III. Quantum Mechanics of Vibration, Rotation, and …master/lecture/PCSS/3_quantum-32.pdf ·...

III. Quantum Mechanics of Vibration, Rotation, and Atomic structure

▶ Many fundamental properties of materials including surface, thin film, bulks can be explained by simple model systems such as harmonic oscillator, rigid rotor, structures of hydrogen atom, and particle-in-a-box problem.

1. The wave nature of particlesby Louis de Broglie in 1825– Electrons with energy of 10 KeV,

Mvh

=λ

Α=×=××

×==

×=×=→=

−−

−

−

&12.0102.1)109.5)(1011.9(

sec)(1062.6

1011.9 sec,/109.521

11731

34

3172

mJMvh

kgMmvMvE

λ

1Prof. S.-J. Park - Physical Chemistry of Solid Surfaces

2. Uncertainty principle by W. Heisenberg in 1926

3. Some quantum mechanical operatorsVariable Operator

Position x xy yz z

Linear momentum Px

Py

Pz

π4hPx x ≥∆∆ or

π4htE ≥∆∆

Spectral broadening Life time of particle in excited state

xih

∂∂

π2

yih

∂∂

π2

zih

∂∂

π2


Variable OperatorAngular Mx

Momentum

My

Mz

)(2 y

zz

yi

h∂∂

−∂∂

πCartesian coordinates

)coscotsin(2 φ

φθθ

φπ ∂

∂−

∂∂

−i

hPolar coordinates

)(2 z

xx

zi

h∂∂

−∂∂

π

)sincot(cos2 φ

φθθ

φπ ∂

∂−

∂∂

ih

)(2 x

yy

xi

h∂∂

−∂∂

π

)(2 φπ ∂

∂i

h


Variables Operator

)sin

1)(sinsin

1(4 2

2

22

2

φθθθ

θθπ ∂∂

+∂∂

∂∂

−hM2 = Mx

2 + My2 +Mz

2

Kinetic energy

)(8 2

2

2

2

2

2

2

2

zyxmh

∂∂

+∂∂

+∂∂

−π

)(21 222

zyx VVVm ++

Potential energy

V(x, y, z) V(x, y, z)


▶ Normalization of wave function

▶ Orthogonality

▶ Quantum mechanical average

∫∫

∞

∞−

∞

∞−=τψψ

τψψ

d

dGG

*

*

)(,0* mldml ≠=∫∞

∞−τψψ

∫∞

∞−=1* τψψ d

(different energy)


▶ (e.g) Average momentum

▶ The Boltzmann Distribution

02sin2

cossin

sin22

sin2

02

02

0

=⎥⎦⎤

⎢⎣⎡⎟⎠⎞

⎜⎝⎛=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛=

∫

∫

a

a

a

av

axn

na

ianh

dxa

xna

xnianh

dxa

xnaxi

ha

xna

P

ππ

ππ

ππ

π

kT

j

iji

eNN )( εε −−

=


4. Particle in a box▶ electron in a quantum confined system

▶ One-dimensional Schrödinger equation.

potential

0 a

π-electrons of a conjugated system of double bonds in a molecule

,3,2,1 ,8

2 ,sin

8

2

22

2

2

2

2

==Ε

==Ψ

ΕΨ=−

nma

hna

Aa

xnA

dxd

mh

n

nπ

ψπ

From normalization ∫∞

∞−=1* τψψ d


Energy level wave function probabilities

π- electronsQuantum well, quantum wire, quantum dot

n=1

n=2

n=3 Ψ3

Ψ2

Ψ1

Ψ32

Ψ22

Ψ12


5. The vibrational energies of a diatomic molecules

Potential

rre

Morse potential

Harmonic oscillator [U=a(r - re)2]


▶Maclaurin series expansion about r = re

Near r = re :

From Hook’s law

...)(21)( 2

2

2

)( +−⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟

⎠⎞

⎜⎝⎛+=

===− e

rre

rrrrrr rr

drudrr

drduUU

ee

ee

22

2

)( )(21

err

rr rrdr

udUe

e−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

=

−

errqkqqU −== ,21)( 2


▶ Quantum mechanical solution of Harmonic oscillator

ΕΨ=Ψ+Ψ

− )21(

82

2

2

2

2

kqdqdh

µπ

km1 m2

µπν k

21

=21

21

mmmm+

=µ

q = r - re

µπkhvv 2

)21( +=Ε ν = 0, 1, 2, 3, …

νhv )21( +=

µπν k

21

=


...3,2,1,0 ),(2

21

==Ψ vqHeN v

q

vv αα

Where, Hν is the Hermite polynomial of degree v,

,2

hkµπ

α =

ν

νν

ν dzedezH

zz

22

)1()(−

−=21

!2 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

πνα

νvN

Selection rule : 1±=∆vFor heteronuclear diatomic molecule (dipole moment change)


6. The Rotational energies of diatomic moleculesRigid Rotor

V = 0, r = d

Hamiltonian H = T + V =

From spherical harmonics

m1d m2

22

2∇−

µh

),()( ϕθmlYrR=Ψ

drY ml =←=Ψ ),( ϕθ

⎥⎦

⎤⎢⎣

⎡∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

−=

−=

2

2

222

2

22

sin1sin

sin1

2

21

ϕθθθ

θθµ

µ

dd

Ld

H

h


Ψ=Ψ EH

...2,1,0 ,2

)1(

),(),()1(2

1

),(),(2

1

2

2

22

22

=+

=

=+

=

ld

llE

EYYlld

EYYLd

ml

ml

ml

ml

µ

ϕθϕθµ

ϕθϕθµ

h

h

We use l for rotational quantum number

Energy depends only on lψ depends on l and m )( lml ≤≤−

IllE

2)1( 2h+

=2

2

dI

mdI i

µ=

=∑ ; moment of inertia

d1 d2

d Center of gravity

Selection rule

1±=∆l

∴Levels are (2l+1)-fold degenerate


7. Hydrogen atomsThe hydrogen atom

Schrödinger equationrZeU

2−=

0)(2 2

22

2

2

2

2

2

=Ψ++∂Ψ∂

+∂Ψ∂

+∂Ψ∂

rZeE

zyx h

µ

0)(22 =Ψ++Ψ∇rZE

U

r

In spherical polar coordinates

z

x

θ

0φ

yr

θcosrz =

ϕθ cossinrx =ϕθ sinsinry =


0)(2sinsin1

sin11

22222

2 =+Ε+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂ ψ

θψθ

θθϕψ

θψ

rz

rrrr

rr

),()()()()(),,( ϕθϕθϕθψ YrRrRr =ΨΘ=

Angular part

0)1(sin

1sinsin

12

2

2 =++∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂ YllYY

ϕθθθ

θθ

Radial part

0)1(2212

22 =⎟

⎠⎞

⎜⎝⎛ +

−++⎟⎠⎞

⎜⎝⎛ R

rll

rzE

drdRr

drd

r


)exp()(cos

),(

ϕθ

ϕθ

imP

Ym

l

ml

=

: surface spherical harmonicsϕϕϕ mimeim sincos +=

Associated Legendre functions

,...2,1,0 ,)1()1(!2

1)( 222 =−−= +

+

lwdwdw

lwP l

mlm

lm

l ml

])2[(2])![(2

)!1()(0

122/3

0

21

30 r

nazLer

naz

lnnlnrR l

lnna

zrl

l

nl++

−+

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡+−−

=

Associated Laguerre polynomial

,)()( sr

ssr d

LdpLρρ

= )()( rr

r

r eddepL ρρ

ρρ −=


Normalized hydrogen-like wave function

22

42

2 hneZE µ

−=

n = 1, 2, 3..l = 0, 1, 2, 3..m = -l, -l + 1,…0, 1, l - 1, l n = 1

n = 2

n = 3n = 4

reV

2

−=

rE


The probability of finding electron in the region of space r + dr, θ + dθ, φ + dφ

What is the probability of electron at r + dr,(in a thin spherical shell centered at the origin)?

The factor eimφ in Ylm(θ,φ) be removed;

Any linear combination of eigenfunctions of a degenerate energy level is an eigenfunction of the Hamiltonian with the same eigenvalue.

[ ] ϕθθϕθτψ ddrdrYrRd mlnl sin),()( 2222 =

[ ] [ ] drrrRddYdrrrR nlm

lnl22

2

0

2

0

22 )(sin),()( =∫∫ ϕθθϕθππ

Normalized spherical harmonics

[ ] drrrRnl22)(= Radial distribution function


One way to obtain a real function is


azrpp

azrppp

azrppp

zeaz

reaz

i

reaz

z

y

x

225

22

225

222

225

222

21

cossin241)(

21

cossin4

1)(2

1

0

11

11

−

−

−

⎟⎠⎞

⎜⎝⎛==

⎟⎠⎞

⎜⎝⎛=+=

⎟⎠⎞

⎜⎝⎛=+=

−

−

πψψ

ϕθπ

ψψψ

ϕθπ

ψψψ

azrp

azrp

azrp

zeaz

yeaz

xeaz

z

y

x

225

2

225

2

225

2

21

41

41

−

−

−

⎟⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛=

πψ

πψ

πψ

Radial part; R(r) of ψ Radial distribution function



7. Electron energy levels in a hydrogen atomAn orbital is characterized by a set of quantum numbers.

1 The principal quantum number n defines the electron shelln = 1, 2, 3, 4

K, L, M, N

2 The azimuthal quantum number l (angular quantum number)defines the sub-shells s,p,d,f,,,,

l = 0, 1, 2, 3… l ≤ n-1s, p, d, f…

3 The magnetic quantum number m corresponds to the possible values of the magnetic moment of the electron in an external magnetic field

-l ≤ m ≤ l

4 An electron in an orbital is characterized by the spin quantum number s, s = ± 1/2

It is convenient to introduce the inner quantum number j which represents the total angular momentum

J = L+S with j = l + s = l ± 1/2 with j > 0 (l+ s), (l+ s) –1,… ⎢l- s ⎢

Spin – orbit coupling

Quantum notation corresponding to the three inner shellsShell n sub- l m number of notation of ĵ number of

shell ( 2l+1 ) orbital orbital states (2 j+1)K 1 s 0 0 1 1s 1/2 2L 2 s 0 0 1 2s 1/2 8

p 1 +1,0,-1 3 2p 1/2,3/2M 3 s 0 0 1 3s 1/2 18

p 1 -1,0,+1 3 3p 1/2,3/2d 2 -2,-1,0, 5 3d 3/2,5/2

+1,+2


In the convenient notation, an atomic energy level or terms is represented by a capitol letter corresponding to the magnitude of , where are the l values of the individual electrons where L = 0, 1, 2, 3Term symbol is S, P, D, F…..In the case of H atom, there is only one electron and L = l

n=2, S, Pn=3, S, P, D

When spin-orbit interaction is considered energy levels should be calculated by considering Es.o

inner quantum number j = l ± sThe spin-orbit interaction energy

∑= ilLrr

ilr

SLJS L +=+12multiplicity

slrH os ⋅⋅= )(. ξ

[ ])1()1()1(21 2

. +−+−+= SSLLJJAhE os

Constant A depends on L and S


n=2: S1/2 D1/2 P3/2

n=3: S1/2 P1/2 P3/2 D3/2 D5/2

For atomic hydrogen, orbital energy level(term) are

2D5/2 2D3/2

2S 2P3/2 2P2D 2S1/2

2P1/2

2S 2P3/2 2S1/2

2P 2P5/2

n = 3

n = 2

“doublet P one” : 2P1

SLJS L +=+12

multiplicity


<Hund’s rule>To decide which of the terms arising from a given atomic

configuration is lowest in energy, we use Hund’s rule:1. The term with the highest multiplicity is lowest in energy

coupling energy is needed for pairing electrons2. If there is more than one term with the highest multiplicity, then the largest value of L lies lowest. Hund’s rule gives only the lowest term, and should not be used to decide the order of the remaining terms.Eg) ground state configuration of carbon: 1s22s22p2 1S, 3P, 1D

Hund’s rule predicts that 3P is the lowest term. agree with exp

For 1s22s22p3 (exp) 5S<3D<3P<1D<3S<1P

Don’t agreeAgree with Hund’s rule


X-ray and spectroscopic notation

Q.Ns X-ray X-ray level spectroscopic leveln l j1 0 1/2 1 K 1s 1/2

2 0 1/2 1 L1 2s 1/2

2 1 1/2 2 L2 2p 1/2

2 1 3/2 3 L3 2p 3/2

3 0 1/2 1 M1 3s 1/2

3 1 1/2 2 M2 3p 1/2

3 1 3/2 3 M3 3p 3/2

3 2 3/2 4 M4 3d 3/2

3 2 5/2 5 M5 3d 5/2


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