Chapter 27Chapter 27Quantum PhysicsQuantum Physics

Need for Quantum PhysicsNeed for Quantum Physics Problems remained from classical Problems remained from classical

mechanics that relativity didn’t explainmechanics that relativity didn’t explain Blackbody RadiationBlackbody Radiation

The electromagnetic radiation emitted by a The electromagnetic radiation emitted by a heated objectheated object

Photoelectric EffectPhotoelectric Effect Emission of electrons by an illuminated metalEmission of electrons by an illuminated metal

Spectral LinesSpectral Lines Emission of sharp spectral lines by gas atoms Emission of sharp spectral lines by gas atoms

in an electric discharge tubein an electric discharge tube

Development of Quantum Development of Quantum PhysicsPhysics

1900 to 19301900 to 1930 Development of ideas of quantum mechanicsDevelopment of ideas of quantum mechanics

Also called wave mechanicsAlso called wave mechanics Highly successful in explaining the behavior of atoms, Highly successful in explaining the behavior of atoms,

molecules, and nucleimolecules, and nuclei Quantum Mechanics reduces to classical Quantum Mechanics reduces to classical

mechanics when applied to macroscopic systemsmechanics when applied to macroscopic systems Involved a large number of physicistsInvolved a large number of physicists

Planck introduced basic ideasPlanck introduced basic ideas Mathematical developments and interpretations Mathematical developments and interpretations

involved such people as Einstein, Bohr, Schrinvolved such people as Einstein, Bohr, Schrödinger, de ödinger, de Broglie, Heisenberg, Born and DiracBroglie, Heisenberg, Born and Dirac

Blackbody RadiationBlackbody Radiation An object at any temperature is An object at any temperature is

known to emit electromagnetic known to emit electromagnetic radiationradiation Sometimes called Sometimes called thermal radiationthermal radiation Stefan’s Law describes the total power Stefan’s Law describes the total power

radiatedradiated The spectrum of the radiation depends The spectrum of the radiation depends

on the temperature and properties of the on the temperature and properties of the objectobject

Blackbody Radiation Blackbody Radiation GraphGraph

Experimental data for Experimental data for distribution of energy in distribution of energy in blackbody radiationblackbody radiation

As the temperature As the temperature increases, the total increases, the total amount of energy amount of energy increasesincreases Shown by the area under Shown by the area under

the curvethe curve As the temperature As the temperature

increases, the peak of increases, the peak of the distribution shifts to the distribution shifts to shorter wavelengthsshorter wavelengths

Wien’s Displacement LawWien’s Displacement Law The wavelength of the peak of the The wavelength of the peak of the

blackbody distribution was found blackbody distribution was found to follow to follow Wein’s Displacement LawWein’s Displacement Law λλmaxmax T = 0.2898 x 10 T = 0.2898 x 10-2-2 m • K m • K

λλmaxmax is the wavelength at the curve’s peak is the wavelength at the curve’s peak T is the absolute temperature of the T is the absolute temperature of the

object emitting the radiationobject emitting the radiation

The Ultraviolet The Ultraviolet CatastropheCatastrophe

Classical theory did not Classical theory did not match the experimental match the experimental datadata

At long wavelengths, the At long wavelengths, the match is goodmatch is good

At short wavelengths, At short wavelengths, classical theory predicted classical theory predicted infinite energyinfinite energy

At short wavelengths, At short wavelengths, experiment showed no experiment showed no energyenergy

This contradiction is called This contradiction is called the the ultraviolet catastropheultraviolet catastrophe

Planck’s ResolutionPlanck’s Resolution Planck hypothesized that the blackbody Planck hypothesized that the blackbody

radiation was produced by radiation was produced by resonatorsresonators Resonators were submicroscopic charged Resonators were submicroscopic charged

oscillatorsoscillators The resonators could only have The resonators could only have discrete discrete

energiesenergies EEnn = n h = n h ƒƒ

n is called the n is called the quantum numberquantum number ƒ is the frequency of vibrationƒ is the frequency of vibration h is h is Planck’s constantPlanck’s constant, 6.626 x 10, 6.626 x 10-34-34 J s J s

Key point is quantized energy statesKey point is quantized energy states

Photoelectric EffectPhotoelectric Effect When light is incident on certain metallic When light is incident on certain metallic

surfaces, electrons are emitted from the surfaces, electrons are emitted from the surfacesurface This is called the This is called the photoelectric effectphotoelectric effect The emitted electrons are called The emitted electrons are called photoelectronsphotoelectrons

The effect was first discovered by HertzThe effect was first discovered by Hertz The successful explanation of the effect was The successful explanation of the effect was

given by Einstein in 1905given by Einstein in 1905 Received Nobel Prize in 1921 for paper on Received Nobel Prize in 1921 for paper on

electromagnetic radiation, of which the electromagnetic radiation, of which the photoelectric effect was a partphotoelectric effect was a part

Photoelectric Effect Photoelectric Effect SchematicSchematic

When light strikes E, When light strikes E, photoelectrons are photoelectrons are emittedemitted

Electrons collected at Electrons collected at C and passing C and passing through the ammeter through the ammeter are a current in the are a current in the circuitcircuit

C is maintained at a C is maintained at a positive potential by positive potential by the power supplythe power supply

Photoelectric Photoelectric Current/Voltage GraphCurrent/Voltage Graph

The current increases The current increases with intensity, but with intensity, but reaches a saturation reaches a saturation level for large level for large ΔV’sΔV’s

No current flows for No current flows for voltages less than or voltages less than or equal to –ΔVequal to –ΔVss, the , the stopping potentialstopping potential The stopping potential The stopping potential

is independent of the is independent of the radiation intensityradiation intensity

Features Not Explained by Features Not Explained by Classical Physics/Wave Classical Physics/Wave TheoryTheory

No electrons are emitted if the No electrons are emitted if the incident light frequency is below incident light frequency is below some some cutoff frequencycutoff frequency that is that is characteristic of the material being characteristic of the material being illuminatedilluminated

The maximum kinetic energy of The maximum kinetic energy of the photoelectrons is independent the photoelectrons is independent of the light intensityof the light intensity

More Features Not More Features Not ExplainedExplained

The maximum kinetic energy of The maximum kinetic energy of the photoelectrons increases with the photoelectrons increases with increasing light frequencyincreasing light frequency

Electrons are emitted from the Electrons are emitted from the surface almost instantaneously, surface almost instantaneously, even at low intensitieseven at low intensities

Einstein’s ExplanationEinstein’s Explanation A tiny packet of light energy, called a A tiny packet of light energy, called a photonphoton, ,

would be emitted when a quantized oscillator would be emitted when a quantized oscillator jumped from one energy level to the next lower jumped from one energy level to the next lower oneone Extended Planck’s idea of quantization to Extended Planck’s idea of quantization to

electromagnetic radiationelectromagnetic radiation The photon’s energy would be E = hThe photon’s energy would be E = hƒƒ Each photon can give all its energy to an electron Each photon can give all its energy to an electron

in the metalin the metal The maximum kinetic energy of the liberated The maximum kinetic energy of the liberated

photoelectron is KE = hphotoelectron is KE = hƒ – Φƒ – Φ Φ is called the Φ is called the work functionwork function of the metal of the metal

Explanation of Classical Explanation of Classical “Problems”“Problems”

The effect is not observed below a The effect is not observed below a certain cutoff frequency since the certain cutoff frequency since the photon energy must be greater photon energy must be greater than or equal to the work functionthan or equal to the work function Without this, electrons are not emitted, Without this, electrons are not emitted,

regardless of the intensity of the lightregardless of the intensity of the light The maximum KE depends only on The maximum KE depends only on

the frequency and the work the frequency and the work function, not on the intensityfunction, not on the intensity

More ExplanationsMore Explanations The maximum KE increases with The maximum KE increases with

increasing frequencyincreasing frequency The effect is instantaneous since The effect is instantaneous since

there is a one-to-one interaction there is a one-to-one interaction between the photon and the between the photon and the electronelectron

Verification of Einstein’s Verification of Einstein’s TheoryTheory

Experimental Experimental observations of a observations of a linear relationship linear relationship between KE and between KE and frequency confirm frequency confirm Einstein’s theoryEinstein’s theory

The x-intercept is The x-intercept is the cutoff the cutoff frequencyfrequency

PhotocellsPhotocells Photocells are an application of the Photocells are an application of the

photoelectric effectphotoelectric effect When light of sufficiently high When light of sufficiently high

frequency falls on the cell, a frequency falls on the cell, a current is producedcurrent is produced

ExamplesExamples Streetlights, garage door openers, Streetlights, garage door openers,

elevatorselevators

X-RaysX-Rays Electromagnetic radiation with short Electromagnetic radiation with short

wavelengthswavelengths Wavelengths less than for ultravioletWavelengths less than for ultraviolet Wavelengths are typically about 0.1 nmWavelengths are typically about 0.1 nm X-rays have the ability to penetrate X-rays have the ability to penetrate

most materials with relative easemost materials with relative ease Discovered and named by Roentgen Discovered and named by Roentgen

in 1895in 1895

Production of X-rays, 1Production of X-rays, 1 X-rays are produced when X-rays are produced when

high-speed electrons are high-speed electrons are suddenly slowed downsuddenly slowed down Can be caused by the electron Can be caused by the electron

striking a metal targetstriking a metal target A current in the filament A current in the filament

causes electrons to be causes electrons to be emittedemitted

These freed electrons are These freed electrons are accelerated toward a accelerated toward a dense metal targetdense metal target

The target is held at a The target is held at a higher potential than the higher potential than the filamentfilament

Production of X-rays, 2Production of X-rays, 2 An electron passes An electron passes

near a target nucleusnear a target nucleus The electron is The electron is

deflected from its deflected from its path by its attraction path by its attraction to the nucleusto the nucleus This produces an This produces an

accelerationacceleration It will emit It will emit

electromagnetic electromagnetic radiation when it is radiation when it is acceleratedaccelerated

Diffraction of X-rays by Diffraction of X-rays by CrystalsCrystals

For diffraction to occur, the spacing For diffraction to occur, the spacing between the lines must be between the lines must be approximately equal to the wavelength approximately equal to the wavelength of the radiation to be measuredof the radiation to be measured

For X-rays, the regular array of atoms For X-rays, the regular array of atoms in a crystal can act as a three-in a crystal can act as a three-dimensional grating for diffracting X-dimensional grating for diffracting X-raysrays

Schematic for X-ray Schematic for X-ray DiffractionDiffraction

A continuous beam of A continuous beam of X-rays is incident on X-rays is incident on the crystalthe crystal

The diffracted radiation The diffracted radiation is very intense in is very intense in certain directionscertain directions These directions correspond These directions correspond

to constructive interference to constructive interference from waves reflected from from waves reflected from the layers of the crystalthe layers of the crystal

The diffraction pattern The diffraction pattern is detected by is detected by photographic filmphotographic film

Photo of X-ray Diffraction Photo of X-ray Diffraction PatternPattern

The array of spots is The array of spots is called a called a LaueLaue pattern pattern

The crystal structure is The crystal structure is determined by determined by analyzing the positions analyzing the positions and intensities of the and intensities of the various spotsvarious spots

This is for NaClThis is for NaCl

Bragg’s LawBragg’s Law The beam reflected from the The beam reflected from the

lower surface travels farther lower surface travels farther than the one reflected from than the one reflected from the upper surfacethe upper surface

If the path difference equals If the path difference equals some integral multiple of some integral multiple of the wavelength, the wavelength, constructive interference constructive interference occursoccurs

Bragg’s LawBragg’s Law gives the gives the conditions for constructive conditions for constructive interferenceinterference 2 d sin 2 d sin θ = m λ, m = 1, 2, θ = m λ, m = 1, 2,

3…3…

The Compton EffectThe Compton Effect Compton directed a beam of x-rays toward a Compton directed a beam of x-rays toward a

block of graphiteblock of graphite He found that the scattered x-rays had a He found that the scattered x-rays had a

slightly longer wavelength that the incident slightly longer wavelength that the incident x-raysx-rays This means they also had less energyThis means they also had less energy

The amount of energy reduction depended The amount of energy reduction depended on the angle at which the x-rays were on the angle at which the x-rays were scatteredscattered

The change in wavelength is called the The change in wavelength is called the Compton shiftCompton shift

Compton ScatteringCompton Scattering Compton assumed Compton assumed

the photons acted the photons acted like other particles like other particles in collisionsin collisions

Energy and Energy and momentum were momentum were conservedconserved

The shift in The shift in wavelength iswavelength is

)cos1(cmh

eo

Compton Scattering, finalCompton Scattering, final The quantity h/mThe quantity h/meec is called the c is called the Compton Compton

wavelengthwavelength Compton wavelength = 0.00243 nmCompton wavelength = 0.00243 nm Very small compared to visible lightVery small compared to visible light

The Compton shift depends on the The Compton shift depends on the scattering angle and not on the scattering angle and not on the wavelengthwavelength

Experiments confirm the results of Experiments confirm the results of Compton scattering and strongly support Compton scattering and strongly support the photon conceptthe photon concept

QUICK QUIZ 27.1An x-ray photon is scattered by an electron. The frequency of the scattered photon relative to that of the incident photon (a) increases, (b) decreases, or (c) remains the same.

QUICK QUIZ 27.1 ANSWER

(b). Some energy is transferred to the electron in the scattering process. Therefore, the scattered photon must have less energy (and hence, lower frequency) than the incident photon.

QUICK QUIZ 27.2A photon of energy E0 strikes a free electron, with the scattered photon of energy E moving in the direction opposite that of the incident photon. In this Compton effect interaction, the resulting kinetic energy of the electron is (a) E0 , (b) E , (c) E0 E , (d) E0 + E , (e) none of the above.

QUICK QUIZ 27.2 ANSWER

(c). Conservation of energy requires the kinetic energy given to the electron be equal to the difference between the energy of the incident photon and that of the scattered photon.

QUICK QUIZ 27.3A photon of energy E0 strikes a free electron with the scattered photon of energy E moving in the direction opposite that of the incident photon. In this Compton effect interaction, the resulting momentum of the electron is (a) E0/c (b) < E0/c (c) > E0/c (d) (E0 E)/c (e) (E Eo)/c

QUICK QUIZ 27.3 ANSWER(c). Conservation of momentum

requires the momentum of the incident photon equal the vector sum of the momenta of the electron and the scattered photon. Since the scattered photon moves in the direction opposite that of the electron, the magnitude of the electron’s momentum must exceed that of the incident photon.

Photons and Photons and Electromagnetic WavesElectromagnetic Waves

Light has a dual nature. It exhibits both Light has a dual nature. It exhibits both wave and particle characteristicswave and particle characteristics Applies to all electromagnetic radiationApplies to all electromagnetic radiation

The photoelectric effect and Compton The photoelectric effect and Compton scattering offer evidence for the particle scattering offer evidence for the particle nature of lightnature of light When light and matter interact, light When light and matter interact, light

behaves as if it were composed of particlesbehaves as if it were composed of particles Interference and diffraction offer Interference and diffraction offer

evidence of the wave nature of lightevidence of the wave nature of light

Wave Properties of Wave Properties of ParticlesParticles

In 1924, Louis de Broglie In 1924, Louis de Broglie postulated that postulated that because photons because photons have wave and particle have wave and particle characteristics, perhaps all forms characteristics, perhaps all forms of matter have both propertiesof matter have both properties

Furthermore, the frequency and Furthermore, the frequency and wavelength of matter waves can wavelength of matter waves can be determinedbe determined

de Broglie Wavelength de Broglie Wavelength and Frequencyand Frequency

The The de Broglie wavelengthde Broglie wavelength of a of a particle is particle is

The frequency of matter waves isThe frequency of matter waves ismvh

hEƒ

The Davisson-Germer The Davisson-Germer ExperimentExperiment

They scattered low-energy electrons from a They scattered low-energy electrons from a nickel targetnickel target

They followed this with extensive diffraction They followed this with extensive diffraction measurements from various materialsmeasurements from various materials

The wavelength of the electrons calculated The wavelength of the electrons calculated from the diffraction data agreed with the from the diffraction data agreed with the expected de Broglie wavelengthexpected de Broglie wavelength

This confirmed the wave nature of electronsThis confirmed the wave nature of electrons Other experimenters have confirmed the Other experimenters have confirmed the

wave nature of other particleswave nature of other particles

QUICK QUIZ 27.4A non-relativistic electron and a non-relativistic proton are moving and have the same de Broglie wavelength. Which of the following are also the same for the two particles: (a) speed, (b) kinetic energy, (c) momentum, (d) frequency?

QUICK QUIZ 27.4 ANSWER

(c). Two particles with the same de Broglie wavelength will have the same momentum p = mv. If the electron and proton have the same momentum, they cannot have the same speed because of the difference in their masses. For the same reason, remembering that KE = p2/2m, they cannot have the same kinetic energy. Because the kinetic energy is the only type of energy an isolated particle can have, and we have argued that the particles have different energies, Equation 27.15 tells us that the particles do not have the same frequency.

QUICK QUIZ 27.5We have seen two wavelengths assigned to the electron, the Compton wavelength and the de Broglie wavelength. Which is an actual physical wavelength associated with the electron: (a) the Compton wavelength, (b) the de Broglie wavelength, (c) both wavelengths, (d) neither wavelength?

QUICK QUIZ 27.5 ANSWER(b). The Compton wavelength, λC =

h/mec, is a combination of constants and has no relation to the motion of the electron. The de Broglie wavelength, λ = h/mev, is associated with the motion of the electron through its momentum.

The Electron MicroscopeThe Electron Microscope The electron microscope The electron microscope

depends on the wave depends on the wave characteristics of electronscharacteristics of electrons

Microscopes can only Microscopes can only resolve details that are resolve details that are slightly smaller than the slightly smaller than the wavelength of the radiation wavelength of the radiation used to illuminate the used to illuminate the objectobject

The electrons can be The electrons can be accelerated to high accelerated to high energies and have small energies and have small wavelengthswavelengths

The Wave FunctionThe Wave Function In 1926 SchrIn 1926 Schrödinger proposed a wave ödinger proposed a wave

equation that describes the manner in equation that describes the manner in which matter waves change in space which matter waves change in space and timeand time

SchrSchrödinger’s wave equation is a key ödinger’s wave equation is a key element in quantum mechanicselement in quantum mechanics

SchrSchrödinger’s wave equation is ödinger’s wave equation is generally solved for the generally solved for the wave functionwave function, , ΨΨ

The Wave Function, contThe Wave Function, cont The wave function depends on the The wave function depends on the

particle’s position and the timeparticle’s position and the time The value of ΨThe value of Ψ22 at some location at at some location at

a given time is proportional to the a given time is proportional to the probability of finding the particle at probability of finding the particle at that location at that timethat location at that time

The Uncertainty PrincipleThe Uncertainty Principle When measurements are made, When measurements are made,

the experimenter is always faced the experimenter is always faced with experimental uncertainties in with experimental uncertainties in the measurementsthe measurements Classical mechanics offers no Classical mechanics offers no

fundamental barrier to ultimate fundamental barrier to ultimate refinements in measurementsrefinements in measurements

Classical mechanics would allow for Classical mechanics would allow for measurements with arbitrarily small measurements with arbitrarily small uncertaintiesuncertainties

The Uncertainty Principle, The Uncertainty Principle, 22

Quantum mechanics predicts that a Quantum mechanics predicts that a barrier to measurements with ultimately barrier to measurements with ultimately small uncertainties does existsmall uncertainties does exist

In 1927 Heisenberg introduced the In 1927 Heisenberg introduced the uncertainty principleuncertainty principle If a measurement of position of a particle is If a measurement of position of a particle is

made with precision made with precision Δx and a simultaneous Δx and a simultaneous measurement of linear momentum is made measurement of linear momentum is made with precision Δp, then the product of the with precision Δp, then the product of the two uncertainties can never be smaller than two uncertainties can never be smaller than h/4h/4

The Uncertainty Principle, The Uncertainty Principle, 33

Mathematically,Mathematically,

It is physically impossible to measure It is physically impossible to measure simultaneously the exact position and simultaneously the exact position and the exact linear momentum of a the exact linear momentum of a particleparticle

Another form of the principle deals Another form of the principle deals with energy and time: with energy and time:

4hpx x

4htE

Thought Experiment – the Thought Experiment – the Uncertainty PrincipleUncertainty Principle

A thought experiment for viewing an electron with a powerful A thought experiment for viewing an electron with a powerful microscopemicroscope

In order to see the electron, at least one photon must bounce off itIn order to see the electron, at least one photon must bounce off it During this interaction, momentum is transferred from the photon During this interaction, momentum is transferred from the photon

to the electronto the electron Therefore, the light that allows you to accurately locate the Therefore, the light that allows you to accurately locate the

electron changes the momentum of the electronelectron changes the momentum of the electron

Scanning Tunneling Scanning Tunneling Microscope (STM)Microscope (STM)

Allows highly detailed Allows highly detailed images with resolution images with resolution comparable to the size comparable to the size of a single atomof a single atom

A conducting probe A conducting probe with a sharp tip is with a sharp tip is brought near the brought near the surfacesurface

The electrons can The electrons can “tunnel” across the “tunnel” across the barrier of empty spacebarrier of empty space

Scanning Tunneling Scanning Tunneling Microscope, contMicroscope, cont

By applying a voltage between the surface By applying a voltage between the surface and the tip, the electrons can be made to and the tip, the electrons can be made to tunnel preferentially from surface to tiptunnel preferentially from surface to tip

The tip samples the distribution of The tip samples the distribution of electrons just above the surfaceelectrons just above the surface

The STM is very sensitive to the distance The STM is very sensitive to the distance between the surface and the tipbetween the surface and the tip Allows measurements of the height of surface Allows measurements of the height of surface

features within 0.001 nmfeatures within 0.001 nm

Limitation of the STMLimitation of the STM There is a serious limitation to the STM There is a serious limitation to the STM

since it depends on the conductivity of since it depends on the conductivity of the surface and the tipthe surface and the tip Most materials are not conductive at their Most materials are not conductive at their

surfacesurface An An atomic force microscopeatomic force microscope has been has been

developed that overcomes this limitationdeveloped that overcomes this limitation It measures the force between the tip and the It measures the force between the tip and the

sample surfacesample surface Has comparable sensitivityHas comparable sensitivity