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

Two opposing concepts

Particle theory

Field theory

Quantum Mechanics

Position, mass, velocity, charge, spin

Functions continuous in space and time

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

X1 X2

t=0

t=0

tfinal

tfinal=(x2-x1)/vparticle

wave

Mechanical Analogy of transmision Mechanical Analogy of transmision

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

X2

t=0

t=0

tfinal

tfinal=(x2-x1)/v

X1

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

SImple Harmonic MotionSImple Harmonic Motion

m

)sin(2/1

2

2

tm

Ax

xdtxdm

dtdvmmaF

Equation of Motion Equation of Motion

Displacement Displacement x

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

Wave: Wave: The propagation of a perturbation. It is also a way to transport Energy.Periodic changes in Space and Time of some Property.There is no net dosplacement of the medium

Shape of the disturbance: Shape of the disturbance:

„Profile“ of the wave such as an instantaneous „snapshot“ of the wave form.It reoeats periodically in Time and Space.Particular „Pattern“: Sinusoidal Waves. These waves are called „Harminic Waves“.

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

One-Dimensional Waves – General ConceptsOne-Dimensional Waves – General Concepts

t o x

Shape of the disturbance Shape of the disturbance

Harmonic WavesHarmonic Waves→ → Sinusoidal WavesSinusoidal Waves

Time, t

T

Temporal Period Temporal Period ),(),( Ttxtx

Space, x

Spatial Period or WavelengthSpatial Period or Wavelength ),(),( txtx

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

Space, x

p

AmplitudeAmplitude

PhasePhase

time, t

Ref

FrequencyFrequency

Tf 1

Angular FrequencyAngular Frequency

T

f

22

Wave NumberWave Number

1

Angular Wave NumberAngular Wave Numberor propagation Numberor propagation Number

2k

fT

v

Phase VelocityPhase VelocityTemporal PeriodTemporal Period

v

T

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

)v(2sin

sin

22sinsin),(

txA

tkxA

tT

xAtkxAtx

Wave FunctionWave Function

Differential Wave EquationDifferential Wave Equation

2

2

22

2

v1

tx

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

Travelling WavesTravelling Waves

)v(2sin txA

)v(2sin txA

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

Travelling / Standing WavesTravelling / Standing Waves

)v(2sin1 txA

)v(2sin2 txA

txA

txtxA

v2cos2sin2

)v(2sin)v(2sin21

Standing Waves:Standing Waves: When the primary and reflected Waves have the same A and speed It vibrates „in place“.Nodes: Places where the medium does not ever vibrate.

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

The Complex RepresentationThe Complex Representation

Complex number:Complex number: 1 with iiyxz

Imaginary y

Real x

|z|

(x+i y)

x=|z| cos y=|z| sin

Euler Formula:Euler Formula: z = |z| (cos + isin |z| ei

Complex Conjugate:Complex Conjugate: z* = |z| (cos - isin |z| e-i

Wave Function:Wave Function: )cos()( tkxAeA tkxi

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

Light as an Electromagnetic WaveLight as an Electromagnetic Wave

+EE

Differential Wave EquationDifferential Wave Equation

2

2

22

2

2

2

22

2

c1c1

tB

xB

tE

xE