WORLD OF MOLECULES

Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework**

Consortium leader

PÁZMÁNY PÉTER CATHOLIC UNIVERSITY

Consortium members

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***

**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben

***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.

PÁZMÁNY PÉTER CATHOLIC UNIVERSITY SEMMELWEIS

UNIVERSITY

WORLD OF MOLECULES

DUAL NATURE OF ELECTRONS

(Molekulák világa)

(Az elektron kettős természete)

KRISTÓF IVÁN

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1. Nucleus 2. Isotopes

3. Tables of isotopes 4. Radioactivity

5. Decay modes

6. Bohr-Sommerfeld model 7. Quantum numbers

8. Electron structure 9. Examples

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World of Molecules: Dual nature of electrons

Previously - Properties of atoms

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World of Molecules: Dual nature of electrons

Previously – Table of Isotopes Types of

decay

http://commons.wikimedia.org/wiki/File:Table_isotopes_en.svg

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World of Molecules: Dual nature of electrons

Previously – decay chain of U-238

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World of Molecules: Dual nature of electrons

Previously - Periodic table – electron configurations

http://en.wikipedia.org/wiki/Periodic_table_%28electron_configurations%29

1. Dual nature of light

2. Particle nature of electron

3. Wave nature of electrons (de Broglie) 4. Particle-wave duality of electrons

5. Schrödinger equation

6. The wave functions of the electron in 1D

7. The wave functions of the electron in a harmonic oscillator 8. The wave functions of the electron in 3D

9. The wave functions of the electron in the Hydrogen atom 10. Short introduction to complex numbers

Table of Contents

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World of Molecules: Dual nature of electrons

Light – electromagnetic wave

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http://en.wikipedia.org/wiki/File:EM_spectrum.svg

World of Molecules: Dual nature of electrons

Light – dual nature of light

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World of Molecules: Dual nature of electrons

• Light has dual nature

• certain experiments can be explained by

• particle like behavior

• e.g. photoelectric effect

• wave like behavior

• e.g. double-slit experiment, interference

• this is termed as wave-particle duality

Light – dual nature of light

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World of Molecules: Dual nature of electrons

Photoelectric effect

• electrons are emitted from a metallic or non- metallic surface due to visible or UV light illumination

• the energy of electrons is directly proportional to the frequency of the photon above a certain

threshold

• can be explained by particle like behavior and

elastic collision

World of Molecules: Dual nature of electrons

Light – photoelectric effect

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World of Molecules: Dual nature of electrons

Light – photoelectric effect

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http://sv.wikipedia.org/wiki/Fil:Fotoelektrisk_effekt3.png

Light – dual nature of light

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World of Molecules: Dual nature of electrons

Double-slit experiment

• polarized light is passing through two parallel slits and hits a screen on the other side

• produces periodical diffraction and interference patterns on the screen

• the distance of the two slits have to be in the range of the wavelength of the photon

• even one photon can produce such interference

pattern

Light – double-slit experiment

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http://en.wikipedia.org/wiki/File:Ebohr1.svg

World of Molecules: Dual nature of electrons

Light – double-slit experiment

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World of Molecules: Dual nature of electrons

Electron – dual nature

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World of Molecules: Dual nature of electrons

• Electron was thought to be a particle

• Bohr’s model of the atom places the electrons as particles on allowed orbits around a tiny

nucleus

• One quantum number is sufficient to describe the electron (principal quantum number)

• could describe the emmission spectral lines of

excited hidrogen gas

Electron – dual nature

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World of Molecules: Dual nature of electrons

• For further refinement to describe more elements electron structure

• Sommerfeld expanded Bohr’s theory with two other quantum numbers

• azimuthal quantum number (l) and magnetic quantum number (m)

• the emission spectral lines of many elements

could be approximated with this refinement

Bohr-Sommerfeld model (1920)

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• The electrons can only travel in special orbits

• at discrete distances from the nucleus

• with specific energies

• The electrons do not lose energy as they travel on these orbits – in contrast with classical

electrodynamics

• The angular momentum of electrons are integer multiples of the reduced Plack’s constant (h/2π)

World of Molecules: Dual nature of electrons

Bohr-Sommerfeld model (1920)

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• Angular momentum

• Radius of orbits

• The circumference of orbits are inversely

proportional to the momentum of the electrons , …

2 , 1 where

2 ,

h =

= n n

r m _{n n}

v π

n

n n m

r v

2 π = h

World of Molecules: Dual nature of electrons

Bohr-Sommerfeld model (1920)

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http://en.wikipedia.org/wiki/File:Bohr_atom_model_English.svg

World of Molecules: Dual nature of electrons

Bohr-Sommerfeld model (1920)

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World of Molecules: Dual nature of electrons

• 4 quantum numbers uniquely represent the state of the electron inside an atom

• n - principal quantum number

describes the electron shell (n=1, 2, ..., 6)

• l - azimuthal q. n. or angular momentum describes the subshell (l =0, 1..., n-1)

• m - magnetic quantum number

describes the subshell’s shape (m= -l, ..., 0, ..., l)

• s - spin quantum number (s =-1/2 or +1/2)

Bohr-Sommerfeld model (1920)

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http://en.wikipedia.org/wiki/File:Sommerfeld_ellipses.svg

World of Molecules: Dual nature of electrons

Bohr-Sommerfeld model (1920)

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World of Molecules: Dual nature of electrons

name symbol meaning Value

Principal quantum number

n Shell (distance from nucleus) n=1,2,3...,6 Azimuthal quantum

number

l Subshell (shape of orbital) l=0,1, ..., n-1 Magnetic quantum

number

m energy shift (orientation of the sub shell's shape)

m=-l, ...,0, ..., l Spin quantum number s Spin of the electron

2 or 1 2

1 +

−

=

s

Wave nature of electrons

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World of Molecules: Dual nature of electrons

• de Broglie (in 1924)

• proposed a new interpretation of electron

• analogue to the particle-wave duality of photons/light

• supposed that electrons as particles could exhibit wave like properties

• further development:

• all particles can have wave properties

Wave nature of electrons

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World of Molecules: Dual nature of electrons

• de Broglie (in 1924)

• also proton and neutron can be assigned a wave property

• introduced wave properties of the electron to the Bohr-Sommerfeld model

• resulted in electrons as standing waves along the circumference of orbitals

• At that time it was a mere assumption

Bohr-Sommerfeld model with de Broglie (in 1924)

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• Angular momentum and wavelength

• Radius of orbits

• The circumference of orbits are integer multiples of the electron’s wavelength

, … 2 , 1 where

2 ,

h =

= n n

r m _{n n}

v π

n r mv

h p

n n

λ = ^h = ⇒ ² π

n

n n m

r v

2 π = h

World of Molecules: Dual nature of electrons

λ

= n

Wave nature of electrons

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World of Molecules: Dual nature of electrons

• If de Broglie’s assumption holds

• then electrons should also have interference patterns in double-slit experiments

• Within years it has been proven that crystals are small enough optical lattices for the electron’s wavelength to produce diffraction patterns

• however, since electrons are different than

photons their diffraction patterns can only be

obtained from a large number of electrons

Light – double-slit experiment

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http://en.wikipedia.org/wiki/File:Ebohr1.svg

World of Molecules: Dual nature of electrons

Light – double-slit experiment

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World of Molecules: Dual nature of electrons

Light – double-slit experiment (intensity plot)

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http://upload.wikimedia.org/wikipedia/commons/7/71/Fentes_de_young_profil_intensite.png

World of Molecules: Dual nature of electrons

Electrons - duble-slit experiment

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World of Molecules: Dual nature of electrons

Duble-slit experiment on single electrons

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http://upload.wikimedia.org/wikipedia/commons/8/83/2Slits-particles_only.jpg

World of Molecules: Dual nature of electrons

Electrons - duble-slit experiment – 10 electrons

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World of Molecules: Dual nature of electrons

Electrons - duble-slit experiment – 200 electrons

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http://upload.wikimedia.org/wikipedia/commons/7/7e/Double-slit_experiment_results_Tanamura_2.jpg

World of Molecules: Dual nature of electrons

Electrons - duble-slit experiment – 6000 electrons

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World of Molecules: Dual nature of electrons

Electrons - duble-slit experiment – 40 000 electrons

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http://upload.wikimedia.org/wikipedia/commons/7/7e/Double-slit_experiment_results_Tanamura_2.jpg

World of Molecules: Dual nature of electrons

Electrons - duble-slit experiment – 140 000 electrons

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World of Molecules: Dual nature of electrons

World of Molecules: Dual nature of electrons

photon

electron

Double-slit experiment

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http://en.wikipedia.org/wiki/File:Single_slit_and_double_slit2.jpg |

http://upload.wikimedia.org/wikipedia/commons/7/7e/Double-slit_experiment_results_Tanamura_2.jpg

Wave nature of electrons

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World of Molecules: Dual nature of electrons

• The diffraction pattern is apparent at approx.

40 000 electrons

• Due to detection methods (only particle like collision can be detected) the exact position of each electron on the screen cannot be

predicted

• if we repeat the measurement it will not look

the same, however the diffraction pattern will

similarly be apparent

Wave nature of electrons

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World of Molecules: Dual nature of electrons

• only the probability of the electron’s appearance on the sceen can be predicted

• the exact location cannot be predicted

• electron does behave like a wave this is clearly indicated on the diffraction patterns

• a single electron’s wave nature cannot be visualized with such an experiment, due to probability functions only a large number of

electrons can show us their wave like properties

Wave-particle duality - concept

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World of Molecules: Dual nature of electrons

Waves

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World of Molecules: Dual nature of electrons

• examples

• Electromagnetic wave

• electron wave

• material wave (e.g. proton, neutron)

• descriptors

– angular frequency ω

• the rate of change of angular displacement

– wave number vector

• the number of wavelengths per unit circumference k

( ) ^{ω , k}

World of Molecules: Dual nature of electrons

where

is the Planck constant is the reduced Planck constant

is the normal vector of a plane

is the wavelength

Formulas for waves

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= W

ω h

ν = W

λ h p

=

n p

k = =

λ π 2

h

π

2

= h

n

λ

Particles

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World of Molecules: Dual nature of electrons

• examples:

• photons

• electrons

• protons

• descriptors

– energy W

• kinetic and potential energy of the particle

– momentum vector

• product of the mass and the velocity vector of the particle

p

( ^{W , p} )

World of Molecules: Dual nature of electrons

where

m is the mass of the particle

v is the velocity vector ω is the angular frequency υ is the frequency of the

wave

Formulas for particles

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ν ω h W = =

k p = ⋅

n v

p = = ⋅

λ

m h

π

2

= h

World of Molecules: Dual nature of electrons

• Erwin Schrödinger found the mathematical model which satisfies the Bohr postulates and results in the de Broglie wave theory.

1. The wave equation has to satisfy the Planck and de Broglie postulates

2. The energy of a particle is made up of its kinetic and potential energy

3. The wave equation has to be linear (superposition principle) to return the interference results correctly

The Schrödinger model (1926)

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, h

h W

p =

= ν

λ

pot 2

2 W

m W = p +

( ) ^{x t ,} ( ) ( ) ^{x t}

ψ = Ψ ⋅ Ψ

1D, time dependent

Erwin Schrödinger

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World of Molecules: Dual nature of electrons

World of Molecules: Dual nature of electrons

• The wave function (Ψ) has to satisfy the time dependent Schrödinger equation

• the range values of the wave function ar complex numbers

• it can always be written in this form (1D, time dep.)

The Schrödinger model (1926)

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( ) ( )

( )

, - j

W t

t

x t x e

ψ

Ψ

= Ψ ⋅

( ) ^W ( ) ( ) ^{x x t x t}

m x t

x ⎟ ⎟ ∀ ∀

⎠

⎞

⎜ ⎜

⎝

⎛ − +

∂ =

∂ , ,

d d , 2

j t _{2 pot}

2

2 ψ

ψ

World of Molecules: Dual nature of electrons

• Ψ(x) satisfies the time independent Schrödinger equation

• where

is the HAMILTONIAN OPERATOR, which includes the kinetic and potential energies of the particle

The Schrödinger model (1926)

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( ) ^{x W Ψ} ( ) ^x

Ψ = ⋅

⋅ H

( ) ^x

x W

m ^{2 pot}

2 2

d d

2 +

−

=

H

World of Molecules: Dual nature of electrons

• are harmonic functions in time

• the function describing the waves should be

changing harmonically in time at every point in space

• the range of the values of the

wave function is complex number

• the function should be continuous, bounded, and derivable twice

Wave functions

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) 2

(

)] j

2 sin(

j )

2

[cos( t t Ae ^t

A πν ⋅ + πν ⋅ = ^{πν ⋅}

1

j = −

World of Molecules: Dual nature of electrons

• the time dependent part of the wave function can be expressed with the

use of the energy of the particle

Wave functions of the electron in a 1D potential well

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→

← a

= 0

x x = a

( ) ^{r , t}

ψ

→

= h

ν W W t

t e

e ^{j ( 2 πν ⋅ )} = ^{j ⋅}

( ) ^{x , t = Ψ} ( ) ^{x ⋅ e - j W t}

ψ

...

2 2 2

1 1 1

1

1 2 8

v 2

2 ma

h m

W p a

m h p

a → = = → = =

λ =

2 2 2

2 2 2

2

2 4 8

2

v ma

h m

W p a

m h p

a → = = → = =

λ =

2 2 2

3 3 3

3

3 9 8

2 2

v 3 3

2

ma h m

W p a

m h p

a → = = → = =

λ =

a

x = 0 x

W

a

x = 0 x

W

W

1

W

2

World of Molecules: Dual nature of electrons

Wave functions of the electron in a 1D potential well

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• The solution of the Schrödinger equation results the W _i energies and the respective Ψ _i wave functions

( ) ^{x = W ⋅ Ψ} ( ) ^x

Ψ

⋅ H

( ) ^x

x W m

where

2 pot 2 2

d d

2 +

−

= H

( ) ( ) ^{, ,} ₂ ^{, ,..., ...,} ( ) ^{, , ... ...}

1

2 1

x x

x

W W

W

n n

Ψ Ψ

Ψ

( ) ^x

Ψ 1

( ) ^x

Ψ 2

a

x = 0 x

W pot

W 1

W 2

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World of Molecules: Dual nature of electrons

Wave functions of the electron in a 1D potential well

( ) ^{x t} ₁ ( ) ^{x e - j W 1 t}

1 , = Ψ ⋅

ψ

( ) ^{x t} ₂ ( ) ^{x e - j W 2 t}

2 , = Ψ ⋅

ψ

( ) _n ( ) ^{W t}

n

n

e x

t

x , = Ψ ⋅ ^{- j} ψ

...

...

( ) ^x

Ψ 1

( ) ^x

Ψ 2

a

x = 0 x

W pot

W 1

W 2

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World of Molecules: Dual nature of electrons

Wave functions of the electron in a 1D potential well

• The full, time

dependent solutions

• The energies can be written / summarized

• here the wave number is

• Solving the equation and applying time dependency

• The general form of the solution

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World of Molecules: Dual nature of electrons

Wave functions of the electron in a 1D potential well

∑ ( )

⁻

=

n

W t n

n

e

t )

^{j2 h}

,

( ψ ^π

ψ r r

where n = 1, 2, 3, ...

n a k n ₌ π

2

2 2

2 1

8

W h n W n

n = ma = ⋅

( ) ^{x t =}

n ,

ψ ^t

W n a

x

A n j

e

sin π −

World of Molecules: Dual nature of electrons

• Where will we find the electron?

• We cannot tell exactly

• Only probability distribution of the electron’s position can be calculated

• The value of ψψ *dV will result the probability of the electron being in a volume dV around an

arbitrarily chosen (x,y,z) point.

• ψ * is the complex conjugate pair of ψ .

Wave functions of the electron in a 1D potential well

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∫ ⁼

V

V 1

* d ψψ

2

.

*

ψ

ψψ =

World of Molecules: Dual nature of electrons

Wave functions of the electron in a harmonic oscillator

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→

←

x 0

x = _{C x}

x F W

x C

W

_pot ^pot

= − ⋅

Δ

= Δ

→

⋅

= 2

1

₂

The potential energy of this system from which the force can be expressed

( ) ^x

W _pot

( ) ^x

F x

W 1

W 2

W 3

( ) ^{C x} ( ) ^{x W} ( ) ^x

x x

m ψ _{+ ⋅ ⋅} ψ ₌ ψ

− ^{2 2} ₂ ²

2 1 d

d 2

The Schrödinger equation takes the following form

( ) ^{x = W ψ} ( ) ^x

ψ

H

World of Molecules: Dual nature of electrons

The solution of the Schrödinger equation

• the energy levels

• constants

• wave functions

Wave functions of the electron in a harmonic oscillator

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0 0

1 0

0

2

..., 1 2 ,

, 3 2

1 ω ω _⎟ ω

⎠

⎜ ⎞

⎝ ⎛ +

=

=

= W W n

W

_n

0

m ,

= C

ω _{α =} ^Cm _,

₀

,

π

= α A

( ) ₀ ^{( )}

²

0 x = A e ^{− α ⋅} x

ψ

₁

( )

⁰

( ) ^{( )}

²

2

2 2

x

e A x

x

− ⋅

⋅

⋅

=

α

α ψ

( ) 0 ( ( ) ² ) ^{( ) 2}

2

2

1 2

x

e x

A x

− ⋅

⋅

−

⋅

=

α

α

ψ

World of Molecules: Dual nature of electrons

If we consider 3 dimensions, the solution of the wave equation is assumed in the following form:

Thus the Hamiltonian and the wave equation takes the following form

The sum of these three separate functions is constant which is only possible if each of the left-hand side expressions are constant by themselves.

Thus the energy can be divided into such three constants:

Wave functions of the electron in a 3D box

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( ^{x , y , z} ) ^ψ

₁

( ) ( ) ( ) ^{x ψ}

₂

^{y ψ}

₃

^z

ψ =

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ² ( ) ^{2 ,}

d d 1 2

d d 1 2

d d 1

p3 2 2 2

3 2 3

2 p2 2

2 2 2

2 p1 2

1 2 1

m W z

m W z

z y z

m W y

y x y

m W x

x

x ⎥ = −

⎦

⎢ ⎤

⎣

⎡ −

⎥ +

⎦

⎢ ⎤

⎣

⎡ −

⎥ +

⎦

⎢ ⎤

⎣

⎡ − ψ

ψ ψ

ψ ψ

ψ

3 2

1

W W

W

W = + +

World of Molecules: Dual nature of electrons

Wave functions of the electron in a 3D box

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( )

⎩ ⎨

⎧

∞

<

<

= <

.

,

<

0 ,

<

0 ,

<

0 if , , 0

pot

,

otherwise

c z b

y a

z x y x W

x

y z

a

b c

( ) ( ) ( ) ( )

c n z b

n y a

n x A

z y

x z

y

x ψ ψ ψ π π π

ψ , , =

_{1 2 3}

= sin

₁

⋅ sin

₂

⋅ sin

₃

A abc z

y c x

n z b

n y a

n x A

a b c

8 1

= d d d sin

sin sin

0 0 0

3 2 2

2 1

2

2

∫ ∫ ∫ ^π ⋅ ^π ⋅ ^π ⇒ =

( ) ^t

n n W n c

n z b

n y a

n x t abc

z y n x

n n

3 2 j 1

e sin

sin 8 sin

; , , 3 2

1

^{1 2 3}

⋅ −

⋅

⋅

= π π π

ψ

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ + +

=

^{2 1}₂^{2 2}₂^{2 3}₂²

8

h 3 2

1 c

n b

n a

n n m

n W n

The potential energy of this system

The form of the solution based on the previous assumption

The constant can be calculated from the assumption that the electron is inside the box

The wave equation solution

The energy levels

World of Molecules: Dual nature of electrons

The possible energy levels of the electron in the box

If the box has symmetry (namely b=a) the possible energy levels will be degenerate:

Degenerate stationary state means that a given energy level has two or more different wave functions; here and has the same energy.

In case of dual symmetry the level of degeneration increases.

Wave functions of the electron in a 3D box

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⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ + +

= +

+

=

_{1 2 3} ^{2 1}₂^{2 2}₂^{2 3}₂²

8

h 3

2

1 c

n b

n a

n W m

W n W

n W n

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ + +

=

^{2 12} ₂ ^{22 3}₂²

3 8

2

1 c

n a

n n

m h n

n W n

3 2 1 n n ψ n

3

1

2 n n

ψ n

World of Molecules: Dual nature of electrons

Dual symmetry results in the following energy expression

Different wave functions represent the different stationary states of the electron, i.e. in a degenerate state the same energy level can house many electrons in different stationary states.

The (n ₁ , n ₂ , n ₃ ) numbers will determine the energy level in a box, thus the lowest energy level will be W ₁ , with (1,1,1) and will have one stationary state.

Wave functions of the electron in a 3D box

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(

2 3²

)

2 2

2 1 2

8 h 3

2

1 n n n

n ma n

W n = + +

a n z a

n y a

n x n a

n

n π π π

ψ 8

₃

sin

₁

sin

₂

sin

₃

3

2

1 = ⋅ ⋅

a z a

y a

x

a π π π

ψ 8 sin sin sin

1 3 , 1 ,

1 = ⋅ ⋅ ⋅

2 2 1

, 1 ,

1 8

3h

W = ma

World of Molecules: Dual nature of electrons

The next energy level W ₂ , (W ₂ =2W ₁ ), with three possible stationary states (2,1,1), (1,2,1) and (1,1,2).

The following energy levels W ₃ =3W ₁ and W ₄ =11/3 W ₁ has also three separate stationary states.

W ₅ =4W ₁ has only one: (2,2,2);

W ₆ =14/3 W ₁ has six different stationary states:

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

Wave functions of the electron in a 3D box

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a z a

y a

x

a π π π

ψ 8 sin 2 sin sin

1 3 , 1 ,

2

= ⋅ ⋅ ⋅

2 2 2

, 1 , 1 1

, 2 , 1 1

, 1 ,

2

8

6h W ma

W

W = = =

a z a

y a

x

a π π π

ψ 8 sin sin 2 sin

1 3 , 2 ,

1

= ⋅ ⋅ ⋅

a z a

y a

x

a π π π

ψ 8 sin sin sin 2

2 3 , 1 ,

1

= ⋅ ⋅ ⋅

World of Molecules: Dual nature of electrons

1D arrangement required 1 quantum number (n)

3D arrangement requires 3 quantum numbers (n ₁ , n ₂ , n ₃ ) Considering the experimental results that electrons have an

intrinsic angular momentum called spin (can be +1/2 or –1/2).

When defining a stationary state for an electron the spin should also be taken into account (this is the spin quantum number) as a separate part of the wave function:

These results double the possible stationary states an electron can have at a given energy level (see the energy diagram).

Wave functions of the electron in a 3D box

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( ) ⎟

⎠

⎜ ⎞

⎝ ⎛+

+

= 2

, 1 ,

3 2 2 1

1

ψ n n n x y z ψ

s

ψ ( ) ⎟

⎠

⎜ ⎞

⎝ ⎛−

−

= 2

, 1 ,

3 2 2 1

1

ψ n n n x y z ψ

s

ψ

World of Molecules: Dual nature of electrons

Wave functions of the electron in a 3D box

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W

W

1

W

2

W

3

W

4

W

5

^W

⁶

ψ ψ W

2 / 1 211 2

/ 1 211 2

/ 1 121 2

/ 1 121 2

/ 1 112 2

/ 1 112 1

2

= 2 W ψ

₊

ψ

₋

ψ

₊

ψ

₋

ψ

₊

ψ

₋

W

2 / 1 221 2

/ 1 221 2

/ 1 212 2

/ 1 212 2

/ 1 122 2

/ 1 122 1

3

= 3 W ψ

₊

ψ

₋

ψ

₊

ψ

₋

ψ

₊

ψ

₋

W

2 / 1 331 2

/ 1 331 2

/ 1 313 2

/ 1 313 2

/ 1 133 2

/ 1 133 1

4

3 11

− +

− +

−

= W ψ

+

ψ ψ ψ ψ ψ

W

2 / 1 222 2

/ 1 222 1

5

= 4 W ψ

₊

ψ

₋

W

⎩ ⎨

= ⎧

− +

− +

− +

− +

− +

− +

2 / 1 321 2

/ 1 321 2

/ 1 312 2

/ 1 312 2

/ 1 231 2

/ 1 231

2 / 1 213 2

/ 1 213 2

/ 1 132 2

/ 1 132 2

/ 1 123 2

/ 1 123 1

6

3 14

ψ ψ

ψ ψ

ψ ψ

ψ ψ

ψ ψ

ψ W ψ

W

The energy diagram of the box taking into account the spin quantum number as well

World of Molecules: Dual nature of electrons

from the potential field the Hamiltonian can be expressed, and the

„one electron problem” can be solved:

e.g. electron in a 1D well e.g. electron in a 3D box

Wave functions of any potential field

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( ) ^{r H}

_pot

( ) ^r

pot

W

W ⇒ = − m Δ + 2

2

→

= W ψ ψ

H

( ) ^{; ,} ( ) ^{; ... ,} ( ) ^{; ...}

, _{1 2 2}

1 ψ r W ψ r W _n ψ _n r

W ^ψ _n ( ) ^{r , t = ψ} _n ( ) ^{r e − j W}

ⁿ

^{t ; n = 1 , 2 ,...}

⇒

∂ =

∂ ψ

t

ψ H j

2 2 2

8 n

ma h W n =

( ) ^{n t}

W a

x n t a

n

x

e j 2 sin

, −

= π

ψ

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ + +

=

^{2 1}₂^{2 2}₂^{2 3}₂²

8

h 3 2

1 c

n b

n a

n n m

n W n

( ) ^t

n n W n c

n z b

n y a

n x t abc

z y n x n n

3 2 j 1

e sin

sin 8 sin

; , , 3 2

1

^{1 2 3}

⋅ −

⋅

⋅

= π π π

ψ

World of Molecules: Dual nature of electrons

• the nucleus is positively charged and positioned in the middle of a sphere, the electrons are „orbiting” in this system, thus

• the potential energy of the electron has a spherical symmetry

• the Schrödinger equation can be written in the following form

Hydrogen atom and hydrogen like atoms

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r U Ze

q

W

_{pot e}

1

4

₀

2

⋅

−

=

= πε

4 0 2

0 2

2 ⎟⎟ =

⎠

⎜⎜ ⎞

⎝

⎛ +

+

Δ ψ

ψ πε

r W Ze

m _e

World of Molecules: Dual nature of electrons

• we can utilize that the system has spherical symmetry

• conversion to spherical coordinates

Hydrogen atom and hydrogen like atoms

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ϕ ϑ , , r

2 2 2

2 2

2

z y

x ∂

+ ∂

∂ + ∂

∂

= ∂

Δ ψ ψ ψ ψ 1 ,

2 , ,

,

ϑ ϕ ≡ Δ + Δ ϑ ϕ

Δ

≡

^{r r}

r

2 2 , 2

2 2

sin sin 1

sin 1 1 ,

ϕ

∂

∂ ϑ ϑ

∂ ϑ ∂ ϑ

∂

∂ ϑ

∂

∂

∂

∂

ϕ

ϑ

⎟⎟ +

⎠

⎜⎜ ⎞

⎝

≡ ⎛ Δ

⎟⎟ ⎠

⎜⎜ ⎞

⎝

≡ ⎛

Δ r r

r r

r

^ψ ( ^{r , ϑ , ϕ} ) ( ) ( ) ( ) ^{= R r ⋅ Φ ϕ ⋅ Θ ϑ}

( ) ψ ( ϑ ϕ )

ψ x , y , z → r , ,

World of Molecules: Dual nature of electrons

• the solution of the Schrödinger equation results in the discrete energy levels of electron orbitals, and the

wave functions of these orbitals

• the energy levels

• the lowest energy level (at n=1) and the

corresponding orbital radius from the Bohr model Hydrogen atom and hydrogen like atoms

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ψ ψ = W H

h ,

8 ₀ ^{2 2 2}

4 2

n e W _n mZ

− ε

=

m 0

1 528 , h 0

eV, 6

, h 13

8

e

₁₀

2 0 2 2 1

2

4

= = = ⋅

−

⋅

e m π r ε

ε

m

_H

World of Molecules: Dual nature of electrons

• The wave functions of electrons in a general form

• where are coupled Legendre polynomials

• and are Laguerre polynomials

Hydrogen atom and hydrogen like atoms

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( ϑ ϕ ) ( ϑ ) ^ϕ

ψ P ^m

nr L r

nr nr r

r A

r

_n ^m

m n

e j 2 cos

e 2 ,

,

1 1

2 1

1

⎟⎟ ⋅ ⋅

⎠

⎜⎜ ⎞

⎝

⋅ ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⋅ ⎛

−

=

₊⁺

( ) ( )

P x

₀

1 P x

₁

x P x

₂

1 x

²

P x

₃

x

³

x

2 3 1 1

2 5 3

( ) = , ( ) = , ( ) = − , ( ) = −

P m

( )

P x x d P x

l

dx

m

m m

l

( )

m

( )

= − 1

^{2 2}

1 2 +

+

L n

L x ( )

k

i

k x L x d

dx L x

i

k

k

i k

i

p p

p i

( ) = − ! ⎛ , ( ) ( )

⎝ ⎜ ⎞

⎠ ⎟ =

∑

=

¹

0

World of Molecules: Dual nature of electrons

• n, l, m are the three quantum numbers

• introducing the spin quantum number, the wave functions of electrons can be written in the following forms

• i.e. the spin quantum number is treated as a separate wave function

• at W _n energy there are 2n ² different quantum states

Hydrogen atom and hydrogen like atoms

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( ϑ ϕ ) ( ϑ ) ^ϕ

ψ P ^m

nr L r

nr nr r

r A

r

_n ^m

m n

e j 2 cos

e 2 ,

,

1 1

2 1

1

⎟⎟ ⋅ ⋅

⎠

⎜⎜ ⎞

⎝

⋅ ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⋅ ⎛

−

=

₊⁺

( ) ⎟

⎠

⎜ ⎞

⎝

⎛ +

+

= 2

, 1 ,

12 n m s

m

n

ψ r ϑ ϕ ψ

ψ ( ) ⎟

⎠

⎜ ⎞

⎝

⎛ −

−

= 2

, 1

2

,

1 n m s

m

n

ψ r ϑ ϕ ψ

ψ

World of Molecules: Dual nature of electrons

Bohr – Sommerfeld model - 2n ² rule

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value name of shell number of electrons in shell Principal quantum

number (n) 1 K 2

2 L 2+6=8

3 M 2+6+10=18

4 N 2+6+10+14=32

5 O 2+6+10+14+18=50

6 P 2+6+10+14+18+22=72

7 Q 2+6+10+14+18+22+26=98

World of Molecules: Dual nature of electrons

• 1s orbital

• n=1, l=0, m=0

• 2s orbital

• n=2, l=0, m=0

• 2p orbital

• n=2, l=1, m=-1

• 2p orbital

• n=2, l=1, m=0

• 2p orbital

• n=2, l=1, m=1

• ...

Hydrogen atom and hydrogen like atoms

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1 3

1 100

e 2

1 r

r r

−

= π ψ

1 1

2 3

1 200

e 2 1 2

4

1 r

r r

r r

−

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ −

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

− 2

= π

ψ

ϕ π ϑ

ψ 1 e 2 sin sin

8

1

₁

1 2 3

1 1

, 1 ,

2

⋅

−

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

−

r

r r

r r

π ϑ

ψ 1 e 2 cos

4

1

₁

1 2 3

1 0

, 1 , 2

r r r

r r

−

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

ϕ π ϑ

ψ 1 e 2 sin cos 8

1

^{3 2} ₁

1 , 1 ,

2

⋅

−

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛ r

r r

r

r

World of Molecules: Dual nature of electrons

1s orbital 2s orbital

Hydrogen atom and hydrogen like atoms

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http://commons.wikimedia.org/wiki/File:S1M0.png | http://commons.wikimedia.org/wiki/File:S2M0.png

World of Molecules: Dual nature of electrons

2p _x orbital 2p _y orbital

Hydrogen atom and hydrogen like atoms

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World of Molecules: Dual nature of electrons

2p _z orbital 3d _xy orbital

Hydrogen atom and hydrogen like atoms

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http://commons.wikimedia.org/wiki/File:P2M1.png | http://commons.wikimedia.org/wiki/File:D3M1.png

World of Molecules: Dual nature of electrons

• the electron’s behavior is described by a wave function

• the wave function’s value itself does not have a physical meaning (has complex value)

• The product ψ ψ* gives the probability of finding the electron at a specific volume

• the position information (as a probability) can be calculated from the wave function

• The solution of the Schrödinger equation results in the discrete energy levels of the given system

• These are the energies the electrons can occupy

• The four quantum numbers uniquely define an electron

Summary

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World of Molecules: Dual nature of electrons

• The following equations do not have solutions in real numbers.

• in mathematics there are numbers beyond the real numbers, these are called complex numbers and these equations have solutions in complex numbers.

• Complex numbers are represented on the complex plane, as vectors having a real and an imaginary part.

Short introduction to complex numbers

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2

1,2

4 13 0

4 16 4 13 4 36

2 2

x x

x

− + =

± − ⋅ ± −

= =

2

1,2

1 0 1 x

x

+ =

= ± −

( )

( ) ( ^{( )} )

1,2

2 3

2 3 2 3

x j

x j x j

= ±

− + ⋅ − − ( ) ( )

x

1,2

j

x j x j

= ±

− ⋅ +

World of Molecules: Dual nature of electrons

Short introduction to complex numbers

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a

Real axis

b a

z = + j ⋅

complex number

r ϕ

⋅ j b

Imaginary axis point on a

complex plane

( ^{cos ϕ j sin ϕ} ) ^{j ϕ}

j r r e

b a

z = + ⋅ = + = ⋅

algebraic form polar form exponential form

World of Molecules: Dual nature of electrons

Short introduction to complex numbers

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absolute value (modulus) 2

2 b

a

r = + = arg( ); tg = , if z ≠ 0 a

z ϕ b

ϕ

argument

Real part

( ) ^{cos ϕ}

Re z = a = r ⋅

Imaginary part

( ) ^{j j sin ϕ}

Im z = ⋅ b = ⋅ r

complex conjugate numbers

( ^{cos ϕ j sin ϕ} ) ^{j ϕ}

j r r e

b a

z = + ⋅ = + = ⋅

( ^{cos ϕ j sin ϕ} ) ^{- j ϕ}

j r r e

b a

z ^∗ = − ⋅ = − = ⋅

b a

z = + j ⋅

b a

z ^∗ = − j ⋅

World of Molecules: Dual nature of electrons

• Mathematical operations with complex numbers Short introduction to complex numbers

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( ^{cos ϕ j sin ϕ} ) ^{j ϕ}

j r r e

b a

z = + ⋅ = + = ⋅

( _{1 2} ) ^j

2 1

2

1 ± z = a ± a + b ± b ⋅

z

( _{1 2} )

2 j 1

2

1 ⋅ z = r ⋅ r ⋅ e ϕ + ϕ

z ₁ ^{: z} ₂ ⁼ ( ^r ₁ ^{: r} ₂ ) ^{⋅ e j ( ϕ 1 − ϕ 2 )}

z

ϕ n n

n r e

z = ⋅ ^j

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World of Molecules: Dual nature of electrons

Next - Properties of chemical bonds, spectroscopy 1. Spectroscopy

2. Absorption spectroscopy 3. Emission spectroscopy

4. Chemical properties of atoms 5. Types of chemical bondings

6. Basic properties of chemical bonds 7. Covalent, ionic and metallic bonds 8. Hydrogen bonds

9. van der Waals forces