Quantum Mechanics of Atoms

量子力學用於原子結構

包立不相容原理也適用於廁所

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薛丁格理論 ( The Schrodinger Theory )

Schrodinger Equation

structure of hydrogen atom

properties of other atoms

periodic table

a system of atoms, crystals

semiconductors

integrated circuits ( I.C. )

氫原子的薛丁格程式 ( Schrodinger Equ. For H Atom )

for an e- in an atom with Z protons :

( Z : atomic number 原子序 , Z = 1 for H atom )

Ep = ————— ( r : distance between e- and protons )- 1 Ze2

4o r

( x )( x,y,z ) , Ep( x ) Ep( x,y,z )

- h2/2m ( 2/x2 + 2/y2 + 2/z2 ) + Ep( x,y,z ) = E

( x,t ) = ( x ) ( t )

( t ) = e( -i E t / h )

– + Ep(x) = Eh2

2m1

( x)

d2( x)

dx2

Ep = ————— ( r : distance between e- and protons )- 1 Ze2

4o r

x

y

z

r e-in spherical coordinates :

Ep( x,y,z ) Ep( r )

/x, /y, /z/r, /, /

Schrodinger equ. becomes :

——— ( r2 —— ) + ————— ( sin—— ) + ——————

+ —— [ E –Ep( r ) ] = 0

1 r2 r

r

1 r2sin

1 2r2sin2

2mh

use separation of variables : ( r,,) = R( r ) ( ) ( ) 代入上式

Schrodinger equ. becomes :

————— ( r2 —— ) - —— r2sin2[ E –Ep( r ) ] –——— ( sin—— )

= ————

- sin2 dR dr

dRdr

2mh2

dd

sind d

1 d2 d2

———— = -ml2

——— ( r2 — ) + —— [ E –Ep( r ) ] = ——— - —————( sin—— )

1 d2 d2

1 dR dr

dRdr

2mr2

h2

ml2

sin2

1 1 dsind

dd

等式兩邊必定等於同一常數

等式兩邊必定等於同一常數

———— = -ml2

——— - —————( sin—— ) = l ( l+1 )

1 d2 d2

ml2

sin2

1 1 dsind

dd

1 dR dr

dRdr

2mr2

h2——— ( r2 — ) + —— [ E –Ep( r ) ] = l ( l+1 )

, , R must be well-behaved functions

(1) must be single-valued

ml = 0, 1, 2, …

(2) must be finite

l = 0, 1, 2, … and l ml

(3) R must be finite

E = En = –—————— , n = 1, 2, 3, … and l < nZ2 e4 m 1

8o2 h2 n2

n : principle quantum number ( 主量子數 )

decide En

l : orbital quantum number ( 角量子數 )

0, 1, 2, ….., n-1

ml : magnetic quantum number ( 磁量子數 )

0, 1, 2, 3, ….., l

量子數的物理意義( Physical Significance of Quantum Numbers )

•n, l, ml decide n l ml

states having different sets of quantum numbers ( different ),

but the same energy degenerate states

0 1 2 3 ls p d f

n

4

3

2

1E

( ml=0 ) ( ml= -1 , 0 , 1 ) ( ml= -2 , -1 , 0 , 1 , 2 )

( for H atom )

degeneratestatesdegeneratestates

degeneratestates

•l angular momentum ( L ) of the atom

similar to p - i h /x , L can be represented as an operator

L2 n l ml= l ( l+1 ) h2 n l ml

L = l ( l+1 ) h ( ~ l h )

( 電子的轉動角動量是量子化 ,

作不連續性的變化 )

•ml z component of L in an external magnetic field directed

along the z-direction

Lz = ml h ( ml,max = l )

( 角動量的方向 , 或者說電子轉動的平面在空間中的變化 , 也是跳躍式

不連續性的變化 )

L

e-

例 :

l = 1 , L = 2 h l = 2 , L = 2 x 3 h = 6 h

2h

1h

ml = 0

-1h

-2h

z

L1h

ml = 0

-1h

z

L

空間量子化( space

quantization )

Zeeman 效應 ( Zeeman Effect )

–均勻磁場中的實驗

i

: magnetic dipole moment associated with the atom’s

angular momentum L

Ep : A potential energy associated with in a magnetic

field B, Ep = ( e / 2m ) B Lz

Etotal = En + Ep

Stern-Gerlach 實驗 ( Stern-Gerlach Experiment )

–不均勻磁場中的實驗

電子自轉 ( Electron Spin )

•electron spin intrinsic angular momentum :

S = s ( s+1 ) h , s = ½

= 3 /2 h

Sz = ms h , ms = ½

•4 quantum numbers to specify

a state ( or a wave-function )

of an electron : n, l, ml, ms

n : 1, 2, 3, …..

l : 0, 1, 2, ..., n-1

ml : 0, 1, 2, …, l

ms : ½

spin up spin down

原子波函數的特徵 ( Features of the Atomic Wavefunctions )

•s states ( l = 0 )2 have spherical symmetry

1s probabilitydistribution

1s orbitrepresentative

shape

1s orbitprobabilitydistributionand shape

•other states axial symmetry, but no spherical symmetry

例 : 2p states ( l = 1 )

例 : 3d states ( l = 2 )

•n,l,mln,l,ml

has

spherical symmetry

例 : all 6 of the p state e- wave

functions add up

spherical symmetry

all 10 of the d state e- wave

functions add up

spherical symmetry

ml

•by looking at the radial probability density P( r )

[ P( r )dr : probability of finding e- between r r+dr ]

given state n , lower l more likely to be found near the nucleus

( lower l lower angular momentum L )

包立不相容原理 ( Pauli’s Exclusion Principle )

no 2 electrons in a system ( an atom or a solid ) can be in the

same quantum state ( have the same n, l, ml, ms )

Pauli ( 1900 –1958 )

1927 Solvay Conference at Belgium

元素週期表 ( Periodic Table )

n l ml ms

1 0 0 ½ 1s2

2 0 0 ½ 2s2

1 -1 ½0 ½ 2p6

1 ½

3 0 0 ½ 3s2

1 1, 0 ½ 3p6

2 -2 ½-1 ½0 ½ 3d10

1 ½2 ½

2 x ( 2l + 1 )

•for H atom : n En ( regardless of l )

•for multi-electron atoms :

( 1 ) n, l En,l

( 2 ) n e- energy higher, E ( E < 0 )

E12 > E23 > E34

( 3 ) for a given n , e- with the lowest l has the lowest energy

E

f ( l = 3 )d ( l = 2 )p ( l = 1 )s ( l = 0 )

1s

2s 2p

3s 3p 3d

4s 4p 4d 4f

5s 5p 5d 5f

6s … …

•elements in a group have similar chemical and optical properties

( because they have similar valence e- configurations )

( outermost e- valence electron , 價電子 )

•elements in a group have similar chemical and optical properties

( because they have similar e- configurations )

•rare gases ( group VIII ) He, Ne, Ar … : monoatomic inert gases

Ne

filled valence

subshell

•alkali elements ( group I )

1 outermost e- ( outermost e- valence electron )

because of the spherical charge distribution of the inner e-

valence e- sees only one “+e”net charge

easy to ionize this valence e- ( chemically active )

ionic compound or metallic solid

例 : Cl + e- Cl- + 3.62 eV ( Cl : 1s22s22p63s23p5 )

Na + 5.14 eV Na+ + e-

5.14 eV –3.62 eV = 1.52 eV can be provided by the

Coulomb’s attraction energy between Na+ and Cl-

NaCl very stable compounds

•scandium to zinc ( 過渡金屬 )

because 4s is lower in energy than 3d

4s filled before 3d

because 4s radius > 3d radius

4s2 electrons shield 3d e- from external influence

4s2 e- participate in bonding

Sc to Zn have similar chemical properties