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V

A

V

A

The Foundation of Quantum Mechanics


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


... 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)


m v

Albert Einstein (1879-1955)

v c 021

mmv c


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


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


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


Kirchhoff Wien

Rayleigh-Jeans

PlanckPlanck


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

-+


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)


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


•

•

•

• —Planck

• —Einstein

• —Bohr RutherfordBohr


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


2. Einstein ( )E=h 1905p=h/ 1917

3. E p ;

( ) ( )I | |2( ) I = N/ ( ) | |2

1.

0

E B tH J D t

DB


4. 1923 Compton X

1927 Nobel

Arthur H. Compton(1892-1962)

Compton

X0.0731nm0.0709nm

X0.0731nm0.0709nm

X0.0731nm0.0709nm

X0.0731nm0.0709nm


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)


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


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


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


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)


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


(Quantum Mechanics)

•

—

•(

Postulate)

•

W. Heisenberg

E. Schrödinger

P. Dirac

1932 1933Nobel

1.3 (State Fuction and Shrödinger Equation)


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)


dx

x+dxx

2

2 dx

• C ( )C

•(stationary state)

i( , ) e ( )Etq t q 2 2( , ) ( )q t q


2. • (single-valued) —| |2

x

(x)


• (continuous) —

x

(x)

x

(x)


• (quadratically integrable) —

| |2d — d| |2d

1( )

x

(x)

x

(x)

| |2d


2*d d 1

2*d d K

' 1K

' '* *1d d 1K (normalization factor)

1K

(well-behaved function)


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)


2. Schrödinger

• m, V ,(x, y, z),

Schrödinger

• , Schrödinger ,( ) ,

E,

m,V (x,y,z)EH

ESchrödinger !


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


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) =


),,(),,(),,(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 )


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


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


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


•

•

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

•


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


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


•Schrödinger

• Schrödinger

•


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


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


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



1.4.4

(FEMO—Free electron molecular orbital model)

Schrödinger

2kd C-C

l = 2kd

44d 4d

d


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


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


3. C—C

C1 C2 C3 C4C2 C3

C4C3C1

2 22

2 21

2 21+2 2

2

C2

x / d


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


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


: 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


(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


1.5.2

III

(1) F (q) (t)

),(),(ˆ tqFtqF

x, y, z zzyyxx ˆˆˆ

(2) G (q) (p) (t)G


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


)()()( 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


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


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


•••

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


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


• 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


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


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


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


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)


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


k = 0

k = 1

k = 2

k = 3


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