PHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and Waves Wave Phenomena i rReflexionReflexionRefractionRefraction ii t n 1 n 2 n 1 sin

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

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

p=w sinmmDiffraction at a latticeDiffraction at a lattice

Diffraction at SlitsDiffraction at Slits


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

Then, every particle with nonzero rest mass m travelling at velocity v has an related wave

Then, every particle with nonzero rest mass m travelling at velocity v has an related wave

= h/ mv = h/ mv

1. The particle property is caused by their mass. 2. The wave property is related with particles' electrical charges. 3. Particle-wave duality is the combination of classical mechanics and electromagnetic field

theory.

1. The particle property is caused by their mass. 2. The wave property is related with particles' electrical charges. 3. Particle-wave duality is the combination of classical mechanics and electromagnetic field

theory.

A central concept of Quantics:Quantics: wave–particle dualitywave–particle duality is the concept that all matter matter and energyenergy exhibits both wavewave -like and particleparticle -like properties.

A central concept of Quantics:Quantics: wave–particle dualitywave–particle duality is the concept that all matter matter and energyenergy exhibits both wavewave -like and particleparticle -like properties.


Schrödinger Miau!

Superposition of two States:Superposition of two States:Broadly stated, a quantum superposition is Broadly stated, a quantum superposition is the combination of all the possible states of a the combination of all the possible states of a systemsystem.

Superposition of two States:Superposition of two States:Broadly stated, a quantum superposition is Broadly stated, a quantum superposition is the combination of all the possible states of a the combination of all the possible states of a systemsystem.

Alife+ DeadAlife+ Dead

DeadDead

AlifeAlife

I don‘t like it and I regret that I got involved in it….

Schrödinger's catSchrödinger's catSchrödinger's catSchrödinger's catIt is a „Gedanken“ (thought experiment) often described as a paradoxIt is a „Gedanken“ (thought experiment) often described as a paradox


Schrödinger wrote:„One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a "blurred model" for representing reality. In itself it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks.“

Schrödinger wrote:„One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a "blurred model" for representing reality. In itself it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks.“


Schrödinger's famous tought experiment poses the question: when does a quantum system stop existing as a mixture of states and become one or the other? (More technically, when does the actual quantum state stop being a linear combination of states, each of which resemble different classical states, and instead begin to have a unique classical description?) If the cat survives, it remembers only being alive. But explanations of experiments that are consistent with standard microscopic quantum mechanics require that macroscopic objects, such as cats and notebooks, do not always have unique classical descriptions. The purpose of the thought experiment is to illustrate this apparent paradox: our intuition says that no observer can be in a mixture of states, yet it seems cats, for example, can be such a mixture. Are cats required to be observers, or does their existence in a single well-defined classical state require another external observer?

Schrödinger's famous tought experiment poses the question: when does a quantum system stop existing as a mixture of states and become one or the other? (More technically, when does the actual quantum state stop being a linear combination of states, each of which resemble different classical states, and instead begin to have a unique classical description?) If the cat survives, it remembers only being alive. But explanations of experiments that are consistent with standard microscopic quantum mechanics require that macroscopic objects, such as cats and notebooks, do not always have unique classical descriptions. The purpose of the thought experiment is to illustrate this apparent paradox: our intuition says that no observer can be in a mixture of states, yet it seems cats, for example, can be such a mixture. Are cats required to be observers, or does their existence in a single well-defined classical state require another external observer?

An interpretation of quantum mechanicsAn interpretation of quantum mechanics. A key feature of quantum mechanics is that the state of every particle is described by a wavefunction, which is a mathematical representation used to calculate the probability for it to be found in a location, or state of motion. In effect, the act of measurement causes the calculated set of probabilities to "collapse" to the value defined by the measurement. This feature of the mathematical representations is known as wave function collapse.

An interpretation of quantum mechanicsAn interpretation of quantum mechanics. A key feature of quantum mechanics is that the state of every particle is described by a wavefunction, which is a mathematical representation used to calculate the probability for it to be found in a location, or state of motion. In effect, the act of measurement causes the calculated set of probabilities to "collapse" to the value defined by the measurement. This feature of the mathematical representations is known as wave function collapse.


The Waves and the Incertainty Principle of HeisenbergerThe Waves and the Incertainty Principle of HeisenbergerThe Waves and the Incertainty Principle of HeisenbergerThe Waves and the Incertainty Principle of Heisenberger

„The measurement of particle position leads to loss of knowledge about

particle momentum and visceversa.“

mv y

y

x

p

2p sin = pypy2p sin = pypy

sin = ±ysin = ±y

py y ≈ 2p= 2h

py y ≈ 2p= 2h

The momentum of the incoming beam is all in the x direction. But as a result of diffraction at the slit, the diffracted beam has momentum p with components on both x and y directions.

The momentum of the incoming beam is all in the x direction. But as a result of diffraction at the slit, the diffracted beam has momentum p with components on both x and y directions.


Rydberg

The Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements.

The Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements.

Continuous spectrum emission lines absorptiom linesContinuous spectrum emission lines absorptiom lines

Rydberg formula for hydrogenRydberg formula for hydrogen

Whereλvac is the wavelength of the light emitted in vacuum,RH is the Rydberg constant for hydrogen,n1 and n2 are integers such that n1 < n2

Whereλvac is the wavelength of the light emitted in vacuum,RH is the Rydberg constant for hydrogen,n1 and n2 are integers such that n1 < n2


Bohr

He suggested that electrons could only have certain classical motions:

1.The electrons can only travel in special orbits: at a certain discrete set of distances from the nucleus with specific energies. 2.The electrons do not continuously lose energy as they travel. They can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency determined by the energy difference ΔE = E2 − E1 = h of the levels according to Bohr's formula where h is

Planck‘s Constant. 3.the frequency of the radiation emitted at an orbit with period T is as it would be in classical mechanics--- it is the reciprocal of the classical orbit period:

c/

He suggested that electrons could only have certain classical motions:

1.The electrons can only travel in special orbits: at a certain discrete set of distances from the nucleus with specific energies. 2.The electrons do not continuously lose energy as they travel. They can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency determined by the energy difference ΔE = E2 − E1 = h of the levels according to Bohr's formula where h is

Planck‘s Constant. 3.the frequency of the radiation emitted at an orbit with period T is as it would be in classical mechanics--- it is the reciprocal of the classical orbit period:

c/

Bohr‘s Atom ModelBohr‘s Atom ModelBohr‘s Atom ModelBohr‘s Atom Model


The angular momentum AA is restricted to be an integer multiple of a fixed unit: A= n h / 2mvr [1]where n = 1,2,3,… and is called the principal quantum number. The lowest value of n is 1. This gives a smallest possible orbital radius of 0.0529 nm. This is known as the BohrBohr radius. Once an electron is in this lowest orbit, it can get no closer to the proton. Starting from the angular momentum quantum rule Bohr Bohr was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogenlyke atoms and ions.Bohr's condition, that the angular momentum is an integer multiple of h/2 was later reinterpreted by de Broglie as a standing wave condition: n r = L [2]the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit.

The angular momentum AA is restricted to be an integer multiple of a fixed unit: A= n h / 2mvr [1]where n = 1,2,3,… and is called the principal quantum number. The lowest value of n is 1. This gives a smallest possible orbital radius of 0.0529 nm. This is known as the BohrBohr radius. Once an electron is in this lowest orbit, it can get no closer to the proton. Starting from the angular momentum quantum rule Bohr Bohr was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogenlyke atoms and ions.Bohr's condition, that the angular momentum is an integer multiple of h/2 was later reinterpreted by de Broglie as a standing wave condition: n r = L [2]the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit.

X=0 X=L

-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


To calculate the orbits requires two assumptions:1. Classical mechanicsClassical mechanics

The electron is held in a circular orbit by electrostatic attraction. The centripetal forcecentripetal force is equal to the Coulom forceCoulom force:

mv2/r = ke2 /r2 [3]where mm is the mass and ee is the charge of the electron. This determines the speed at any radius:

v=√(ke2 /mr) [4]It also determines the total energy at any radius: E=mv2/2 - ke2 /r = -ke2 /2r [5]The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton. For larger nuclei, the only change is that ke2 is everywhere replaced by Z ke2 where Z is the number of protons.

To calculate the orbits requires two assumptions:1. Classical mechanicsClassical mechanics

The electron is held in a circular orbit by electrostatic attraction. The centripetal forcecentripetal force is equal to the Coulom forceCoulom force:

mv2/r = ke2 /r2 [3]where mm is the mass and ee is the charge of the electron. This determines the speed at any radius:

v=√(ke2 /mr) [4]It also determines the total energy at any radius: E=mv2/2 - ke2 /r = -ke2 /2r [5]The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton. For larger nuclei, the only change is that ke2 is everywhere replaced by Z ke2 where Z is the number of protons.


2. . Quantum rule

Substituting the expression for the velocity [4] in the expression for the angular momentum [1] gives an equation for r in terms of n:

√(ke2mr) = nh/2so that the allowed orbit radius at any n is:

r = n2 h2 / (4 ke2 m) [7]The smallest possible value of r is 0.51 x 10-10m (n=1) is called the Bohr Bohr

radiusradius The energy of the n-th level is determined by the radius (replacing [7] in [5] :

E= (2ke2) 2 m/(2n2 h2 )The combination of natural constants in the energy formula is called the Rydberg

energy RH (n=1): RH = (2ke2) 2 m/(2h2 )

2. . Quantum rule

Substituting the expression for the velocity [4] in the expression for the angular momentum [1] gives an equation for r in terms of n:

√(ke2mr) = nh/2so that the allowed orbit radius at any n is:

r = n2 h2 / (4 ke2 m) [7]The smallest possible value of r is 0.51 x 10-10m (n=1) is called the Bohr Bohr

radiusradius The energy of the n-th level is determined by the radius (replacing [7] in [5] :

E= (2ke2) 2 m/(2n2 h2 )The combination of natural constants in the energy formula is called the Rydberg

energy RH (n=1): RH = (2ke2) 2 m/(2h2 )


electronselectronsPhotonsPhotons

UncertaintyUncertainty

WavesWavesWavesWaves

diffractiondiffraction

EnergyEnergy

EinsteinEinstein

BohrBohr

HeisenbergHeisenberg

PlanckPlanck

SchrödingerSchrödinger

de Brogliede Broglie




The book of nature you can only understand, if you havepreviously learnt its language and the letters. It is written inmathematical language and the letters are geometrical figures,and without these means it is impossible for human beings tounderstand even a word of it.

Galileo GalileiGalileo Galilei, 16th century

The book of nature you can only understand, if you havepreviously learnt its language and the letters. It is written inmathematical language and the letters are geometrical figures,and without these means it is impossible for human beings tounderstand even a word of it.

Galileo GalileiGalileo Galilei, 16th century


Dirac Notation, Hilbert SpaceDirac Notation, Hilbert SpaceDirac Notation, Hilbert SpaceDirac Notation, Hilbert Space

Dirac Hilbert

The mathematical concept of a Hilbert spaceHilbert space, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces.

The mathematical concept of a Hilbert spaceHilbert space, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces.

Bra-ket notationBra-ket notation is a standard notation for describing quantum states composed of angle brackets and vertical bars. It can also be used to denote abstract vectors. It is so called because the inner product of two states is denoted by a bracketbracket, , consisting of a left part, , called the brabra, and a right part, , called the ket. The notation was invented by Paul DiracPaul Dirac and is also known as Dirac notationDirac notation. .

Bra-ket notationBra-ket notation is a standard notation for describing quantum states composed of angle brackets and vertical bars. It can also be used to denote abstract vectors. It is so called because the inner product of two states is denoted by a bracketbracket, , consisting of a left part, , called the brabra, and a right part, , called the ket. The notation was invented by Paul DiracPaul Dirac and is also known as Dirac notationDirac notation. .

Mathematically, a pure quantum state is typically represented by a vectorvector in a Hilbert space.Hilbert space.Mathematically, a pure quantum state is typically represented by a vectorvector in a Hilbert space.Hilbert space.


An Operator transforms the original vector |u> in general by changing its amount (magnitude) and its direction into a new vectorAn Operator transforms the original vector |u> in general by changing its amount (magnitude) and its direction into a new vector

OperatorsOperatorsOperatorsOperators

Eigenvalue: The Operator transforms the original vector |u> in particular by changing its amount (magnitude) (elongation or compression) but mantaining its direction into a new vector

Eigenvalue: The Operator transforms the original vector |u> in particular by changing its amount (magnitude) (elongation or compression) but mantaining its direction into a new vector

Hermitian Operator: which fulfil the condition: (always real Eigenvalues)Hermitian Operator: which fulfil the condition: (always real Eigenvalues)

uOuuuO ˆˆ uOuuuO ˆˆ

If in addition an Hermitian Operator also satisfies the condition:

AdjugateAdjugate

If in addition an Hermitian Operator also satisfies the condition:

AdjugateAdjugate*ˆˆ OO *ˆˆ OO


Conclusions, Summary, Some of more important Conclusions, Summary, Some of more important PostulatesPostulatesConclusions, Summary, Some of more important Conclusions, Summary, Some of more important PostulatesPostulates

•Duality Behaviour: Wavelike Wavelike ↔ Particlelike↔ Particlelike: Associated with any particle is a wavefunction having wavelength related to a particle momentum by: = h/p = h/√(2m(E-V)) = h/p = h/√(2m(E-V)) (de Broglie)

•Duality Behaviour: Wavelike Wavelike ↔ Particlelike↔ Particlelike: Associated with any particle is a wavefunction having wavelength related to a particle momentum by: = h/p = h/√(2m(E-V)) = h/p = h/√(2m(E-V)) (de Broglie)

•Wave Function:: its absolute square is proportional to the probability densityprobability density for finding the

particle. It describes a state as completely as possible. First Postulate First Postulate

•Wave Function:: its absolute square is proportional to the probability densityprobability density for finding the

particle. It describes a state as completely as possible. First Postulate First Postulate

•Wave Function:Wave Function:: : it is an it is an eigenfunctioneigenfunction of of Schrödinger‘s EquationSchrödinger‘s Equation, which can be , which can be

constructedconstructed from the classical wave equationclassical wave equation requiring: = h/p = h/√(2m(E-V)) = h/p = h/√(2m(E-V))

•Wave Function:Wave Function:: : it is an it is an eigenfunctioneigenfunction of of Schrödinger‘s EquationSchrödinger‘s Equation, which can be , which can be

constructedconstructed from the classical wave equationclassical wave equation requiring: = h/p = h/√(2m(E-V)) = h/p = h/√(2m(E-V))

•Wave Function:: to be acceptable: it must be single-valued, continuous, nowhere infinite, with

a piecewise continuous first derivative, square-integrable.

•Wave Function:: to be acceptable: it must be single-valued, continuous, nowhere infinite, with

a piecewise continuous first derivative, square-integrable.

•Operator:ÔÔ: For any Observable there is an operator which is constructed from the classical

expression according to a simple recipe. Second PostulateSecond Postulate

•Operator:ÔÔ: For any Observable there is an operator which is constructed from the classical

expression according to a simple recipe. Second PostulateSecond Postulate

•Wave Function: - Normalization:•Wave Function: - Normalization:

1V* d

1V* d

, ,


•Operator:ÔÔ: The EigenvaluesEigenvalues for such an OperatorOperator are the possiblepossible values we can measure for

that quantity. Fourth PostulateFourth Postulate

•Operator:ÔÔ: The EigenvaluesEigenvalues for such an OperatorOperator are the possiblepossible values we can measure for

that quantity. Fourth PostulateFourth Postulate

•Operator:ÔÔ: The act of measuringmeasuring the quantity forces the system into a state described by an

EigenfunctionEigenfunction of the Operator

•Operator:ÔÔ: The act of measuringmeasuring the quantity forces the system into a state described by an

EigenfunctionEigenfunction of the Operator

•Time-Dependent Schrödinger Equation: The State functions State functions or Wave functions Wave functions satisfy the equation:

•Both, HamiltonianHamiltonian and Wave functionsWave functions are time-dependent. Third PostulateThird Postulate

•Time-Dependent Schrödinger Equation: The State functions State functions or Wave functions Wave functions satisfy the equation:

•Both, HamiltonianHamiltonian and Wave functionsWave functions are time-dependent. Third PostulateThird Postulate

EHtzyxti

htzyx

ˆ),,,(2

),,,(ˆ ψ ψ H

EHtzyxti

htzyx

ˆ),,,(2

),,,(ˆ ψ ψ H

• If the Hamiltonian OperatorHamiltonian Operator for a system is time.independent, stationary eingenfunctionsstationary eingenfunctions exist of the form:

The time-dependent exponential does not affect the measurable properties of a system in this state and is almost always completely ignored in any time-independent problem.

• If the Hamiltonian OperatorHamiltonian Operator for a system is time.independent, stationary eingenfunctionsstationary eingenfunctions exist of the form:

The time-dependent exponential does not affect the measurable properties of a system in this state and is almost always completely ignored in any time-independent problem.

)/2exp(),,,( hiEtzyx )/2exp(),,,( hiEtzyx

•Expectation Value mm of an Operator MM: Fifth PostulateFifth Postulate

It is equivalent to the arithmetic average of all the possible measured values of a property times their frequency of occurrence.

•Expectation Value mm of an Operator MM: Fifth PostulateFifth Postulate

It is equivalent to the arithmetic average of all the possible measured values of a property times their frequency of occurrence.

MdMm ˆ,Vˆ*

MdMm ˆ,Vˆ*


•Completeness of Eigenfunctions of an Operator MM: Sixth PostulateSixth Postulate

The Eigenfunctions for any quantum mechanical operator corresponding to an observable variable constitute a complete set.complete set.•Briefly, a series of functions Briefly, a series of functions having certain restrictions is said to be having certain restrictions is said to be completecomplete if an arbitrary if an arbitrary

function function having the same restrictions can be expressed in terms of the series: having the same restrictions can be expressed in terms of the series: •

•Completeness of Eigenfunctions of an Operator MM: Sixth PostulateSixth Postulate

The Eigenfunctions for any quantum mechanical operator corresponding to an observable variable constitute a complete set.complete set.•Briefly, a series of functions Briefly, a series of functions having certain restrictions is said to be having certain restrictions is said to be completecomplete if an arbitrary if an arbitrary

function function having the same restrictions can be expressed in terms of the series: having the same restrictions can be expressed in terms of the series: •

iii

iii cc ,

iii

iii cc ,

•Time-Dependent Schrödinger Equation:

•Time-independent Schrödinger Equation:

•Time-Dependent Schrödinger Equation:

•Time-independent Schrödinger Equation:

EHtzyxti

htzyx

ˆ),,,(2

),,,(ˆ ψ ψ H

EHtzyxti

htzyx

ˆ),,,(2

),,,(ˆ ψ ψ H

EHzyxVm

zyx

ˆ),,(ˆ2

),,(ˆ2

ψ ψ

H EHzyxVm

zyx

ˆ),,(ˆ2

),,(ˆ2

ψ ψ

H

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PHYSICAL CHEMISTRY - ADVANCED MATERIALS Particles and Waves Wave Phenomena i rReflexionReflexionRefractionRefraction ii t n 1 n 2 n 1 sin