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Young/FreemanYoung/Freeman

University Physics University Physics

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Ch 39 The Wave Nature Ch 39 The Wave Nature

of Particlesof Particles

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39.1 Wave Properties of Particles39.1 Wave Properties of Particles

�� In 1924, Louis de Broglie postulated that In 1924, Louis de Broglie postulated that because because

photons have wave and particle characteristics, perhaps all photons have wave and particle characteristics, perhaps all

forms of matter have both wave and particle properties.forms of matter have both wave and particle properties.

�� Like photons a Like photons a freefree particle should behave like a particle should behave like a

wave (h =6.6 10wave (h =6.6 10--3434 J.sJ.s is the Planck constant)is the Planck constant)

-- with frequency f = E/h with frequency f = E/h

-- withwith wavelength wavelength λλ = = h/mvh/mv

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39.2 39.2 Is this postulate true?Is this postulate true?Confirmation: Confirmation: The DavissonThe Davisson--GermerGermer ExperimentExperiment..

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The electron beam is scattered by Bragg diffraction

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Electrons behave as light wavesElectrons behave as light waves

�� We know that in the case of XWe know that in the case of X--rays the diffraction angles are rays the diffraction angles are due to constructive scattering due to constructive scattering from parallel planes. The radii from parallel planes. The radii of the circles are determined by of the circles are determined by the Bragg law the Bragg law nnλλ = 2dsin= 2dsinθθ

�� Using for electrons this Bragg Using for electrons this Bragg law with law with λλ calculated by the De calculated by the De Broglie relation gives a perfect Broglie relation gives a perfect fitfit……

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XX--ray diffraction (left) / electron ray diffraction (left) / electron

diffraction (right)diffraction (right)

Powder crystal gives rise to rings.

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39.4 39.4 The electron microscope The electron microscope p = p = h/h/λλ

�� As the resolution of a As the resolution of a microscope is limited by the microscope is limited by the wavelength of the radiation wavelength of the radiation Ruska (1944) thought of Ruska (1944) thought of building an electron building an electron microscope as then the microscope as then the wavelength could be made wavelength could be made small enough so as to have small enough so as to have a high resolution. a high resolution.

�� Typically electrons are Typically electrons are accelerated over 100keV. accelerated over 100keV.

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An electron microscope An electron microscope

1000 x higher resolution1000 x higher resolution

ResolutionResolution

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Resolving Power of MicroscopesResolving Power of Microscopes

�� To see or resolve an object, we need to use light of To see or resolve an object, we need to use light of

wavelength no larger than the object itselfwavelength no larger than the object itself

�� Since the wavelength of visible light is about 0.4 to Since the wavelength of visible light is about 0.4 to

0.7 0.7 µµm,m,

an ordinary microscopean ordinary microscope

can only resolve objectscan only resolve objects

as small as this, such asas small as this, such as

bacteria but not virusesbacteria but not viruses

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Two further experiments to show Two further experiments to show

wave propertieswave properties

�� Single slit experiment double slit experimentSingle slit experiment double slit experiment

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39.3 Single slit diffraction by a beam 39.3 Single slit diffraction by a beam

of electrons: eof electrons: electrons do behave as

light waves.�� A beam of electrons A beam of electrons

is scattered by a is scattered by a

screen with one split.screen with one split.

�� We observe an We observe an

interference pattern interference pattern

as with light. as with light.

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How can we understand interference How can we understand interference

when the electrons come one by one?when the electrons come one by one?

�� When the beam intensity gets When the beam intensity gets

very weak we see individual very weak we see individual

spots an the screen. spots an the screen.

�� Gradually the interference Gradually the interference

pattern is build up. pattern is build up.

�� We do not know in advance We do not know in advance

where an electron will go.where an electron will go.

�� All electrons have the same All electrons have the same

probabilityprobability for going for going

somewheresomewhere. This results in the . This results in the

observed diffraction pattern.observed diffraction pattern.© 2005 Pearson Education

Quantum Mechanics

allows to calculate the

probability where an

electron will go.

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The slit causes broadening of the The slit causes broadening of the

electron waveelectron wave

�� When the electrons come through the When the electrons come through the slit the component of slit the component of ppyy varies mostly varies mostly (for 85%) between (for 85%) between ppyy = = ±± ppxxθθ11. .

�� In the case of light it is known that In the case of light it is known that approximately approximately θθ11== λλ/a. /a. So for So for electrons in the electrons in the ±± θθ11 direction direction ppyy ≈≈ ±±ppxxλλ/a. This implies an uncertainty in /a. This implies an uncertainty in ppyy of the order of the order ∆∆ppyy ≥≥ ppxxλλ/a./a.

�� Because of De Broglie law Because of De Broglie law ppxx ≈≈ h/h/λλit followsit follows →→ aa ∆∆ppyy = = ∆∆yy ∆∆ppyy ≥≥ hh

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A Heisenberg uncertainty relationA Heisenberg uncertainty relation

�� We do not know in advance the component We do not know in advance the component ppyy , , there is an inherent uncertainty there is an inherent uncertainty ∆∆ppyy. .

�� We do not know in advance the y component, We do not know in advance the y component, there is an inherent uncertainty there is an inherent uncertainty ∆∆yy==a.a.

�� Expression Expression ∆∆yy∆∆ppyy ≥≥ hh is an example of a is an example of a HeisenbergHeisenberg uncertainty relationuncertainty relation. Such relations . Such relations abound in Quantum Physicsabound in Quantum Physics

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Two slit diffraction with electrons: Two slit diffraction with electrons:

Electrons do behave as light wavesElectrons do behave as light waves

Quantum Mechanics allows to calculate the

probability where an electron will go.

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Through which slit did the electron come?Through which slit did the electron come?

�� Say we let the electrons come Say we let the electrons come

one by one. Then we want to one by one. Then we want to

know through which slit each know through which slit each

electron came.electron came.

�� If we detect the electron path the If we detect the electron path the

interference pattern disappears interference pattern disappears

and we observe the particle and we observe the particle

result R1.result R1.

�� If we do not detect the path we If we do not detect the path we

observe the wave result R2.observe the wave result R2.

R2

R1

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Quantum Mechanics: a new theory Quantum Mechanics: a new theory

of matter: the pioneers of matter: the pioneers ±±±±±±±±19251925

W. Heisenberg Erwin Schrödinger

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But !!! But !!!

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A lot of booksA lot of books……

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About Quantum MechanicsAbout Quantum Mechanics

�� Quantum Mechanics was invented about 1925Quantum Mechanics was invented about 1925

�� Its basic concept is a complex valued wave function to Its basic concept is a complex valued wave function to

describe a particle. All the knowledge about the particle describe a particle. All the knowledge about the particle

is hidden in this wave function.is hidden in this wave function.

�� The wave function varies with time according to the The wave function varies with time according to the

SchrSchröödinger equation.dinger equation.

�� The theory is highly mathematical.The theory is highly mathematical.

�� Quantum Mechanics is now used in various fields with Quantum Mechanics is now used in various fields with

much success. much success.

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39.5 A wave function to describe a 39.5 A wave function to describe a

particle (1 dimension)particle (1 dimension)

� The wave function ψ(x,t) for a particle contains all of the information about that particle.

� The quantity |ψ(x,t)|2 is called the position probability distribution function.

� It determines the relative probability of finding a particle near a given position x at a given time t.

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What is the information about the What is the information about the

position? position?

�� QM : we can never perfectly predict where a particle QM : we can never perfectly predict where a particle

will be found. Using the wave function we can only will be found. Using the wave function we can only

predict a probability to find the particle somewhere. predict a probability to find the particle somewhere.

�� QM says: QM says: P(x,x+dxP(x,x+dx)=|)=|ΨΨ(x)|(x)|22dxdx. .

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Different types of wave functions.Different types of wave functions.

Where is the particle?Where is the particle?

�� A A peaked distributionpeaked distribution

�� A double peaked A double peaked

distributiondistribution

�� A wide distributionA wide distribution

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Average position and uncertaintyAverage position and uncertainty

�� If there is a peaked If there is a peaked

probability we can probability we can

usefully talk about usefully talk about

-- an average position an average position

-- a region of great a region of great

certainty with width certainty with width ∆∆xx

of the probabilityof the probability

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Rules for calculation Rules for calculation

average position <x> and width average position <x> and width ∆∆∆∆∆∆∆∆xx

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Heisenberg uncertainty relationHeisenberg uncertainty relation

(can be derived from theory)(can be derived from theory)

�� If the position probability distribution is narrow If the position probability distribution is narrow peaked, the calculated momentum probability peaked, the calculated momentum probability distribution is very broad. This implies that both speed distribution is very broad. This implies that both speed and position cannot simultaneously be well known.and position cannot simultaneously be well known.

∆∆xx ∆∆vv ≥≥ ħħ/2m/2m

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HeisenbergHeisenberg’’s uncertainty relations uncertainty relation

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Normalization and ProbabilityNormalization and Probability

The wave function must also be normalized so that The wave function must also be normalized so that

the probability of the particle being somewhere on the probability of the particle being somewhere on

the the xx axis is 1. This implies that axis is 1. This implies that ΨΨ(x(x→→infinity,tinfinity,t)=0.)=0.

The probability The probability PP((xx) ) dxdx of a particle being between of a particle being between xx and and x x + + dxdx

is given in the equationis given in the equation

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The Schrodinger wave equationThe Schrodinger wave equation

�� In 1926 SchrIn 1926 Schröödinger proposed a dinger proposed a wave equationwave equation

that describes how the wave function changes in that describes how the wave function changes in

time.time.

�� SchrSchröödingerdinger’’s wave equations wave equation is a key element in is a key element in

quantum mechanicsquantum mechanics

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Schrodinger equation and Schrodinger equation and

Hamiltonian (1 dimension)Hamiltonian (1 dimension)

�� If we know the wave If we know the wave function at a given time function at a given time we can use this equation we can use this equation to calculate the wave to calculate the wave function later on.function later on.

�� H is the Hamiltonian of H is the Hamiltonian of the system, it is the the system, it is the energy in terms of x energy in terms of x and p=(and p=(ħħ/i)ð/ðx/i)ð/ðx

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Example of time evolution: tunnelingExample of time evolution: tunneling

�� A particle approaches a A particle approaches a

barrier: classically it barrier: classically it

bounces back.bounces back.

�� In quantum mechanics In quantum mechanics

there is a chance to there is a chance to

bounce back and a bounce back and a

chance to go through the chance to go through the

barrier.barrier.

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��A computer A computer

calculationcalculation

�� Change of wave Change of wave

function function ΨΨ(x,t(x,t) with ) with

time is calculated time is calculated

according to the according to the

Schrodinger equation.Schrodinger equation.

�� Video shows time Video shows time

dependent probability dependent probability

distribution |distribution |ΨΨ(x,t)|(x,t)|22

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Stationary states IStationary states I

)()()()(

2 2

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xExxUdx

xd

mψψ

ψ=+−

ℏ

�� There are particular wave There are particular wave functions for which the functions for which the position distribution does not position distribution does not change.change.

�� These wave functions are of the These wave functions are of the form shown on the left where form shown on the left where ψψ(x(x) is a solution of the ) is a solution of the equation equation

H H ψψ(x(x) =E ) =E ψψ (x) (x)

�� This equation is called the time This equation is called the time independent Schrodinger independent Schrodinger equation.equation.

ℏ/),,(),,,(

iEtezyxtzyx

−=Ψ ψ

U(x) = the potential energy

E = the total energy

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Stationary states II: propertiesStationary states II: properties

�� The solution of H The solution of H ψψ(x(x) =E ) =E ψψ(x(x) is a particular ) is a particular

wave function which describes a particle in a wave function which describes a particle in a

‘‘statestate’’ with energy E.with energy E.

�� Solving this equation shows that all E values are Solving this equation shows that all E values are

not possible, energy is not possible, energy is quantisedquantised in quantum in quantum

mechanics. Examples will be discussed later on.mechanics. Examples will be discussed later on.

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Use of stationary states Use of stationary states

� All (complex valued ) wave functions can be constructed

by superposing stationary-state wave functions. These can

represent particles that are localized in a certain region and

still have wave properties, giving it both particle and wave

aspects.

� General wave function, with complex coefficients cn

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Uncertainty in energyUncertainty in energy

� Probability is everywhere in QM.

� A particle with a general wave function

has not a precise energy. It has an energy En with

probability |cn|2. Mostly the energy is quantised.

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Uncertainty in momentum p = Uncertainty in momentum p = mvmv

� Also a momentum measurement is uncertain.

� A particle with a general wave function Ψ(x,t) has a momentum probability in an interval [p,p+dp] given by P(p,p+dp,t).

� Momentum is not quantised, is similar to position.

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ENDEND

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THE END.

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Two questions on the MachTwo questions on the Mach--ZehnderZehnder

interferometer.interferometer.

�� In the next slides the concept of a beam splitter In the next slides the concept of a beam splitter

is introduced. Then a Machis introduced. Then a Mach--ZehnderZehnder

interferometer is described.interferometer is described.

�� The behavior of particles in such an The behavior of particles in such an

interferometer is very intriguing and shows right interferometer is very intriguing and shows right

away the peculiar quantum behavior we are not away the peculiar quantum behavior we are not

used to in our daily life.used to in our daily life.

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A beam splitter in actionA beam splitter in action

�� Particles enter from Particles enter from the left.the left.

�� A beam A beam splitter splitter reflects and transmits reflects and transmits 50% of the particles50% of the particles

�� A particle is detected A particle is detected in T or in R, NEVER in T or in R, NEVER in both detectors.in both detectors.

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More beam splittersMore beam splitters

�� Each time a stream of Each time a stream of

particles reaches a particles reaches a

beamsplitterbeamsplitter 50% of the 50% of the

particles is reflected and particles is reflected and

50% is transmitted.50% is transmitted.

�� In each detector TR, RT, In each detector TR, RT,

TT or RR we find 25% TT or RR we find 25%

of the particlesof the particles

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The MachThe Mach--ZehnderZehnder interferometer: interferometer:

interferenceinterference�� Four beam splitters are Four beam splitters are

arranged on the corners of a arranged on the corners of a square.square.

�� There are now two detectors: There are now two detectors: one detects the particles one detects the particles scattered following TR or scattered following TR or RT, the other detects the RT, the other detects the particles following the TT or particles following the TT or RR path.RR path.

�� Observation:Observation:

The particles arrive ONLY in The particles arrive ONLY in the TR/RT detector !the TR/RT detector !

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Question 1 for all students.Question 1 for all students.(You may discuss this problem with your friends)(You may discuss this problem with your friends)

�� Try to find an explanation in terms of a wave Try to find an explanation in terms of a wave

description of the fact that only the RT/TR description of the fact that only the RT/TR

detector registers the particles.detector registers the particles.

(Hint(Hint: think of phase shifts which : think of phase shifts which e.g. occure.g. occur when when

a wave is reflected from a a wave is reflected from a surface.)surface.)

�� Try to describe the experiment in terms of Try to describe the experiment in terms of

particles.particles.

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Extended MachExtended Mach--ZehnderZehnder: :

A bigger mysteryA bigger mystery

�� Now we make one of the Now we make one of the paths longer. paths longer.

→→ ssome particles arrive ome particles arrive in the TT/RR detector.in the TT/RR detector.

�� For a particular path For a particular path enlargement all particles enlargement all particles come in the TT/RR come in the TT/RR detector!detector!

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Question 2 for all students.Question 2 for all students.(You may discuss this problem with your friends)(You may discuss this problem with your friends)

�� Try to find an explanation in terms of a wave Try to find an explanation in terms of a wave

description of the fact that for a particular path description of the fact that for a particular path

enlargement the TT/RR detector registers all the enlargement the TT/RR detector registers all the

particles.particles.

�� Try to describe the experiment in terms of Try to describe the experiment in terms of

particles.particles.