2 55
pi tan z z/z20 z2 z0.
zk Ek. pi z0 = 8 3 3 pi . z0 pi/2 ().
2.2.3
V (x) = a(x), a > 0. E < 0
~2
2m
d2
dx2 a(x) = E
x 6= 0 : ~
2
2m
d2
dx2= E
d2
dx2=
2m
~2(E) = k2
k2 = 2mE~2
> 0, k2 =2m
~2|E|
(x) ={Bekx, x < 0Fekx, x > 0
:
(i) B = F (x) = Bek|x|
(ii) (x) pi pi V (x) pi, x = 0.(x) x = 0 (x = 0) pi, (x) pi B,E.
Schrodinger pi +.
~2
2m
+
d
dx
(d
dx
)dx a
+
(x)(x) = E
+
(x) dx
~2
2m
(d
dx(x = +) d
dx(x = )
) a(0) = 0,
0.d
dx
= Bk, ddx
= Bk, (0) = B ( 0)
~2
2m(2k) = a ~
2
mk = a k = ma
~2
k2 = (ma)2
~4=
2m
~2|E| |E| = ma
2
2~2
pi: +
(x)(x) = 1
+
B2e2k|x| dx = 1 2B2 +
0
e2kx dx = 1
B2 = k B =k =
ma
~
(x) =ma
~ema~2 |x|
( E < 0).
-
56 Schrodinger - pi
. E > 0
~2
2m a(x) = E
x 6= 0 ~2
2m = E = 2mE
~2
= k2 (x) = Aeikx +Beikx, x < 0
k2 =2mE
~22(x) = Fe
ikx +Geikx, x > 0
G = 0: pi pipipi pi , pi pi .
: 1(0) = 2(0+) A+B = Fd1dx
x=0
= ik(AB) d2dx
x=0+
= ikF
Schrodinger:
~2
2m[ikF ik(AB)] = a(A+B)
ik(A+B) ik(AB) = 2ma~2
(A+B)
2ikB = 2ma~2
(A+B) B = i mak~2
(A+B)
= mak~2
B = i 1 iA
R =
|B|2|A|2 =
2
1 + 2,
T = 1R = 11 + 2
=|F |2|A|2
2.3
pi pi pi pi pi pi . :
V (x) =1
2kx2
md2x
dt2= kx x = 2x
2 =k
m x(t) = A cost+B sint
x(t) = x0 sin(t+ )
E =1
2kx2 +
1
2mv2 =
1
2m2x20
-
2.3 57
2.3.1 , pi Hermite
Schrodinger
~2
2m
d2
dx2+
1
2m2x2 = E
d2
dx2+
(2m
~2E m
22
~2x2)
= 0
=
m
~x =x, =
m
~' 1
()2
~ =Joule sec
sec ' Joule
~m
=Joule secKgr (sec)1
' Nt msec2
Kgr
~m' Kgr msec2
m sec2Kgr ' (m)
2
2 =m
~x2
ddx
=d
d
d
dx=
d
d
d2
dx2=
d2
d2
d2
d2+
(2m
~2E 2
) = 0
d2
d2+
(2E
~ 2
) = 0
pi () , , pipi < < +.
pi
() () 0 . pi () , pi pi pi. pipi 2E
~ 2. pi pi pi :
d2
d2 2 = 0
pi pi pi () = e2/2.
pi:d
d= e2/2
d2
d2= 2e
2/2 e2/2
pipi,
d2
d2= 2e
2/2 = 2
-
58 Schrodinger - pi
d2
d2 2 = 0 ..
:
() = e2/2y()
y() pi pi , () 0 . () pi pi pipi :
d
d= e2/2y() + e2/2 dy
d
d2
d2= 2e
2/2y() e2/2y() e2/2 dyd e2/2 dy()
d+ e
2/2 d2y
d2
d2
d2= e
2/2
[d2y
d2 2dy
d+ (2 1)y()
] Schrodinger :
d2y
d2 2dy
d+
(2E
h 1)y = 0
: 2Eh 1 = 2
E =( +
1
2
)~
d2y
d2 2dy
d+ 2y = 0
pi Hermite pi Hermite.
Parity pi.:
y() =
+k=0
kk
k.
y() =k
kk(k 1)k2
y() =k
kkk1
=k2======k=+2k1=+1
=0 +2(+2)(+1)
+k=0
akk(k 1)k2
pi k=2 2
+k=0
kkk + 2
+k=0
kk = 0
+k=0
[k+2(k + 2)(k + 1) + k(2 2k)
]k = 0,
k+2 = 2(k )k(k + 2)(k + 1)
-
2.3 59
() 0 6= 0, 1 = 0
y() = 0[1 2
2!2 +
22
4!( 2)4 + . . .
]0, 2 = 2
20, 4 =
2(2 )3 4 2 =
4(2 )02 3 4
, 0 6= 0.()
0 = 0, 1 6= 0 pi pi 3 5 . . . pi , 1 6= 0.
y() = 1
[ 2( 1)
3!3 +
22
5!( 1)( 3)5 + . . .
] parity
pi . () .
= = n n = 0, 1, 2, 3, . . ., y() pi.
pi :
(i) En = ~(n+
1
2
)
(ii) y0() = 0, n = 0,
y1() = 1, n = 1,
y2() = 0(1 22), n = 2,
y3() = 1
( 2
33), n = 3,
0 = pi
1 = pi
0 = pi
1 = pi
n() = e2/2yn() = cne
2/2Hn()
Hn() pi Hermite pi. +
n(x)n(x) dx =1
+
n()n() d
Hermite:
H0() = 1, 0 = 1
H1() = 2, 1 = 2
H2() = 42 2, 0 = 2
H3() = 83 12, 1 = 12
...
pi Hermite
Hn() = (1)nHn() pi :
Hn() = (1)ne2dn(e
2)
dn
-
60 Schrodinger - pi
pi Hermite pi pi pi. : +
Hn()Hm()e
2 d = 0, m 6= n
pi : +
n()m() d = nm
pin() = cnHn()e
2/2
cn =14pi
1
2nn!
En =
(n+
1
2
)~
n(x) =4
pi
1
2nn!ex
2/2Hn(x)
=m
~
n = 0 0(x) = 4
piex
2/2
n = 1 1(x) = 4
pi
2xex
2/2
n = 2 2(x) = 4
pi
123
(4x2 2)ex2/2
...
(i) n(x) pi pi pi, pi En < V (x), .
(ii) pi pi n = 0 n = 1 n = 2, . . ..
(iii) x ' pi 0(x) ' 12
(x)2 ' 12
~m
(iv) E0 =
1
2~
(x) (p) ' ~2
-
2.3 61
E = p2
2m+
1
2m2x2
H =p2
2m+
1
2m2x2 [x,H] = 2i~
2mp
[p,H] = i~m2x, n|[x,H]|n = 0 p = 0
n|[p,H]|n = 0 x = 0x = 0, p = 0
p2 = (p)2 ' ~2
4(x)2
x2 = (x)2
E ' ~2
8m
1
(x)2+m2
2(x)2
E, dEd(x)
= 0
(x)2 = ~2m
E ' 12~ pi.
2.3.2
:
H =p2
2m+
1
2m2x2 = ~
2
2m
d2
dx2+
1
2m2x2
pi pi
=
m
~x
pi
H = ~[
1
22 1
2
d2
d2
] . a a a
a =
m
2~x+
ipx2m~
=12 +
12
d
d
a =m
2~x ipx
2m~=
12 1
2
d
d
x, px -pi . a, a ( )
aa =1
2
( d
d
)=
1
2
{2 d
d +
d
d d
2
d2
}=
1
2
(2 d
2
d2
)+
1
2
[,
d
d
]
-
62 Schrodinger - pi
: [,d
d
]= 1
aa = 12
(2 d
2
d2
) 1
2
N = aa
H = ~(N +
1
2
)
[a, a] =1
2
[ +
d
d, d
d
]=
1
2[, ] +
1
2
[d
d,
] 1
2[,
d
d] 1
2
[d
d,
d
d
]=
1
2(1
2
)= 1
[a, a] = 1[N, a] = [aa, a] = a[a, a] + [a, a]a = a
[N, a] = [aa, a] = a[a, a] + [a, a]a = a
aa aa = 1 aa 1 = aa N = aa 1 pi N ,
N = H = ~(+ 1
2
)
pi N . N :
(x)N(x) dx =
aa dx =
(a)(a) dx 0
( ). N ,
N = N(a) = ( 1)api:
[N , a] = a
Na aN = a Na = aN a N =
N(a) = aN a = ( 1)a
N(a2) = Na(a) = (aN a)a = aN(a) a2 = ( 2)a2N(a3) = Na(a
2) = (aN a)a2 = aN(a2) a3 = a( 2)a2 a3 = ( 3)a3 N(a) = ( )a
0. ( > ) , . pi pi k : ak = 0. l > k, al = alkak = 0
-
2.3 63
Nak1 = ( (k 1))ak1 N = aa pi
aak = ( (k 1)) ak1pi ak = 0 = k 1 =
= n,
En =
(n+
1
2
)~
n = 0, 1, 2, . . . ,.k = n+ 1 = + 1
= 0 k = 1 0, a0 = 0 = 1 k = 2 1, a21 = 0
0:
N0 = 0 a(a0) = 0
a0 = 0 12
( +
d
d
)0 = 0
d0d
= 0 0() = ce2/2
0(x) = ce
m~ x2
2
pi : +
0(x)0(x) dx =1m
~
+
0()0()d = 1
+
e2
d =pi c = 1
4pi
c = 4m
pi~
2.1. En pi .
pi. 1,2 :
d21dx2
=2m
~2(V (x) E)1 1
11 =
2m
~(V (x) E)
d22dx2
=2m
~2(V (x) E)2 1
22 =
2m
~2(V (x) E)
11
1 =1
22 21 = 12
d
dx(2
1 12) = 0
21 12 = = 1,2,12 0 x
1
1=
22 d
dx(ln 1) =
d
dx(ln 2)
1 = c2
-
64 Schrodinger - pi
pi H = ~(N + 12
), H -
N .
pi n(): Na aN = a
N(a0) = aN0 + a0 = 0 + a0 = a0 a0 N 1 = 1 1 ' a0
N(a1) = aN1 + a1 = 2(a1)
a1 = (a)20 N 2 = 2
2 ' (a)20 pi pi.
pi n ' (a)n0
pi
n =1n!
(a)n0
pi pi : : {
an =n+ 1n+1
an =nn1
pi : {an = nn+1an = nn1
Nn = nn
aan = nn(aa 1)n = nn aan = (n+ 1)n
+
n(aan)dx = (n+ 1)
nndx
=1
= (n+ 1)
+
n(aan)dx =
(an)(an)dx = nn
n+1n+1dx
=1
= nn
nn = n+ 1 n =n+ 1 +
n(a
an)dx = n
nndx = n +
n(aan)dx =
(an)
(an)dx = nn
n1n1dx = nn
nn = n n =n
n = cn(a)n0
-
2.3 65
n+1 = 1n+1an
1 = a0
2 =12a1 =
12
(a)20
3 =13a2 =
13
12
(a)30
...
n =1nan1 =
1n
1n 1(a
)2n2 = . . . =1n!
(a)n0
cn = 1n!
1 = a0 =
c2
( d
d
)e
2/2 =c2
(2)e2/2
1 =c2H1()e
2/2
,
n = c1
2nn!Hn()e
2/2, c = 4m
pi~
pi Hn() pi Hermite n.a a ( ) N = a+a .
(m, Nn) = nmn
(m, an) =
n+ 1(m,n+1) =
n+ 1m,n+1
(m, an) =n(m,n1) =
nm,n1
an =n+ 1n+1 1
2
( d
d
)n =
n+ 1n+1 (2.11)
an =nn1 1
2
( +
d
d
)n =
nn1 (2.12)
(2.11) (2.12) :
22n =
n+ 1n+1 +
nn1
:22
dnd
=nn1
n+ 1n+1
: pi V n = n|V |n
V n = n, 12m2x2n
m
~xn = n =
2
2
(n+ 1n+1 +
nn1
)m
~x2n = (n) =
2
2
(n+ 1n+1 +
nn1
)
-
66 Schrodinger - pi
m
~x2n =
2
2
2
2
[n+ 1
n+ 2n+2 +
n+ 1
n+ 1n +
nnn +
nn 1n2
]m
~x2n =
1
2
[(n+ 1)(n+ 2)n+2 + (n+ 1)n + nn +
n(n 1)n2
]V n = 1
2m2n, x2n = 1
2~n, m~ x
2n
=~4
[n|(n+ 1)n+ n|nn]
=~4
(n+ 1 + n) =~4
(2n+ 1)
V n = ~2
(n+
1
2
)T n = 1
2mp2n = 1
2mn, P 2n = . . .
pn = i~dndx
= i~m
~dnd
= i~m
~
2
2
[nn1
n+ 1n+1
]p2n = ~2m~
2
2
[n
dn1d
n+ 1dn+1d
]= . . .
1
2mp2n = ~
4
2
[n
dn1d
n+ 1dn+1d
]= . . .
n|H|n = En = T n + V n
-
3
3.1
3.1.1 pi pi
pi - pi pi pi pi, .
H = H0 + V
pi pi pi pi H0 E(0)n H0 V pi pipi E(0)k E
(0)k1.
: pi pi . pi pi H0.
H0(0)n = E
(0)n
(0)n
H0 pi . pi .
(0)n ,(0)m = nm
pi pi :
H = H0 + V
V pi . pi :
(H0 + V )n = Enn
pi :
n = (0)n +
(1)n +
(2)n + . . .
En = E(0)n + E
(1)n + E
(2)n + . . .
pi E(1)n > E(2)n > E(3)n > . . .. pi pi pi
n ' (0)n En ' E(0)n . pi pi pi pi
n ' (0)n + (1)n
-
68
En ' E(0)n + E(1)n pi
n ' (0)n + (1)n + (2)nEn ' E(0)n + E(1)n + E(2)n
! pi pi . pi . E(1)n = 0 pi pipi E(2)n . pi- pi pi, pi (0)n pi , pi (0)k . pi pi pi pipi- pi pi. pi pi pi
(H0 + V )n = Enn, = 1,
= 0 H0
(0)n = E
(0)n
(0)n
n, En .
n = (0)n +
k 6=n
Cnk()(0)k
pi Cnk() pi
n = (0)n +
k 6=n
nk(0)k +
2k 6=n
bnk(0)k + . . .
n =
(0)n +
(1)n +
2(2)n + . . .
pi pi :
En = E(0)n + E
(1)n +
2E(2)n + . . .
pi 0 1. pipi pi .pi pi: (H0+V )n = Enn n En , .
(H0 + V ){
(0)n + (1)n +
2(2)n + . . .}
={E(0)n + E
(1)n +
2E(2)n + . . .}{
(0)n + (1)n +
2(2)n + . . .}
H0(0)n = E(0)n (0)n 0 = 1 V(0)n +H0(1)n = E(0)n (1)n + E(1)n (0)n
V(1)n +H0(2)n = E
(0)n
(2)n + E
(1)n
(1)n + E
(2)n
(0)n 2
...
pi pi pi pi pi.. V =V1 + V2 V1 > V2. pi pi V1 pi , V2 pi V1. pi pi pi :
V = V1 + 2V2
-
3.1 69
3.1.2 pi
:
V(0)n +H0(1)n = E
(0)n
(1)n + E
(1)n
(0)n
pi pi pi :
(1)n =k
ank(0)k , k 6= n
V(0)n +k 6=n
ankE(0)k
(0)k = E
(0)n
l 6=n
anl(0)l + E
(1)n
(0)n (3.1)
pi pi (3.1) (0)n
(0)n , V(0)n + 0 = 0 + E(1)n (0)n ,(0)n
(0)n ,(0)n = 1 (0)n ,(0)k = 0 k 6= n
E(1)n = (0)n , V(0)n (3.2)
pi pi (3.1) (0)m m 6= n
(0)m , V(0)n + anmE(0)m = E(0)n anm
anm = (0)m , V
(0)n
E(0)n E(0)m
(3.3)
pi pi pi pipi pi V (0)n
Vkn = (0)k , V(0)n
V =
V11 V12 V1NV21 V22 V 2N...
...VN1 VN2 VNN
n = (0)n +
k 6=n
Vkn
E(0)n E(0)k
(0)k (3.4)
En = E
(0)n + Vnn (3.5)
pi |Vnn| |E(0)n E(0)n+1|
|Vnn| |E(0)n E(0)n1|
pi akn pi pi E(0)k 6= E(0)n k 6= n, .
3.1.3
pi pi :
V(1)n +H0(2)n = E
(0)n
(2)n + E
(1)n
(1)n + E
(2)n
(0)n (3.6)
(1)n =
k 6=n
ank(0)k , ank =
-
70
(2)n =
l 6=n
bnl(0)l , E
(1)n =
pi pi (3.6) (0)n
(0)n , V(1)n + 0 = 0 + 0 + E(2)n (0)n ,(0)n
E(2)n = (0)n , V(1)n
E(2)n =k 6=n
ank(0)n , V(0)k
E(2)n =k 6=n
(0)k , V(0)n (0)n , V(0)k E
(0)n E(0)k
E(2)n =k 6=n
VknVnk
E(0)n E(0)k
pi
pi pi (3.6) (0)k k 6= n :
(0)k , V(1)n + (0)k , H0(2)n =
= E(0)n (0)k ,(2)n + E(1)n (0)k ,(1)n + E(2)n (0)k ,(0)n
l 6=n
anlVkl + E(0)k bnk = E
(0)n bnk + E
(1)n ank + 0
bnk(E(0)n E(0)k ) =
l 6=n
anlVkl E(1)n ank + 0
E(1)n = (0)n , V(0)n = Vnn
anl =(0)l , V(0)n E
(0)n E(0)l
=Vln
E(0)n E(0)l
bnk =l 6=n
VklVln
(E(0)n E(0)k )(E(0)n E(0)l )
VnnVkn(E
(0)n E(0)k )2
|Vkn| |E(0)k E(0)n | k, n
, pi.
pi pi (1)n
V(0)n +H0(1)n = E
(0)n
(1)n + E
(1)n
(0)n
(H0 E(0)n )(1)n = (E(1)n V )(0)npi H0, E(0)n ,(0)n , V, E(1)n . pi (1)n , pi- E(2)n = (0)n , V(1)n .
-
3.1 71
3.1.4
1. pi :
V (x) =
, x < 0 x > 2abx, 0 x 2a
pi En pi .
b = 0 pi pi
E(0)n = n2 pi
2~2
8ma2 (0)n =
1
asin(npix
2a
)E(1)n = (0)n , V(0)n = b
1
a
2a0
x sin2(npix
2a
)dx
=b
a
4a2
n2pi2
npi0
y sin2 ydy = bay sin2 ydy =
1
4
(sin2 y 2y sin y cos y)+ y2
4
2. m pi :
V (x) =
, x < 0
1
2kx2 + bx3, x > 0
pi pi -.
U(x) = bx3
V0(x) =1
2kx2 =
1
2m2x2
n(x) = C1
2nn!Hn()e
2/2
piC = 4
m
pi~ =
m
~x =ax
En = ~(n+
1
2
). pi pi x = 0 -
.
pi Hermite n pi pi n . pi pi pin = 2k + 1 k = 0, 1, 2, . . .
k = 0
1(x) =12
4
m
pi~(2ax)eax
2/2
E1 = 32~.
-
72
pi :
E1 =
0
1(x)U(x)1(x)dx = U11
=4
2
m
pi~m
~b
0
x5eax2
dx
0
xkeax2
dx =(k+1
2
)2a(k+1)/2
pi (z) pi :
(z + 1) = z(z) (
1
2
)=pi
0
x5eax2/2dx =
(6/2)
2a6/2=
(3)
2a3=
2!
2a3=
1
a3
E1 = 2bm
pi~m
~
(~m
)3=
bpi
(~m
)3/2 pi pipi :
|E1| 2~
3bpi
(~m
)3/2 ~
3.2
pi () -. . pi . pi pi pi E(0)n E(0)k pi En. pi n pi , - pi E(0)k , pi E
(0)n E(0)k = 0 pi.
pi pi . pi pi . pi , (0)n = |n (0)l = |l, . N . H = H0 + V
H0|k = E(0)k |k, k = 1, 2, . . . , n, lE(0)n =E
(0)l
, . . .
pi pi :
n = (0)n +
k 6=n
ank(0)k =
(0)n + anl
(0)l +(1)n
l = (0)l +
k 6=l
alk(0)l =
(0)l + aln
(0)n +(1)l
(1)n =
k 6=n,l ank
(0)k ,
(1)l =
k 6=l,n alk
(0)k
amk =Vkm
E(0)m E(0)k
, Vkm = (0)k |V |(0)m
-
3.2 73
pi (0)n ,(0)l pi n,l H pi pi . pi H = H0 + V 2 pi H0. pi pi pi. (0)n (0)l pi pi pipi :
n = cnn
(0)n + cnl
(0)l
l = cll(0)l + cln
(0)n
(H0 + V )n = Enn, (H0 + V )l = Ell
:
n =
Ni=1
cni(0)i , n = 1, 2, . . . , N
(0)i . (H0 +V )n = Enn , pipi pi (0)k :
k|H0|n+ k|V |n = Enk|ni
E(0)n cnik|i+i
cnik|V |i = Eni
cnik|i
E(0)n cnk Encnk = i
cniVki
cnk(En E(0)n ) =i
cniVki = cnkVkk +i6=k
cniVki
cnk[Vkk + E
(0)n En
]+i 6=k
cniVki = 0
pi n = 1, 2, . . . , N k = 1, 2, . . . , N . n k = 1 :
cn1
[V11 + E
(0)n En
]+ cn2V12 + cn3V13 + . . .+ CnNV1N = 0
k = 2 :
cn1V21 + cn2
[V22E
(0)n En
]+ cn3V23 + . . .+ cnNV2n = 0
k = 3 :
cn1V31 + cn2V32 + cn3
[V33E
(0)n En
]+ . . .+ cnNV3N = 0
... k = N :
cn1VN1 + cn2VN2 + cn3VN3 + . . .+ CnN
[VNN + E
(0)n En
]= 0
[V11 + E
(0)n En
]V12 V13 . . . V1N
V21
[V22 + E
(0)n En
]V23 . . . V2N
V31 V32
[V33 + E
(0)n En
]. . . V3N
...VN1 VN2 VN3
[VNN + E
(0)n En
]
cn1cn2cn3...
cnN
= 0
-
74
pi N N , N + 1 En, cn1, . . ., cnN . - , pipi :
detV + (E(0)n En)1 = 0 N En N .
cn , pi , pi.
pi pi.
n|H0|n+ n|V |n = Enn|ncnnE
(0)n + cnnVnn + cnlVnl = cnnEn
l|H0|n+ l|V |n = Enl|ncnlE
(0)n + cnnVln + cnlVll = Encnl
{cnnVnn + cnlVnl = cnn(En E(0)n )cnnVln + cnlVll = cnl(En E(0)n )
{(
Vnn VnlVln Vll
)(cnncnl
)= Wn
(cnncnl
)pi
Wn = En E(0)n Wn () 2 2 pi V . :
det
Vnn Wn VnlVln Vll Wn = 0
(Vnn + E(0)n En)(Vll + E(0)n En) VnlVln = 0 , .
En() = E(0)n +1
2(Vnn + Vll) 1
2
[(Vnn Vll)2 + 4VnlVln
]1/2 En(+)
|n = cnn|(0)n + cnl|(0)l En()
|l = cll|(0)l + cln|(0)n (cnn, cnl) (cll, cln) pi V En En(+) En() . pi , pi cnl cnn .pi , cnn.
En = En(+) El = En() n,l pi pipi .
(H0 + V )n = Enn Vn = Enn H0n = Enn E(0)n n = Wnn Vl = Wll , pi Wn = En E(0)n , Wl = El E(0)n . pi :
V |n = Wn|n, V |l = Wl|l l|V = Wll|, V
l|V |n ={Wnl|nWll|n
Wl 6= Wn l|n = 0 V .
E(1)n = Wn = n|V |n = VnnE
(1)l = Wl = l|V |l = Vll
-
3.2 75
H = ~2
2m
(2
x2+
2
y2
)+
1
2m2(x2 + y2) + cxy, c m2
() pi En n c = 0.() c 6= 0, pi pi .
:
() n(x, y) = n1(x)n2(y)En = En1 + En2
~2
2m
2n1x2
+m2
2x2n1 = En1n1
~22m
2n2y2
+m2
2y2n2 = En2n2
piEn1 = ~
(n1 +
1
2
), En2 = ~
(n2 +
1
2
) En = ~(n+ 1)
n = n1 + n2,
{n1 = 0, 1, 2 . . .
n2 = 0, 1, 2, . . .
n = 0, 1, 2, . . . E0 = ~, , 0(x, y) = 0(x)0(y). E1 = 2~ {
10(x, y) = 1(x)0(y) = 1
01(x, y) = 0(x)1(y) = 2
n E(0)n = ~(n + 1), n1 pi pi n1 = 0, 1, 2, . . . , n, n+ 1 .
()
V11 = 1|V |1 =
1(x)0(y)cxy1(x)0(y)dxdy
V11 = 0, V22 = 0
V12 = 1|V |2 = c
1(x)0(y)xy0(x)1(y)dxdy
= c
( +
1(x)x0(x)dx
)2= c
( +
1(x)
~
2m1(x)dx
)2= c
~2m
= V21
x0(x) =
~
2m1(x)
det
2~ E1 c~
2mc~
2m2~ E1
= 0(2~ E1)2 c
2~2
4m22= 0
[2~ E1 c~
2m
] [2~ E1 + c~
2m
]= 0
-
76
E
(+)1 = 2~ +
c~2m
E()1 = 2~
c~2m
(+) = c(+)1 1 + c
(+)2 2, E
(+)1 = E
(0)1 +W
() = c()1 1 + c()2 2, E
()1 = E
(0)1 W
0 c~2mc~2m
0
(c(+)1c(+)2
)= W
(c(+)1
c(+)2
), W =
c~2m
c~2m
c(+)2 =
c~2m
c(+)1 c(+)1 = c(+)2
:
0 c~2mc~2m
0
(c()1c()2
)= W
(c()1
c()2
) c()2 = c()1
pi |c1|2 + |c2|2 = 1 pi |c1| = |c2| = 1/
2.
3.3 pi
pi pi pi pi .
H = H0 + V (r, t) = H0 + V (t)
pi V (t) pi , pipi . pi pi pi H0, pi pi (0)k , E
(0)k .
pi pi t = 0, pi
(0)n = (0) (t) t > 0, pi pi |n |m.
, (0)n pi , :
(t) =m
cm(t)(0)m
cm(t) = eiE(0)m t/~mn.
, cm(t) , pi :
(t) =m
am(t)eiE(0)m t/~(0)m
(H0 + V (t)
) = i~
t
: an(0) = 1, ak(0) = 0 k 6= n. : cm(t) = (0)m |(t) |m, pi |n, t:
Pnm(t) =(0)m |(t)2
-
3.3 pi 77
3.3.1 pi pi pi pi,
cm(t) = (0)m ,(t)dcmdt
= (0)m ,
t = 1
i~(0)m , H
i~dcmdt
=k
ck(t)(0)m |H(0)k
=k
ck(t)E(0)k (0)m |(0)k +
k
ck(t)(0)m |V (t)(0)k
i~dcmdt
= E(0)m cm(t) +k
Vmkck(t)
ck(t) = ak(t)eiE(0)k t/~
i~damdt
eiE(0)m t/~ + (i~)
(iE
(0)m
~
)*0
cm(t) = E(0)m
*0cm(t) +
k
Vmkck(t)
i~dam(t)dt
=k
Vmkak(t)ei(E(0)m E(0)k )t/~
pi :am(t) = a
(0)m (t) + a
(1)m (t) + a
(2)m (t) + . . .
:a(0)n (t) = 1, a
(0)k (t) = 0 k 6= n
a(1)m (t = 0) = 0, a
(2)m (t = 0) = 0 m
i~
da(0)m
dt= 0 a(0)m = = 0, m 6= n
i~da
(1)m
dt=k
Vmkeiwmkta
(0)k , wmk =
E(0)m E(0)k~
pi pi :
i~da
(1)m
dt= Vmn(t)e
iwmnt
a(1)m (t) = i
~
t0
Vmn(t)eiwmnt
dt
Vmn(t) =
(0)m (r)V (r, t)
(0)n (r)d
3x = (0)m |V (t)|(0)n
:
Pnm =1
~2
t0
Vmn(t)eiwmnt
dt2
: T 0 < t < T .
Vmn(t) = Vmn = (0)m |V (r)|(0)n
a(1)m (t) =Vmn~wmn
(1 eiwmnt)
Pnm = |a(1)m (t)|2 =4|Vmn|2
(E(0)m E(0)n )2
sin2
[E
(0)m E(0)n
2~t
]
-
78
-
4 Coulomb - Schrodinger
4.1 - pi
pi me qe = e pi qpi = e mpi = pi. pi :
r
O
rre
E =P 2pi
2mpi+
P 2e2me
+ V (r)
pi r = re rpiV (r) = +
1
4pi0
qpiqe|r| =
e2
4pi0r
:
1. :R =
mere +mpirpime +mpi
, M = me +mpi
P = Pe +Ppi = P
2. pi pi:
P =mpimempi +me
(Ve Vpi) = drdt
=mpiPe mePpimpi +me
pi = mpimempi +me
. , pi :
L = rpi Ppi + re Pe = rdr
dt
:P 2pi
2mpi+
P 2e2me
=P 22M
+P 22
E = E +
P 22
+ V (r) E
(R, r) (re, rpi)
rpi =M
MR me
Mr = R me
Mr
re =M
MR+
mpiM
r = R+mpiM
r
-
80 Coulomb - Schrodinger
r
E
r0
V(r)
(E
-
4.2 - pi 81
4.2 - pi
:
H =P 2pi
2mpi+
P 2e2me
+ V (r)
Ppi i~pi, Pe i~epi pi pi (xpi, ypi, zpi) (xe, ye, ze) . ,pi pi
H =P 22M
+P 22
+ V (r)
P i~R, P i~rpi pi pi (X,Y, Z) (x, y, z) r( ).
(re, rpi, t) (R, r, t)M = me +mpi, =
mempime +mpi
Mpi ' 1836me ' 0, 995mepi Schrodinger :[
~2
2M2R
~2
22r + V (r)
](R, r, t) = i~
t
(R, r, t) = U(R, r)eiE
t/~
[ ~
2
2M2R
~2
22r + V (r)
]U(R, r) = EU(R, r)
E.pi pi (
1
Mpi2pi +
1
Me2e)(
1
M2R +
1
2r)
pi.
4.3
pi , pi , pi - :
U(R, r) = u(R)(r)
( ~
2
2M2Ru
) +
[ ~
2
22r + V (r)
]u = E
u
u(R)(r) pi
1
u
( ~
2
2M2u
)+
1
{ ~
2
22 + V
}= E
R, r, pipi
~
2
2M2u(R) = ERu(R)
~2
22(r) + V (r)(r) = E(r)
ER + E = E
-
82 Coulomb - Schrodinger
pi pi M ER, :
u = Aeik0R, k20 =2M
~2ER
P = ~k0
pi , pi pi , , ( pi) .
~2
22(r) + V (|r|)(r) = E(r)
pi , pi (r, , ) . pi pi pi (r, , ):
2 = 1r2
r
(r2
r
)+
1
r2
{1
sin
(sin
)+
1
sin2
2
2
} Schrodinger :
~2
2
1
r2
r
(r2
r
) ~
2
2r2
[1
sin
(sin
)+ sin2
2
2
]+V (r)(r) = E(r)
(r) = (r, , )
pi pi . pi pipi pi :
(r, , ) = R(r)()() = R(r)Y (, )
Y (, ) . A, B, :
A(r) = ~2
2
1
r2
r
(r2
r
)+ V (r)
B(r) =1
2r2
(, ) = ~2[
1
sin
(sin
)+
1
sin2
2
2
] H = A(r) + B(r)(, )
H = (AR)Y + B(Y )R = ERY
ARR
+BR
R
Y
Y= E
Y (, )Y
=R
BR
(E AR
R
)(r)
(, ) (r)
YY
= Y = Y (, )
pi , pi
RBR
(E AR
R
)= ER AR = BR
-
4.4 83
AR+ BR = ER
~2
2
1
r2
r
(r2R
r
)+
[
2r2+ V (r)
]R = ER ( )
~2[
1
sin
(sin
Y
)+
1
sin2
2Y
2
]= Y ( )
, Y (, ) = ()(). pi pi . pipi ,
= ~2 = ~2l(l + 1)
pi pi.
4.4
.
Y (, ) = ()(), = ~2
, Y :
sin
d
d
(sin
d
d
)+ sin2 = 1
d2
d2
d2
d2= m2
{ = eim, m 6= 0 = c+D, m = 0
() pipi pi , (pi) - (r, , ) (r, , + 2pi)
m = D 0
() = eim m = 0,1,2, . . . :
1
sin
d
d
(sin
d
d
)+
( m
2
sin2
) = 0
= cos dd
=d
d
d
d= sin d
d 1
sin
d
d= d
d
sin2 = 1 cos2 = 1 2
:d
d
[(1 2)d
d
]+
( m
2
1 2)
= 0
= () = ()
m = 0:
(1 2)d2
d2 2d
d+ = 0
-
84 Coulomb - Schrodinger
pi pi .
() =
+k=0
akk
+k=0
{k(k 1)akk2 k(k 1)akk 2kakk +akk
}= 0
+=0
{( + 2)( + 1)a+2 ( + 1)a + a
} = 0
a+2 = ( + 1) ( + 2)( + 1)
a
: +k=0 k(k 1)akk2, k = 0 k = 1
+k=0
k(k 1)akk2 =+k=2
k(k 1)akk2 ==0
( + 2)( + 1)a+2, k = + 2
:
1. a0 6= 0, a1 = 0 : 0, 2, 4, 6, . . .
2. a0 = 0, a1 6= 0 pi : 1, 3, 5, 7, . . .
pi = 1 pi . pi. pi. pi , pi :
= l(l + 1), l =
. . l pi a1 = 0, a0 6= 0, pi l pi pi a0 = 0, a1 6= 0. l pi Legendre Pl() = Pl(cos ).
l = 0(
a2 = 0
a4 = 0, . . .
) P0 = a0
l = 1(
a3 = 0
a5 = 0, . . .
) P1 = a1
l = 2 a2 = 62a0 = 3a0 P2 = a0 3a02, a4, a6, . . . .
...
pi a0, a1 :
Pl() = 12ll!
dl
dl
[(1 2)l
], pi Rodrigues
(pi l, l = , pi l = pi)
m 6= 0 : pi pi :
lm() = (1 2)|m|/2 d|m|
d|m|Pl()
-
4.4 85
Pl() = pi l |m| l !! m = 0,1,2, . . . ,l
: 11Pl()Pl()d =
2
2l + 1ll
d = sin d
pi
0
sin Pl(cos )Pl(cos )d =2
2l + 1ll
pi Legendre pi (1, 1).
f(x) =l
alPl(x), al =2l + 1
2
11f(x)Pl(x)dx
:dPl+1
dx dPl1
dx= (2l + 1)Pl
P0() = 1
P1() =
P2() =1
2(32 1)
P3() =1
2(53 3)
...
(, ) l,m :
Ylm(, ) = Almeim(1 cos2 )|m|/2 d
l+|m|
d cos l+|m|(1 cos2 )l
pi0
sin d
2pi0
dY lm(, )Ylm(, ) = llmm
m = |m| > 0, Alm = (1)l+m
2ll!
(2l + 1)(l m)!
4pi(l +m)!
m = |m| < 0, Alm = (1)l
2ll!
(2l + 1)(l |m|)!
4pi(l + |m|)! l |m| m = 0,1,2, . . . ,l
, pi 0 pi, 0 2pi
g(, ) =+l=0
lm=l
BlmYlm(, )
Blm =
2pi0
pi0
Y lm(, )g(, ) sin dd
Yl,|m| = (1)mY l,|m|
-
86 Coulomb - Schrodinger
Y00 =14pi
Y10 =
3
4picos
Y1,1 =
3
8piei sin
Y20 =
5
16pi(3 cos2 1)
Y2,1 =
15
8piei cos sin
Y2,2 =
15
32pie2i sin2
...
pi l m .
4.5
= l(l + 1)~2 pi . = r, pipi 2/~2 2:
d2R
d2+
2
dR
d+
2
~2
(E
2+
e2
4pi0
)R l(l + 1)
2R = 0
: 2 = 8E/~2 > 0 E < 0 2~2
E
2= 1
4
:
n =2
~2e2
4pi0=
e2
4pi0~
2E
pi E:
E = e4
32pi220~21
n2
:
d2R
d2+
2
dR
d+
{n
1
4 l(l + 1)
2
}R = 0
pi pi pi pi +, R pipipi. :
d2R
d2 R
4= 0 R() = Ae/2 +Be/2
pipi .
A = 0
:
R() = e/2F ()
-
4.5 87
F () pi , R() 0 .
F () = s+k=0
akk, a0 6= 0
R() pi pi F ():
d2F
d2+
(2
1)
dF
d+
[n 1 l(l + 1)
2
]F () = 0
pi F , ak. F () pipi 0.
F () =
+k=0
akk+s
dF
d=k
(s+ k)akk+s1,
d2F
d2=k
(s+ k)(s+ k 1)k+s2
d2F
d2=k
{s(s 1) + 2sk + k(k 1)} akk+s2
F :
k
s(s 1) + 2sk + k(k 1) (s+k)(s+k1)
akk+s2 + 2k
(s+ k)akk+s2
k
(s+ k)akk+s1 + (n 1)
k
akk+s1 l(l + 1)
k
akk+s2 = 0
pi pipi
k = 0:s2
{s(s 1) + 2s l(l + 1)
}a0 = 0
s(s+ 1) = l(l + 1){s = l
s = (l + 1) s = (l + 1) F () ' 1/l+1, pi pi pi 0. l = 0 pi..:
F () ' 1 2F ' 2
(1
)= ()4pi
pi pi. pi . pi s = l.
+k=1
(s+k)(s+k+1)
s(s 1) + 2sk + k(k 1) + 2s+ 2kl(l + 1)
akk+s2+
+k=0
{n s k 1} akk+s1 = 0
pi k = + 1 pi pi 0.
=0
{(s+ + 1)(s+ + 2) l(l + 1)} a+1+s1
+
+k=0
{n s k 1} akk+s1 = 0
-
88 Coulomb - Schrodinger
+k=0
{[(s+ k + 1)(s+ k + 2) l(l + 1)
]ak+1 +
[n s k 1
]ak
}k+s1 = 0
ak+1 = n s k 1(s+ k + 1)(s+ k + 2) l(l + 1)ak
pi s = l. pi pi, n
n l k 1 = 0 ( s = l) n = (l + 1) + k
n l pi n 1, k = 0. n, l , n (l + 1) k
k pi , kmax = n (l + 1), n (l + 1) kmax = 0 akmax+1 = 0,pi akmax+2, akmax+3, . . . , pi pi n (l + 1).
Rnl() =[lkmaxk=0
akk
]e/2, kmax = n (l + 1)
n, ' 1/n2 l = 0, 1, 2, . . . , n 1. Schrodinger ( n) :
nlm(, , r) = Rnl(r)Ylm(, )
:
En = 13, 6n2
eV
pi 1 eV 1, 6 1019 joule. :
=e2
2pi0~21
n
pi n. l m pi m = 0,1,2, . . . ,l, (2l + 1) ., ( n) pi pi (l,m).
n1l=0
(2l + 1) = n2
pi :
0
2pi0
pi0
(Rnl)2Y lm(, )Ylm(, ) sin ddr
2dr = 1
pi pi pi pi pi r r + dr :[Rnl(r)
]2r2dr
pi dV :
nlmnlmdV
pi pi nlmnlm = 0, pi pi pi pi.
-
4.5 89
= r
=e2
2pi0~21
n, =
mempimpi +me
= 1n
1
1 +mempi
mee2
2pi0~2=
1
n
2
1 +mempi
mee2
4pi0~2
pi
a0 =4pi0~2
mee2= 5, 292 1011 m
pi pi Bohr.
= 1n
2(1 +
mempi
)a0
=2
n
1
a0
pi a0 =(
1 +mempi
)a0, 1 +
mempi
= 1, 00054.
= 2n
r
a0
1
R10(r) = 2
(1
a0
)3/2e/2
R20(r) =1
2
2
(1
a0
)3/2(2 )e/2
R21(r) =1
2
6
(1
a0
)3/2e/2
R30(r) =1
9
3
(1
a0
)3/2(6 6+ 2)e/2
R31(r) =1
9
6
(1
a0
)3/2(4 )e/2
R32(r) =1
9
30
(1
a0
)3/22e/2
R
R
R21
R31
R30
R10
R20
1 pi: +0
rker/dr = k!k+1
-
90 Coulomb - Schrodinger
-
5 - spin -
5.1 -
pi :
L = r p
pi p , p = i~
L = (i~)r = (i~)x y zx y zx y z
. : Lx = ypz zpy = (i~)[yz zy]Ly = zpx xpz = (i~)[zx xz]
Lz = xpy ypx = (i~)[xy yx]
pi r,p Lx, Ly, Lz.[Lx, Ly] = i~Lz[Ly, Lz] = i~Lx[Lz, Lx] = i~Ly
pi pi L jkl :
[Lj , Lk] = i~jklLl
j, k, l = 1, 2, 3 123 = 1
1,2,3. (1) .
123 = 231 = 312 = 1
132 = 321 = 213 = 1kk` = 0 k, ` = 1, 2, 3
1
23
+1-1
-
92 - spin -
L2 = L2x + L2y + L
2z
Lx, Ly, Lz V (r) = V (r), .
[L2, Lx] = 0
[L2, Ly] = 0
[L2, Lz] = 0
[L2, Lx] = [L2y, Lx] + [L
2z, Lx]
= Ly[Ly, Lx] + [Ly, Lx]Ly + [Lz, Lx]Lz + Lz[Lz, Lx]
= i~LyLz i~LzLy + i~LyLz + i~LzLy = 0
[Lk, H] = i~(r F)k
V (r) = V (r) F = F (r)r r F = 0 , . :
[Lk, rj ] = i~kjlrl (r1 = x, r2 = y, r3 = z)
[Lk, pj ] = i~jklpl
pi.
[Lx, y] = [ypz zpy, y] = [ypz, y] [zpy, y]= 0 z[py, y] = z(i~) = i~z
pi :
Lk = kjlrjpl
V (r) = V (r) pi
.
x = r sin cos
y = r sin sin
z = r cos r = x sin cos+ y sin sin+ z cos
= x cos cos+ y cos sin z sin = x sin+ y cos
: x, y, z x, y, z.pi {
r = r =
= r r
+ 1
r
+
1
r sin
-
5.1 93
L = r p = (i~)r = (i~)[
1
sin
]
L = xLx + yLy + zLz
, pi x, y, z , :
Lz = (i~)
Ly = (i~)[cos
cos
sin sin
]Lx = (i~)
[ sin
cos
sin cos
]
pi L2
L2 = L2x + L2y + L
2z
L2z = (i~)22
2= ~2
2
2
pi L2x:
L2x = Lx(Lx)
= (i~)2{
sin
+
cos
sin cos
}{
sin
+
cos
sin cos
}
= (~2)
sin2 2
2+ sin cos
0
cos
sin
+cos
sin cos cos
+
cos
sin
:0cos sin
2
+
(cos
sin
)2cos
:0( sin)
+
(cos
sin
)2cos2
2
2
L2y:
L2y = (i~)2{
cos
cos
sin sin
}{
cos
cos
sin sin
}
= (~2)
cos2 2
2 cos sin
*0
(cos
sin
)
+
cos
sin sin2
cos sin
sin cos
0
2
+
(cos
sin
)2
:0sin cos
+
(cos
sin
)2sin2
2
2
-
94 - spin -
:
(L2x + L2y + L
2z) = (~2)
{sin2
2
2+ cos2
2
2+
cos
sin cos2
+cos
sin sin2
+
(cos
sin
)2cos2
2
2
+
(cos
sin
)2sin2
2
2+2
2
}
= (~2){2
2+
cos
sin
+
cos2
sin2
2
2+2
2
}= (~2)
{1
sin
(sin
)+
1
sin2
2
2
}
L2 = (~2)[
1
sin
(sin
)+
1
sin2
2
2
] pi pi , Ylm(, ) ~2l(l + 1).
L2Ylm = ~2l(l + 1)Ylm
LzYlm = m~Ylm
m = 0,1,2, . . . ,l.pi: Ylm(, ) = Almeim(1 cos2 )|m|/2 d
l+|m|(1 cos2 )ld cos l+|m|
5.2 pi
Jx, Jy, Jz , pi :
[Jx, Jy] = i~Jz
[Jy, Jz] = i~Jx
[Jz, Jx] = i~Jy
J2 :
J2 = J2x + J2y + J
2z
pi [J2,J] = 0, piJ = xJx + yJy + zJz
J+ = Jx + iJyJ = Jx iJy = J+ pi :
[Jz, J+] = [Jz, Jx + iJy] = [Jz, Jx] + i[Jz, Jy]
= i~Jy + i(i~)Jx = ~(Jx + iJy) = ~J+
[Jz, J] = ~J
-
5.2 pi 95
pi.([Jz, J+])
= (~J+) = ~J+ = ~J
([Jz, J+])
= (JzJ+) (J+Jz) = JJz JzJ = [Jz, J]
JzJ+ =
