Quantum Mechanics – Applications and Atomic Structures

Quantum Mechanics – Applications and Atomic Structures

Jeong Won Kang

Department of Chemical Engineering Korea University

Applied Statistical Mechanics

Lecture Note - 3

Subjects

 Three Basic Types of Motions (single particle)

 Translational Motions

 Vibrational Motions

 Rotational Motions

 Atomic Structures

 Electronic Structures of Atoms

 One-electron atom / Many-electron atom

Free Translational Motion

ψ _Eψ dx

d

m =

− ^{2 2} ₂ 2

 ^Hψ = ^Eψ

 All values of energies are possible (all values of k)

 Momentum

 Position  See next page

2 2 2

2 dx d H = − m

ikx ikx

k = Ae + Be⁻

ψ

OR

Solutions

m E_k k

2

2 2

=

 k p

e^ikx → _x = + e^−ikx → p_x = −k

Probability Density

B = 0

A = 0

A = B

Ae

ψ =

Be

ψ =

kx Acos

= 2

ψ

2 2

= A

ψ

2 2

= B

ψ

kx A

2

= 4 cos

ψ

nodes

A particle in a box

m

Consider a particle of mass m is confined between two walls

L x x

V

L x V

=

=

∞

=

<

<

=

and 0 for

0 for

0

ψ ψ _{V x} ψ _E dx

d

m + =

− ( )

2 ²

2

2

kx D

kx

k = Csin + cos

ψ m

E_k k

2

2 2

=

All the same solution as the free particle but with different boundary condition

A particle in a box

 Applying boundary condition

 Normalization

0 ) (

0 ) 0 (

=

=

=

= L x x ψ ψ

kx D

kx

k = Csin + cos

ψ

= 0 D

0 sinkL =

C ^{kL = n}π

1,2,3,...

n ) / sin(

)

(x =C n x L =

n π

ψ

2 1 )

/ (

sin

2 0

2 2

0

2 =





2 /

2 1



 



=

C L 2 sin( / ) n 1,2,3,...

) (

2 / 1

=



 



= n x L

x L

n π

ψ ^{En n}_8mL^h

2 2

= n cannot be zero

A particle in a box

- Properties of the solutions

 n : Quantum number (allowable state)

 Momentum

1,2,3,...

n ) / 2 sin(

) (

2 / 1

=



 



=  n x L

x L

n π

ψ

L k n

) -e L (e

L i x L n

x ^{ikx -ikx}

n

π π

ψ  =



 



= 



 



=  2

2 ) 1 / 2 sin(

) (

2 / 1 2

/ 1

 k p

e^ikx → _x = + e^−ikx → p_x = −k

Probability : 1/2 Probability : 1/2

A particle in a box

- Properties of the solutions

 n cannot be zero (n = 1,2,3,…)

 Lowest energy of a particle (zero-point energy)

 If a particle is confined in a finite region (particle’s location is not indefinite), momentum cannot be zero

 If the wave function is to be zero at walls, but

smooth, continuous and not zero everywhere, the it must be curved  possession of kinetic energy

mL E h

8

2 1 =

Wave function and probability density

) / (

2 sin )

(

2

n x L

x L π

ψ ^



 



= 

With n→∞, more uniform distribution:

corresponds to classical prediction

“correspondence principle”

Orthogonality and bracket notation

' 0

* =



1

0

' = n n =

n n

) ' (

0

' '

* _n d n n n n

n = = ≠



' _nn'

n

n ⁼δ Orthogonality

Dirac Bracket Notation

<n| “BRA”

|n’> “KET”

Kronecker delta

Wave functions corresponding to different energies are orthogonal

Orthonormal

=Orthogonal + Normalized

Motion in two dimension

ψ ψ

ψ E

y x

m  =



 





∂ + ∂

∂

− ² ∂² ₂ ² ₂ 2

 ψ(x,y) = X(x)Y(y)

X dx E

X d

m = ^X

− ^{2 2} ₂ 2



Y dx E

Y d

m = ^Y

− ^{2 2}₂ 2



Y

X E

E E = +

1 1 2

/ 1

1

2 sin )

1(

L x n x L

X_n π



 



=

2 2 2

/ 1

2

2 sin )

2(

L y n y L

Y_n π



 



=

2 2 1

1 2

/ 1

2 2 / 1

1

, 2 2 sin sin

) ,

2(

1 L

y n L

x n L

y L

n x

n

π

ψ _ π



 



 



 



= 

m h L n L

E_{n n} n

2

2 2

2 2 2 2

1 2 1 , ₂

1 

 



 +

= Separation of Variables

Solution

The wave functions for a particle confined to a rectangular surface

2 ,

2 ₂

1 = n =

n 1

, 1 ₂

1 = n =

n n₁ = n1, ₂ = 2 n₁ = n2, ₂ =1

Degeneracy

 Degenerate :

 Two or more wave functions correspond to the same energy

 If L₁=L₂ then

2 2 2

,

1 8

5 mL E = h

L y L

x y L

x π π

ψ1,2^{( , ) 2} ^{sin sin 2}



 



= 

L y L

x y L

x π π

ψ2,1^{( , ) 2} ^{sin 2 sin}



 



=  ²

2 1

,

2 8

5 mL E = h

the same energy

with different wave functions

Degeneracy

L y L

x y L

x π π

ψ1,2^{( , ) 2} ^{sin sin 2}



 



= 

L y L

x y L

x π π

ψ2,1^{( , ) 2} ^{sin 2 sin}



 



= 

degenerate not

are functions wave

the ,

if L

≠ L

The degeneracy can be traced to the symmetry of the system

Motion in three dimension

ψ ψ ψ

ψ _E

z y

x

m  =



 





∂ + ∂

∂ + ∂

∂

− ² ∂² ₂ ² ₂ ² ₂ 2



3 3 2

2 1

1 2

/ 1

3 2 / 1

2 2 / 1

1 ,

, 2 2 2 sin sin sin

) , ,

3(

2

1 L

y n L

y n L

x n L

L z L

y

n x

n n

π π

ψ _ π



 



 



 



 



 



=

m h L n L

n L

E_{n n n} n

2

2 2 3 2 3 2

2 2 2 2

1 2 1 ,

, _{2 3}

1 

 



 + +

= Solution

Tunneling

 If the potential energy of a particle does not rise to

infinity when it is in the wall of the container and E < V , the wave function does not decay to zero

 The particle might be found outside the container (leakage by penetration through forbidden zone)

Use of tunneling

 STM (Scanning Tunneling Microscopy)

5 nm

uf

Si(111)-7x7

Vibrational Motion

 Harmonic Motion

 Potential Energy

 Schrödinger equation

kx F = −

2

2 1 kx V =

ψ ψ

E dx kx

d

m + =

− ^{2 2} ₂ ²

2 1 2



m₁ m₂

k : force constant

2 1

2 1

m m

m m

= + μ

reduced mass

Solutions

α y = x Wave Functions

Energy Levels

4 / 2 1



 



=

mk α 

,...

2 , 1 , 0

2)

( 1

2 / 1

=



 



=  +

= v

m v k

E_v ω ω

2

2/

)

( ^y

v v

v = N H y e⁻ ψ

Energy Levels

,...

2 , 1 , 0

2)

( 1

2 / 1

=



 



=  +

= v

m v k

E_v ω ω

1 − = ω

+ v

v E

E

2 1

0 = ω

E

Reason for zero-point energy  the particle is confined : position is not uncertain, momentum cannot be zero

2

2 1 kx V =

Potential energy

Quantization of energy : comes from B.C

±∞

=

= x

v 0 at ψ

Wave functions

Quantum tunneling

The properties of oscillators

 Expectation values

 Mean displacement and mean square displacement

 Mean potential energy and mean kinetic energy

 Virial Theorem

 If the potential energy of a paticle has the form V=ax^b, then its mean potential and kinetic energies are related by

dx v

v ^∞



∞

−

Ω

= Ω

=

Ω ^ˆ ψ^*ˆ ψ

2 / 1 2

) )(

2 ( 1

v mk

x = + 

=0 x

2 / 1 2

) ) (

2 ( 1 2 1 2

1

v mk kx

V = = + 

V b E_K = 2

Ev

V 2

= 1 ^{EK Ev}

2

= 1

Rotation in Two Dimensions : Particles on a ring

 A Rotational motion can be described by its angular momentum J

 J : vector

 Rate at which a particle circulates

 Direction : the axis of rotation

 A particle of mass m constrained to move in a circular path of radius r in xy-plane

 V = 0 everywhere

 Total Energy = Kinetic Energy locity

angular ve :

inertial of

moment

: ²

ω

ω

mr I

I I J

=

=

I E J

pr J

m p

E

z z

2 2 /

2 2

=

±

=

=

Rotation in Two Dimension

Wave Functions

Energy Levels

,...

2 , 1 , 0 ± ±

l = m

I m I

E_m J^{z l}

l 2 2

2 2

2 = 

=

2 /

)1

2 ) (

(φ π

ψm_l ^imlφ

= e

Schrödinger eqn.

) ( )

2

(φ π ψ φ

ψm_l + = m_l

φ ψ ψ

2 2

2 2

 IE d

d =

B.C

The wave function have to be single-valued

Wave functions

 States are doubly degenerate except for m_l = 0

The real parts of wave functions

Rotation in Three dimension

 A particle of mass m free to move anywhere on the 3D surface or a sphere

 Colatitude (여위도) : θ

 Azimuth (방위각) : φ

Rotation in Three dimension

l l

l l

m_l = , −1, −2,...,− harmonics

spherical :

) ,

,m_l (θ φ Yl

Wave Functions

Energy Levels ,...

, 2 , 1 ,

= 0 l

l I l

E = ( +1) 2 Schrödinger eqn.

ψ ψ ^E m ∇ =

− ²

2



Wave functions

(2l + 1) fold degeneracy

Orbital momentum quantum number

Magnetic Quantum number

l l

l l

m_l = , −1, −2,...,− ,...

, 2 , 1 ,

= 0 l

Location of angular nodes

Wave functions

Angular Momentum

 The energy of a rotating particle

 Classically,

 Quantum mechanical

 Angular Momentum

 Magnitude of angular momentum

 Z-component of angular momentum I

E J 2

2

=

,...

2 , 1 , 0 ) 2

1 (

2

= +

= l

l I l

E 

{

}

=

 ml

=

Space Quantization

 The orientation of rotating body is quantized : rotating body may not take up an arbitrary

orientation with respect to some specified axis

Space Quantization

 The vector model

 l_x, l_y, l_z do not commute with each other. (l_x, l_y, l_z are complementary observables)

 uncertainty principle forbids the simultaneous, exact specification of more than one component (unless l=0).

 -If l_z is known, impossible to ascribe values to l_x, l_y.

l

= h/2 πi {-sinφ(∂/∂θ) -cotθ cosφ(∂/∂ϕ)}

l

= h/2 πi {cosφ(∂/∂θ) -cotθ sinφ(∂/∂ϕ)}

l

= h/2 πi (∂/∂ϕ)

Angular momentum L (l

, l

, l

)

The Vector Model

The vector

representing the

state of angular

momentum lies

with its tip on any

point on the mouth

of the cone.

Experiment of Stern-Gerlach

 Otto Stern and Walther Gerlach (1921)

 Shot a beam of silver atoms through an inhomogeneous magnetic field

 Evidence of space quantization

The classically expected result

The Observed outcoming using silver atoms

Spin

 Stern and Gerlach

 observed two bands using silver atoms (2)

 The result conflicts with prediction : (2l+1) orientation (l must be an integer)

 Angular momentum due to the motion of the electron about its own axis : spin

 Spin magnetic number : m_s

 m_s =s, s-1,…,-s

 Spin angular momentum

 Electron spin

 s = ½

 Only two states



 0.866 4

3 ^1/2

=



 



=

Fermions and Bosons

 Electrons : s=1/2

 Photons : s=1

Particles with half-integral spin (s=1/2)

Elementary particles that constitute matters Electrons, nucleus Fermions

Particles with integral spin (s=0,1,…)

Responsible for the forces that binds fermions Photons Bosons

Atomic Structure

and Atomic Spectra

 Electronic Structure of Atoms

 One Electron Atom : hydrogen atom

 Many-Electron Atom (polyelectronic atom)

 Spectroscopy

 Experimental Technique to determine electronic structure of atoms.

 Spectrum

• Intensity vs. frequency (ν), wavelength (λ), wave number (ν/c)

Topics

Structure and Spectra of Hydrogen Atoms

 Electric discharge is passed through gaseous hydrogen , H₂ molecules and H atoms emit lights of discrete frequencies

Spectra of hydrogenic atoms

 Balmer, Lyman and Paschen Series ( J. Rydberg)

 n₁ = 1 (Lyman)

 n₁ = 2 (Balmer)

 n₁ = 3 (Paschen)

 n₂ = n₁ + 1, n₁ +2 , …

 R_H = 109667 cm ^-1 (Rydberg constant)

 Ritz combination principle

 The wave number of any spectral line is the difference between two terms



 



 −

= ₂

2 2

1

1

~ 1

n R_H n

ν

2 1

~ =T −T

2 ν n T_n = R^H

The wave lengths can be correlated by two integers

 Two different states

Spectra of hydrogenic atoms

 Ritz combination principle

 Transition of one energy level to another level with emission of energy as photon

 Bohr frequency condition

2

1

hcT

hcT hv

E = = −

Δ

원자에서 방출되거나 흡수된 electromagnetic radiation 은 주어진 특정 양자수로 제한된다.

따라서 원자들은 몇 가지 주어진 상태만을 가질 수 있음을 알 수 있다.

남은 문제는 양자역학을 이용하여 허용된 에너지 레벨을 구하는 것이다.

The Structure of Hydrogenic Atoms

 Coulombic potential between an electron and hydrogen atom ( Z : atomic number , nucleus charge = Ze)

 Hamiltonian of an electron + a nucleus r

e V Ze

0 2

4π

−

=

r Ze m

V m E

E

H _N

N e

e nucleus

K electron

K

0 2 2

2 2

2 ,

, 2 2 4

ˆ

ˆ ^{+ + = − ∇ − ∇ −} πε

=  

Electron coordinate

Nucleus coordinate

Separation of Internal motion

 Full Schrödinger equation must be separated into two equations

 Atom as a whole through the space

 Motion of electron around the nucleus

 Separation of relative motion of electron from the motion of atom as a whole (Justification 13.1)

r H Ze

0 2 2

2

4 2μ ^{∇ −} πε

−

=  me mN me

1 1

1

1 = + ≈

μ

The Schrödinger Equation

ψ ψ ^E

H = ψ ψ

ψ πε

μ ^{r E}

Ze =

−

∇

−

0 2 2

2

4 2



Y l

l

Y ( 1)

2 = − +

Λ

2 2

0 2

2 ) 1 (

4 r

l l r V_eff Ze

μ

πε

+ +

−

=

ER R

dr V dR R dr

R d

eff =

−



 



 +

− 2

2 ²

2 2



μ

) , ( ) ( )

, ,

( θ φ θ φ

ψ ^r = ^{R r Y}

Separation of Variable :

Energy is centrosymmetric Energy is not affeced by

angular component (Y) (Justification 13.2)

R : Radial Wave Equation New solution required

Y : All the same equation as in Chap.12 (Section 12.7, the particle on a sphere) Spherical Harmonics

The Radial Solutions

0

2 a

= Zr ρ

ER R

dr V dR R dr

R d

eff

=

 −



 



 +

− 2

2

2 2

μ



Solution

n l

n l l

n l

n

L e

N n r

R

( )

ρ _

( ρ )



 



= 

r) in l exponentia (decaying

r) in polynomial (

)

( r = ×

R

...

3 , 2 , 1

with 32

4 2

=

−

= n

n e

e E

Z

π  μ

Allowed Energies

Radial Solution

2 2 0 0

4 e a m

e

πε 

=

Reduced distance Bohr Radius

Associated Laguerre polynomials

Hydrogenic Radial Wavefunctions

The Radial Solutions

Atomic Orbital and Their Energies

 Quantum numbers

 n : Principal quantum number (n=1,2,3,…)

• Determines the energies of the electron

 an electron with quantum number l has angluar momentum

 An electron with quantum number m_l has z-component of angular momentum

ml

l n ,,

...

3 , 2 , 1

with 32 ² ₀^{2 2 2}

4 2

=

−

= n

n e

e E_n Z

π  μ

{

}

±l

±

±

= 0, 1, 2,..., m

with

m_l _l

Shell

Sub Shell

 Bound / Unbound State

 Bound State : negative energy

 Unbound State : positive energy (not quantized)

 Ryberg const. for hydrogen atom

 Ionization energy

 Mininum energy required to remove an electron form the ground state

32 ² ₀^{2 2}

4

 e hcR_H ^He

π

− μ

=

32 ² ₀^{2 2}

4

1 e 

hcR e

E _H ^H

π

= μ

−

=

Energy of hydrogen at ground state n=1

hcRH

I =

Ionization Energy

The energy levels

Shells and Subshells

 Shell

 n = 1 (K), 2 (L), 3 (M), 4(N)

 Subshell ( l=0,…. , n-1)

 l = 0 (s), 1 (p), 2 (d), 3(f), 4(g), 5 (h), 6 (i)

 Examples

 n = 1

• l = 0  only 1s (1)

 n = 2

• l = 0, 1  2s (1) , 2p (3)

 n = 3

• l = 0, 1, 2  3s (1), 3p (3), 3d (5)

 number of orbitals in n_th shell : n²

 n² –f old degenerate

m_l s are limited to the value -l,..,0, …+l

Ways to depicting probability density

Electron densities (Density Shading)

Boundary surface

(within 90 % of electron probability)

Radial Nodes

 Wave function becomes zero ( R(r) = 0 )

 For 2s orbital :

 For 3s orbital : Z a r = 2 ₀ /

Z a

r =1.90 ₀ / r = 7.10a₀ / Z

0

0

Radial Distribution Function

 Wave Function

 probability of finding an electron in any region

 Probability Density (1s)

 Probability at any radius r = P (r ) dr

 Radial Wave Function : R(r )

 Radial Distribution Function : P(r)

 dr 을 곱하면 확률이 된다.

 1s orbital 에 대하여 ,

/ 0

2 2Zr a

e⁻ ψ ∝

2

2 ( )

)

(r r R r P =

ψ 2

2

4 2

)

(^r π^r ψ P =

/ 0

2 2 3 0

4 3

)

( r e ^{Zr a}

a r Z

P = ⁻

Radial Distribution Function

Bohr radius

Wave function Radial Distribution Function

p-Orbital

 Nonzero angular momentum

 l > 0

 p-orbital

 l = 1

• m_l = 0

• m_l = + 1

• m_l = -1

rl

ψ ∝

r

ψ ∝

pz

ψ

py

ψ

px

ψ

D-orbital

 n = 3

 l = 2

 m

= +2, +1, 0,1,2

Spectroscopic transitions and Selection rules

 Transition (change of state)

 All possible transitions are not permissible

 Photon has intrinsic spin angular momentum : s = 1

 d orbital (l=2)  s orbital (l=0) (X) forbidden

• Photon cannot carry away enough angular momentum

...

3 , 2 , 1 with 32 ² ₀^{2 2 2}

4 2

=

−

= n

n e

e E_n Z

π 

n₁, l₁,m_l1 n₂, l₂,m_l2 μ

Photon

hv E = Δ

Selection rule for hydrogenic atoms

 Selection rule

 Allowed

 Forbidden

 Grotrian diagram

1 , 0 ±

= Δml

±1

= Δl

Structures of Many-Electron Atoms

 The Schrödinger equation for many electron-atoms

 Highly complicated

 All electrons interact with one another

 Even for helium, approximations are required

 Approaches

 Simple approach based on H atom

• Orbital Approximation

 Numerical computation technique

• Hartree Fock self consistent field (SCF) orbital

Pauli exclusion principle

 Quantum numbers

 Principal quantum number : n

 Orbital quantum number : l

 Magnetic quantum number : m

 Spin quantum number : m

 Two electrons in atomic structure can never have all four quantum numbers in common

 All four quantum number  “Occupied”

Penetration and Shielding

 Unlike hydrogenic atoms 2s and 2p orbitals are not degenerate in many-electron atoms

 electrons in s orbitals generally lie lower energy than p orbital

 electrons in many-electron atoms experiences repulsion form all other electrons  shielding

 Effective nuclear charge, shielding constant

σ

−

= Z Z_eff

constant shielding

:

charge nuclear

effective :

σ

Zeff

Penetration and Shielding

 An s electrons has a greater “penetration”

through inner shell than p electrons

 Energies of electrons in the same shell

 s < p < d < f

 Valence electrons

 electrons in the outer most shell of an atom in its ground state

 largely responsible for chemical bonds

The building-up principle

 The building-up principle (Aufbau principle )

 The order of occupation of electrons

 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s

 There are complicating effects arising from electron-electron repulsion (when the orbitals have very similar energies)

 Electrons occupy different orbitals of a given subshell before doubly occupying any one of them

 Example : Carbon

• 1s² 2s² 2p²  1s² 2s² 2p_x¹2p_y¹ (O) 1s² 2s² 2p_x² (X)

 Hund’s maximum multiplicity principle

 An atom in its ground state adopts configuration with the greatest number of unpaired electrons

favorable unfavorable

3d and 4s orbital

 Sc atom (Z=21)

 [Ar]3d₃

 [Ar]3d₂4s₁

 [Ar]3d₁4s₂

 3d has lower energy than 4s orbital

 3d repulsion is much

higher than 4s (average

distance from nucleus is

smaller in 3d)

Ionization energy

 Ionization energy

 I₁ : The minimum energy required to remove an electron from many- electron atom in the gas phase

 I₂ : The minimum energy required to remove a second electron from many-electron atom in the gas phase

Li :2s¹ Be :2s²

B : 2s²p¹

Self-consistent field orbital

 Potential energy of electrons

 Central difficulty  presence of electron-electron interaction

 Analytical solution is hopeless

 Numerical techniques are available

•  Hartree-Fock Self-consistent field (SCF) procedure (HF method)

 ⁺ 

−

=

i o i i j

e

r

e r

e V Ze

'

,

2 2

4 2

1

4 π ε π ε

Schrödinger equation for Neon Atom 1s

2s

2p

, 2p electrons

) (

) ) (

( )

( 4

) ) (

( )

( 2

) 4 (

) 2 (

1 2 2

1 2

12

2 2

* 2 2

1 2 2

12

2

* 2 2

1 2 1 2

1 2 2 1 2

r E

r r d

r e r

r r d

r r

e r r Ze m r

p p

i i

p i

o

p i

i i

o

p o

p e

ψ

ψ ψ τ

ψ πε

ψ ψ τ

ψ πε

πε ψ

ψ

=



 





 

− 

 



  + 

−

∇

−







Sum over orbitals (1s, 2s)

Kinetic energy of electron

Potential energy (electron – nucleus)

Columbic operator

(electron – electron charge density)

Exchange operator (Spin correlation effect)

HF Procedure

 There is no hope solving previous eqn. analytically

 Alternative procedure (numerical solution)

Assume 1s, 2s orbital

Solve 2p

Solve 2s

Solve 1s

Compare E and wave function

Converged

Solution

Example

 Orbitals of Na using SCF calculation

Quiz

 What is a degeneracy ?

 Explain four quantum number.

 Why transition between two states are not always allowed ?

 Explain difficulties of solving Schrödinger

equations for many-electron atoms.