physical chemistry - advanced materials particles and waves standing waves wave function...

PHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALSPHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and WavesParticles and WavesParticles and WavesParticles and Waves

Standing WavesStanding WavesStanding WavesStanding Waves Wave FunctionWave FunctionWave FunctionWave Function

Differential Wave EquationDifferential Wave EquationDifferential Wave EquationDifferential Wave Equation

Something more about….Something more about….Something more about….Something more about….

X=0 X=L

Standing WavesStanding Waves

Boundary Conditions:Boundary Conditions:

Standing WavesStanding Waves

Boundary Conditions:Boundary Conditions:

0),(),0( tLxtx 0),(),0( tLxtx

-1,0

-0,5

0,0

0,5

1,0

Y A

xis

Title

X Axis Title

sin(x/L) sin(2x/L) sin(3x/L)

0 L

)()(),( tTxXtx )()(),( tTxXtx Separation of variables:Separation of variables:Separation of variables:Separation of variables:

Wave Function:Wave Function:Wave Function:Wave Function:

2

2

2

2

2

2

2

2

)()(

)()(

t

tTxX

t

x

xXtT

x

2

2

2

2

2

2

2

2

)()(

)()(

t

tTxX

t

x

xXtT

x


2

2

22

2

v

1

tx

2

2

22

2

v

1

tx

constant)(

)(v

1)(

)(

12

2

22

2

t

tT

tTx

xX

xXconstant

)(

)(v

1)(

)(

12

2

22

2

t

tT

tTx

xX

xX

constantv

)(

)(v

1)(

)(

12

2

2

2

22

2

dt

tTd

tTdx

xXd

xXconstant

v

)(

)(v

1)(

)(

12

2

2

2

22

2

dt

tTd

tTdx

xXd

xX

Equivalent to two ordinary (not partial) differential equations:

)()(

)(v

)(

22

2

2

2

2

2

tTdt

tTd

xXdx

xXd

)()(

)(v

)(

22

2

2

2

2

2

tTdt

tTd

xXdx

xXd

Space: f(x)Space: f(x) TIme: f(t)TIme: f(t)

)cos()sin(sin)()(),( tBtAL

xntTxXtx nnnn


xntTxXtx nnnn

Space: X(x)Space: X(x) Time: T(t)Time: T(t)


EigenEigenvaluevalue Condition:Condition:EigenEigenvaluevalue Condition:Condition: Ln 2

Ln

2

EigenEigenfunctions:functions:EigenEigenfunctions:functions: )cos()sin(sin)()(),( tBtAL

xntTxXtx nnnnn


xntTxXtx nnnnn

n=0, ±1, ±2, ±3……n=0, ±1, ±2, ±3……

General solution: General solution: Principle of superpositionPrinciple of superpositionGeneral solution: General solution: Principle of superpositionPrinciple of superpositionSince any linear Combination of the Eigenfunctions would also be a solution

Since any linear Combination of the Eigenfunctions would also be a solution

00

)cos()sin(sin),(n

nnnnn

n tBtAL

xntx

00

)cos()sin(sin),(n

nnnnn

n tBtAL

xntx

Fourier SeriesFourier SeriesFourier SeriesFourier Series


Fourier SeriesFourier SeriesFourier SeriesFourier Series

Any arbitrary function f(x) of period can be expressed as a Fourier Series

Any arbitrary function f(x) of period can be expressed as a Fourier Series

0

)2

cos()2

sin()()(n

nn

nxB

nxAxfxf

0

)2

cos()2

sin()()(n

nn

nxB

nxAxfxf

Y A

xis

Titl

e

X Axis Title

n

n

nxiCxfxf

2exp)()(

n

n

nxiCxfxf

2exp)()(

REALREAL

Fourier SeriesFourier Series

REALREAL


COMPLEXCOMPLEX


COMPLEXCOMPLEX



Wave PhenomenaWave PhenomenaWave PhenomenaWave Phenomena

ir

ReflexionReflexionReflexionReflexion RefractionRefractionRefractionRefraction

i

t

n1

n2

n1 sin (i) = n2 sin (t)

InterferenceInterferenceDiffractionDiffractionInterferenceInterferenceDiffractionDiffraction

Diffraction is the bending of a wave around an obstacle or through an opening.

Diffraction is the bending of a wave around an obstacle or through an opening.

p=w sinw

p=d sin

d

bright fringes bright fringes

mm

bright fringes bright fringes

mm

The path difference must be a multiple of a wavelength to insure constructive interference.

The path difference must be a multiple of a wavelength to insure constructive interference.

Wavelenght dependence


Interference and Diffraction: Interference and Diffraction: Huygens constructionHuygens constructionInterference and Diffraction: Interference and Diffraction: Huygens constructionHuygens constructionIntensity pattern that shows the combined effects of both diffraction and interference when light passes through multiple slits.

Intensity pattern that shows the combined effects of both diffraction and interference when light passes through multiple slits.

m=0

m=2

m=1


Black-Body RadiationBlack-Body RadiationBlack-Body RadiationBlack-Body Radiation

kTU

dfUfc

VdffE

f

f

23

4)(

kTU

dfUfc

VdffE

f

f

23

4)(

A blackbody is a hypothetical object that absorbs all incident electromagnetic radiation while maintaining thermal equilibrium.

A blackbody is a hypothetical object that absorbs all incident electromagnetic radiation while maintaining thermal equilibrium.


2

2

22

L

d

dn

Ln

n

L

2

2

22

L

d

dn

Ln

n

L

1D1D1D1D4

4

V

d

dn 4

4

V

d

dn

3D3D3D3D

2

v ;

v

fdf

d

f

2

v ;

v

fdf

d

f

23v

4f

V

df

dn

df

d

d

dn

23v

4f

V

df

dn

df

d

d

dn

Since there are many more permissible high frequencies than low frequencies, and since by Statistical Thermodynamics all frequencies have the same average Energy, it follows that the Intensity I of balck-body radiation should rise continuously with increasing frequency.

Since there are many more permissible high frequencies than low frequencies, and since by Statistical Thermodynamics all frequencies have the same average Energy, it follows that the Intensity I of balck-body radiation should rise continuously with increasing frequency.

Breakdown of classical mechanical principles when applied to radiation

!!!Ultraviolet Catastrophe!!!!!!Ultraviolet Catastrophe!!!

Black-Body Radiation: Black-Body Radiation: classical theoryclassical theoryBlack-Body Radiation: Black-Body Radiation: classical theoryclassical theory Radiation as Electromagnetic WavesRadiation as Electromagnetic Waves


The Quantum of Energy – The Planck Distribution LawThe Quantum of Energy – The Planck Distribution Law

Physics is a closed subject in which new discoveries of any

importance could scarcely expected….

However… He changed the World of Physics…

Classical Mechanics

Matter Discrete

Energy Continuous

Nature does not make a Jump

Planck: Quanta

E = hE = h

Energy Continuous

hx 10-34 Joule.sechx 10-34 Joule.sec

An oscillator could adquire Energy only in discrete units called QuantaAn oscillator could adquire Energy only in discrete units called Quanta

!Nomenclature change!: →f!Nomenclature change!: →f

dkTc

dEkTh

de

d

c

hdE

kTh

8

)(,

1

8)(

3

2

/3

3

dkTc

dEkTh

de

d

c

hdE

kTh

8

)(,

1

8)(

3

2

/3

3

Max Planck


Photoelectric Effect: EinsteinPhotoelectric Effect: EinsteinPhotoelectric Effect: EinsteinPhotoelectric Effect: Einstein

Metal

• Below a certain „cutoff“ frequency of incident light, no photoelectrons are ejected, no matter how intense the light.

• Below a certain „cutoff“ frequency of incident light, no photoelectrons are ejected, no matter how intense the light.

• Above the „cutoff“ frequency the number of photoelectrons is directly proportional to the intensity of the light.• Above the „cutoff“ frequency the number of photoelectrons is directly proportional to the intensity of the light.

• As the frequency of the incident light is increased, the maximum velocity of the photoelectrons increases.• As the frequency of the incident light is increased, the maximum velocity of the photoelectrons increases.

• In cases where the radiation intensity is extremely low (but photoelectrons are emited from the metal without any time lag.

• In cases where the radiation intensity is extremely low (but photoelectrons are emited from the metal without any time lag.

The radiation itself is

quantized

Fluxe

Fluxe

1 2

1

2


• Increasing the intensity of the light would correspond to increasing the number of photons.• Increasing the intensity of the light would correspond to increasing the number of photons.

Energy of light: E = hEnergy of light: E = h

Kinetic Energy = Energy of light – Energy needed to escape surface (Work Function):

½ mev2= hh

Kinetic Energy = Energy of light – Energy needed to escape surface (Work Function):

½ mev2= hh

PhotonPhotonPhotonPhoton

• Increasing the frequency of the light would correspond to increasing the Energy of photons and the maximal velocity of the electrons.• Increasing the frequency of the light would correspond to increasing the Energy of photons and the maximal velocity of the electrons.

: It depends on the Nature of the Metal : It depends on the Nature of the Metal


Light as a stream of Photons?

E = hdiscrete

Light as a stream of Photons?

E = hdiscrete

Light as Electromagnetic Waves?

E = |Eelec|2 = (1/|Bmag|2 continuous

Light as Electromagnetic Waves?

E = |Eelec|2 = (1/|Bmag|2 continuous

The square of the electromagnetic wave at some point can be taken as the

Probability DensityProbability Density for finding a Photon in the volume element around that point.

The square of the electromagnetic wave at some point can be taken as the

Probability DensityProbability Density for finding a Photon in the volume element around that point.

Energy having a definite and smoothly varying distribution. (Classical)Energy having a definite and smoothly varying distribution. (Classical)

A smoothly varying Probability Density for finding an atomistic packet of Energy. (Quantical)A smoothly varying Probability Density for finding an atomistic packet of Energy. (Quantical)Albert Einstein

Zero rest mass!!


The Wave Nature of MatterThe Wave Nature of MatterThe Wave Nature of MatterThe Wave Nature of Matter

De Broglie

All material particles are associated with Waves

(„Matter waves“

E = h

E = mc2

E = h

E = mc2

mc2 = h= hc

or: mc = h/

mc2 = h= hc

or: mc = h/

A normal particle with nonzero rest mass m travelling at velocity vA normal particle with nonzero rest mass m travelling at velocity v mv = p= hmv = p= h


Source

Source

Electron DiffractionElectron Diffraction

Experim

entalE

xperimental

Expected

Expected

Electron DiffractionElectron Diffraction

Amorphous Material Crystalline MaterialAmorphous Material Crystalline Material

Conclusion: Under certain circunstances an electron behaves also as a Wave!Conclusion: Under certain circunstances an electron behaves also as a Wave!

physical chemistry - advanced materials particles and waves standing waves wave function...

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