1.1 (The Old Quantum Theory) +e -O0ª.#9!1Ñ V 1.1.1 1b 61 ... · Nankai University () ,—|
Transcript of 1.1 (The Old Quantum Theory) +e -O0ª.#9!1Ñ V 1.1.1 1b 61 ... · Nankai University () ,—|
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V
A
V
A
The Foundation of Quantum Mechanics
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1.1 (The Old Quantum Theory)
1.1.1 (Classical Mechanics)
Newton)Maxwell
(Boltzmann &Gibbs)
“The more important fundamental laws and facts of physicalscience have all been discovered, and these are now so firmlyestablished that the possibility of their ever being supplanted inconsequence of new discoveries is exceedingly remote.... Ourfuture discoveries must be looked for in the sixth place of decima”
Albert. A. Michelson( )speech at the dedication of Ryerson Physics Lab, U. of Chicago 1894
“There is nothing new to be discovered in physics now. All thatremains is more and more precise measurement”
- Kelvin, Lord William Thomson Albert A. Michelson became the firstAmerican to receive a Nobel Prize inphysics, 1907
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... The beauty and clearness of the dynamical theory, whichasserts heat and light to be modes of motion, is at presentobscured by two clouds.The first came into existence with the undulatory theory oflight ... it involved the question 'How could the Earth movethrough an elastic solid, such as essentially is theluminiferous ether?'The second is the Maxwell-Boltzmann current doctrineregarding the partition of energy ...
Kelvin 1900 4 27 (in the meeting of the Royal Institution of Great Britain)
Michelson-Morley
Kelvin, Lord William Thomson(1824-1907)
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m v
Albert Einstein (1879-1955)
v c 021
mmv c
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1.1.2 (Blackbody Radiation and Quantization of Energy)
1859 Kirchhoff —
0 1 2 3 4 5 60
1
2
3
4
5
6
1200K
1400K
1600K
1800K
/ 10
3 J
/10-6m
2000K
3max 2.898 10 K mT
Wilhelm Wien(1864-1928)
1911 Nobel
1893 Wien
1896 WienMaxwell
/5
8 e hc kThc
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0 5 100
2
4
Planck distribution Rayleight-Jeans law Wein distribution
/ 10
3 Jm-4
/10-6m
20 30 40 50 60
Rayleigh (1904 Nobel ) JeansRayleigh-Jeans
Ultravioletcatastrophe
1900 10 Planck
5 /
8 1e 1hc kT
hc Rayleigh-JeansWein
4
8 kT
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1900 12 14 Planck
E0 E=n 0
0 = hv
Planck 1918 Nobel
Max Karl Ernst Ludwig Planck(1858-1947)
h = 6.62606896 10-34 J·s
Planck
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Kirchhoff Wien
Rayleigh-Jeans
PlanckPlanck
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Hertz 1887:
1905 Einstein1916 Millikan
1.1.3 Photoelectric Effect and Einstein’s Explaination
kine
tic e
nerg
y of
ejec
ted
elec
tron
20
12
mv h W
Robert A. Millikan(1868-1953)
Einstein 1921 Nobel
Millikan 1923 Nobel
h
A
V
-+
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1.1.4 BohrBohr’s Theory for the Hydrogen Atom
500 1000 1500 2000 2500 3000/nm
2 2
1 12
Rn
2 21 2
1 1Rn n
1109677.58 cmR
1885 Balmer
1889 Rydberg
1908 Paschen (n1=3)
1914 Lyman (n1=1)
1922 Brackett (n1=4)
1924 Pfund (n1=5)
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hv = E" E' ( )
(h/2 )
Niels Bohr(1885-1962)
a0= 52.92pm(Bohr ) Rydberg
Bohr 1922 Nobel
n=1n=2
n=3
E=hv+Ze
Bohr 1913 Rutherford Planck Einstein
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•
•
•
• —Planck
• —Einstein
• —Bohr RutherfordBohr
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1.2 (Wave-Particle Duality of Matter)
1.2.1 (Wave-Particle Duality of Matter)
(Newton) 1704 (Opticks)
(Christian Huygens) 1690 (Traite de la Lumiere)
· (Thomas Young) 1807
(Augustin Fresnel ) 1819
(J. C. Maxwell)1856-1865
(Gustav Hertz) 1887
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2. Einstein ( )E=h 1905p=h/ 1917
3. E p ;
( ) ( )I | |2( ) I = N/ ( ) | |2
1.
0
E B tH J D t
DB
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4. 1923 Compton X
1927 Nobel
Arthur H. Compton(1892-1962)
Compton
X0.0731nm0.0709nm
X0.0731nm0.0709nm
X0.0731nm0.0709nm
X0.0731nm0.0709nm
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1 (De Broglie’s Hypothesis)
1923
De Broglie 1923 9-10 3 1924(Recherches sur la Théorie des
Quanta)
1.2.2 (Wave-Particle Duality of Matter)
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Prince Louis-Victor Pierre Raymond de Broglie
(1892-1987)
de Brogliep E
v ( )
E = hvP = h/
de BroglieEinstein
De Broglie 18(1910) (Paul Langevin)
LangevinEinstein Einstein de Broglie
Langevinde Broglie 1929
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0.01kg 1000m/s9.11 10-31kg 5 106m/s de Broglie
= h/mv = 6.626 10-25Å= 1.46Å
Clinton Joseph Davisson(1881-1958) & Lester Halbert Germer (1896-1971)
10-4cm 104Å; Å1924 11 29 de Broglie
de Broglie…”
2. —de Broglie
1925 Davisson Germer
(Ni ) 1927
de Broglie
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George Paget Thomson(1892-1975)
N
S
N
S
Max Born(1882-1970)
Thomson 1927
de Broglie
X
Davisson Thomson 1937 Nobel
3. de Broglie1926 Born (The Born interpretation)
Born 1954 Nobel
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1.2.3 ( The Uncertainty Principle )1927 Heisenberg
2xx p 2E t
( 1000ms-1 1%)0.01kg ( )
mmskg
sJm
hxx
331
34
1062.61001.0
1062.6
9.1 10-31kg ( )
mmskgsJ
mhx
x
5131
34
103.710101.9
1062.6( Å)
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OP AP = OC = /2sin =OC/AO= / x
px = psin = p / xx px = p = h
x px h
xp
px
O
P
A
x
yO
A
C
••
• Bohn
1 ˆ ˆ[ , ]2
A B A B 2xx p
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(Quantum Mechanics)
•
—
•(
Postulate)
•
W. Heisenberg
E. Schrödinger
P. Dirac
1932 1933Nobel
1.3 (State Fuction and Shrödinger Equation)
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1.3.1 I (Postulate 1)1. I.
•(q, t) (q, t) q1, q2, …,
qf t (statefunction or wave function)
• * d (| |2d ) t ft, q1 q1 q1+dq1, q2 q2 q2+dq2 ,…, qf
qf qf+dqf
• | (x,y,z,t)|2d t(x, y, z) d (d = dxdydz)
| (x, y, z, t)|2 t (x,y, z)
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dx
x+dxx
2
2 dx
• C ( )C
•(stationary state)
i( , ) e ( )Etq t q 2 2( , ) ( )q t q
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2. • (single-valued) —| |2
x
(x)
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• (continuous) —
x
(x)
x
(x)
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• (quadratically integrable) —
| |2d — d| |2d
1( )
x
(x)
x
(x)
| |2d
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2*d d 1
2*d d K
' 1K
' '* *1d d 1K (normalization factor)
1K
(well-behaved function)
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1.3.2 II (Postulate 2)
1. II m E( ) (Schrödinger)
2
2
2
2
2
22
zyx= h/2
2 2 2 2
2 2 2 2 ( , , ) ( , , )8
h V x y z E x y zm x y z
22 ( ,
2, ) ( , , )x y z E x yV
mz
H EHamilton
m(x, y, z)
V2 Laplace (del squared, nabla squared)
E (x, y, z)
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2. Schrödinger
• m, V ,(x, y, z),
Schrödinger
• , Schrödinger ,( ) ,
E,
m,V (x,y,z)EH
ESchrödinger !
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Schrödingerde Broglie
m E Schrödinger
2 2
2d ( )
2 dV x E
m x
V 2
2 2d 2d
m E Vx
ie cos isinkx kx kx 2
)(2 VEmk
k2
mp
mkEVE k 22
222
hhkp2
2
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1.4 (The Particle in a Box)
1.4.1 (Particle in a One-Dimensional Box)
0 l x
V(x)III III
x 0 V(x) = 0< x < l V(x)=0 x l V(x) =
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),,(),,(),,(2 2
2
2
2
2
22
zyxEzyxzyxVzyxm
2 2
2
d ( ) ( ) ( )2 d
V x x E xm x
0 l x
V(x)III III
(I,III) V(x)=2 2
2d ( ) ( ) ( ) ( )
2 dx E x x
m x
2
21 d ( )( )
dxx
x
(x) = 0 (x 0 x l )
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B 0
0 l x
V(x)III III
2 2
2d ( ) ( )
2 dx E x
m x
22
2d ( ) ( ) 0
dx k x
x
kxBkxAx sincos)(
(0) = 0 Acos0 + B sin0 = 0 A = 0
(l) = 0 B sin kl = 0 sin kl = 0
22 2mEk
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0 l x
V(x)(n = 0, 1, 2, … )
n 0n= |m|
III III
sinkl = 0
nkl
sin nB xl
2 2 2
0 0( ) d sin d 1
l l n xx x B xll
B 2
2 sin n xl l
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2 sin n xl l
n =1, 2, …2 2
28nn hEml
(quantum number), ,
2 22k mE2 2
2kE
m
k n l
2 2
8nn hE
ml
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•
•
E4=16E1
E3=9E1
E2=4E1
n=1
n=4
n=3
n=2
0
lE1=h2/8ml2
| |2
2
2
2
22
2
22
1 8)12(
88)1(
mlhn
mlhn
mlhnEEE nn
m l
m l
•
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nBohr
• (node) x=0 x=l | (x)|2 = 0
8ml 2h2
E1=
E2= 4E1
E3= 9E1
E4= 16E1
n = 1
n = 2
n = 3
n = 4 --
-
-
++
++
+
+
l0l0
| (x)|2 (x)
x
x
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de Broglie:
2nl
lnhhp2 2
222
82 mlhn
mpE n = 1,2,…
• (Orthonormality)
i j
l
0
i iixc )(
*
0
0d 1l
i j iji jx i j
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•Schrödinger
• Schrödinger
•
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O
ax
b y
8
88
8
V(x, y)
1.4.2 (Particle in a Two-Dimensional Box)
),(),(2 2
2
2
22
yxEyxyxm
)()(),( yYxXyx
22
2
2
2 2)()(
1)()(
1 mEy
yYyYx
xXxX
2
2 2
2
2 2
21 d ( )( ) d
21 d ( )( ) d
x
y
mEX xX x x
mEY yY y y
E = Ex + Ey
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2
22
8mahnE x
x
2
22
8mbhn
E yy
nx=1, 2, 3…
ny=1, 2, 3…
2( , ) sin sin yx n yn xx ya bab
2
2
2
22
8 bn
an
mhEEE yx
yx
nx=1, 2, 3… ny=1, 2, 3…
+
O
a
bb
a
O
+-
O
a
b
+
-+
+-
-
O
a
b
1,1 1,2 2,1 2,2
2( ) sin xn xX xa a2( ) sin yn y
Y yb b
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2 2( , , ) sin sin sinyx zn yn x n zx y z
a b cabc
1.4.3 (Particle in a Three-Dimensional Box)
2
22
2
22
2
22
,, 888 mchn
mbhn
mahn
E zyxnnn zyx
nx=1, 2, 3… ny=1, 2, 3… nz=1, 2, 3…
a = b = c = l
degeneracy5
10
222
113131311
122212221
112121211
E/(h
2 /8m
l2 )
111
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1.4.4
(FEMO—Free electron molecular orbital model)
Schrödinger
2kd C-C
l = 2kd
44d 4d
d
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1.
22
2d44
4d
4d2d 2d2
2
2
2
2 28 (2 )
8
hEm dhmd
2 2 2
2 2
2
2
22 28 (4 ) 8 (4 )58 8
h hEm d m d
hmd
E1=1/4
E2=1
E1=1/16
E2=1/4
E3=9/16
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2.
n( ) 2 4 6 8 10 12 14(nm) 193 217 268 304 334 364 390
2 22 2
2 2 2 2
(2 1)( 1)8 (2 ) 8 (2 )
hc h k hk km k d m k d
hkcdkm
)12()2(8 22 k
2 4 6 8 10 12 14
200
300
400
/ nm
k
• C 2k2kd
• 2k• HOMO k• LUMO k+1
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3. C—C
C1 C2 C3 C4C2 C3
C4C3C1
2 22
2 21
2 21+2 2
2
C2
x / d
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4.
nm400 430430 460460 490490 570570 600600 630630 780
x1 + x2 = b
R2N CH CH CH NR2k
ax1 x2
2 22 2
2 2
(2 5)[( 3) ( 2) ]8 8
h k hE k kml ml
hkbkacm
hkcml
Ech
)52()(8
)52(8 22
l = ka + b 2k+4HOMO k+2 LUMO k+3
k 1 2 3 a = 247.8pmb=561.4pm
4 5
(nm) 309 409 511 612 713
=245pm
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1.5 (Physical observable and its corresponding operators)
1.5.1 1.
gfAsinx
c c csinx
x d/dx cosxx -cosx
x x+ x+sinx
xsin
( )dx
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: fBfA ˆˆ BA ˆˆ
: fBfAfBA ˆˆ)ˆˆ(
: )ˆ(ˆˆˆ fBAfBA )ˆ(ˆˆ 2 fAAfA
ABBA ˆˆˆˆ d dˆ ˆ( )d d
A Bf x f x fx x
ABBA ˆˆˆˆ
2.
(commute): ABBA ˆˆˆˆ
f
ˆ ˆ ˆˆ ˆ ˆ[ , ] 0A B AB BA
d dˆˆ( ) ( ) 1d d
B Af xf x fx x
ˆ ˆ3, d dA B x ˆ ˆˆ ˆABf BAf
ˆ ˆ, d dA x B x
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(linear operator) : gAcfAcgcfcA ˆˆ)(ˆ2121
c1, c2 f g
(hermitian operator) : 1 2
dˆ id
Ax
*1 2i 0
d/dx
* * * *1 2 1 2 1 2 2 1
di d i d i id
x dx
*** 1 11 2 2 2
d ddi d i d i dd d d
x x xx x x
* *1 2 2 1
ˆ ˆd ( ) dA A * *ˆ ˆd ( ) dA A
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1.5.2
III
(1) F (q) (t)
),(),(ˆ tqFtqF
x, y, z zzyyxx ˆˆˆ
(2) G (q) (p) (t)G
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ˆ iii
pq
ˆ iypy
ˆ ixpx
ˆ izpz
m V(x, y, z)T 222
21
zyx pppm
T
2
2
2
2
2
22222
2ˆˆˆ
21ˆ
zyxmppp
mT zyx
),,(),,(ˆ zyxVzyxV
),,(2
ˆ2
2
2
2
2
22
zyxVzyxm
H
E = T + V
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)()()( xyzxyz
zyx
ypxpkxpzpjzpypipppzyxkji
prM
:
Mx = ypz zpy
My = zpx xpz
Mz = xpy ypx
ˆ ixM y zz y
ˆ iyM z xx z
ˆ izM x yy x
M2 = Mx2 + My
2 + Mz2
22222ˆ
xy
yx
zx
xz
yz
zyM
p
r
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1.5.3 fafA
( , eigenvalue)
(eigenfunction)3 3d e 3e
dx x
x EH
IVÂ (Â = a ) A
aa
lxn
lxn sin2)(
22 2
2
dˆdxpx
2 2 2 2 22 2
2 2 2
d 2 2ˆ ( ) sin sin ( )d 4 4x n n
n x n h n x n hp x xx l l l l l l
2
22
4lhn
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aA ****ˆ aA* * *ˆ d d dA a a
* * * *ˆ( ) d ( ) d dA a aa = a*
:
* *ˆ d di j j i jA a
* * * * *ˆ( ) d ( ) d d dj i j i i i i j i i jA a a a
iii aA jjj aA
ai aj* d 0i j
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•••
ˆ i ixxp x xx x
ˆ i ( ) i ixp x x xx x
ˆ ˆ ˆ[ , ] ix x xx p xp p x
ˆ ˆ( ) ix xxp p x
x
px
y
pypz
z
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1.5.4 (The principle of Superposition States and Average Values)
(1) i ( i = 1,2,…,n) n
n
iiinn cccc
12211
(2)A ( )
( )*
*
ˆ d
d
AA * ˆ dA A
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• Â* * *
* * *
ˆ d d d
d d d
A a aA a =
•n
iiic
1iii aA
* 2*
1 1 1*
* 2
1 1 1
ˆ( ) ( )d | |ˆ d
d ( ) ( )d | |
n n n
i i i i i ii i i
n n n
i i i i ii i i
c A c c aAA
c c c
i
n
ii acA
1
2|||ci|2 i
A ai
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c12/(c1
2+ c22) E1 1
c22/(c1
2+ c22) E2 2
= c1 1+c2 2 , ( c1, c2
, 1, 2 ,
1 1 1 2 2 2ˆ ˆH E H E
1 1 2 2 1 1 2 2 1 1 1 2 2 2ˆ ˆ ˆ ˆ( )H H c c c H c H E c E c
E1 E2 H*
1 1 2 2 1 1 2 2 1 1 2 2 1 1 1 2 2 2
* 2 2 2 21 1 2 2 1 1 2 2 1 1 1 2 2 1 2 2
2 2 2 21 1 1 2 2 2 1 2 1 1 2 1 2 2 1 2
2 2 2 21 1 1 2 1 2 2 2
ˆ( ) ( )d ( )( )d
( ) ( )d ( 2 )d
d d d d
d 2 d d
c c H c c c c c E c EE
c c c c c c c c
c E c E c c E c c E
c c c c
c2 21 1 2 2
2 21 2
E c Ec c
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nnn cxx
2( ) sinnn xx
l l
2*
0 0 0
20
0
2 1 2ˆ d sin d 1 cos d
1 1 2sin2 2 2
l l l
n n
ll
n x n xx x x x x x xl l l l
n x lx xdl n l l
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1.6
1.6.1 (The One-Dimentional Harmonic Oscillator)
1( )2nE n h n = 0,1,2,
2 22
2
d 1ˆ2 d 2
H kxm x
12
km
2
11 122 21( ) ( )e
2 !x
n nnx H xn
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n=3
n=2
n=1
x
V(x) (x)
n=0
V(x)
E3=7hv/2
E2=5hv/2
E1=3hv/2
x
E0=hv/2
| (x)|2
1. E0 = hv/2 2. (E<V | |2 >0)
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1.6.2 (Partical on a ring)
2
22
2mrkEk
k = 0, 1, 2,
x
yr(r, )
2 2 2
2 2ˆ
2H
m x y
x = r cos y = r sin
cos
sin
1( ) cos
1( ) sin
k
k
k
ki1( ) e
2k
2 2 2 2
2 2 2
d dˆ2 d 2 d
Hmr I
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k = 0
k = 1
k = 2
k = 3
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1.6.3 The rigid rotor
m1 m2
r1 r2
R2 2 2 21 21 1 2 2
1 2
m mI m r m r R Rm m
2 22ˆˆ
2 2pH
2 22 2
2ˆ
2 2H
R I
22
2 2
1 1 sinsin sin
, ( , )JJ MY
2
( 1)2JE J J
I