General Physics (PHY 2140)
Lecture 27Lecture 27
� Modern Physics
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� Modern Physics
�Quantum Physics�Blackbody radiation�Plank’s hypothesis
Chapter 27
http://www.physics.wayne.edu/~apetrov/PHY2140/
Quantum PhysicsQuantum Physics
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Introduction: Need for Quantum PhysicsIntroduction: Need for Quantum Physics
Problems remained from classical mechanics that relativity Problems remained from classical mechanics that relativity didn’t explain:didn’t explain:
Blackbody RadiationBlackbody Radiation�� The electromagnetic radiation emitted by a heated objectThe electromagnetic radiation emitted by a heated object
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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 in an electric discharge tubeEmission of sharp spectral lines by gas atoms in an electric discharge tube
Development of Quantum PhysicsDevelopment of Quantum Physics
1900 to 19301900 to 1930�� Development of ideas of quantum mechanicsDevelopment of ideas of quantum mechanics
Also called wave mechanicsAlso called wave mechanicsHighly successful in explaining the behavior of atoms, molecules, and nucleiHighly successful in explaining the behavior of atoms, molecules, and nuclei
Quantum Mechanics reduces to classical mechanics when applied Quantum Mechanics reduces to classical mechanics when applied to macroscopic systemsto macroscopic systemsInvolved a large number of physicistsInvolved a large number of physicists
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Involved a large number of physicistsInvolved a large number of physicists�� Planck introduced basic ideasPlanck introduced basic ideas�� Mathematical developments and interpretations involved such people as Mathematical developments and interpretations involved such people as
Einstein, Bohr, Schrödinger, de Broglie, Heisenberg, Born and DiracEinstein, Bohr, Schrödinger, de Broglie, Heisenberg, Born and Dirac
26.1 Blackbody Radiation26.1 Blackbody Radiation
An object at any temperature is known to emit An object at any temperature is known to emit electromagnetic radiationelectromagnetic radiation�� Sometimes called Sometimes called thermal radiationthermal radiation�� Stefan’s Law describes the total power radiatedStefan’s Law describes the total power radiated
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�� The The spectrumspectrum of the radiation depends on the temperature and of the radiation depends on the temperature and properties of the objectproperties of the object
4P AeTσ=emissivityStefan’s constant
BlackbodyBlackbody
BlackbodyBlackbody is an is an idealized systemidealized system that absorbs incident that absorbs incident radiation of all wavelengthsradiation of all wavelengthsIf it is heated to a certain temperature, it starts If it is heated to a certain temperature, it starts radiate radiate electromagnetic waves of all wavelengthselectromagnetic waves of all wavelengthsCavity Cavity is a good realis a good real--life approximation to a blackbodylife approximation to a blackbody
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Cavity Cavity is a good realis a good real--life approximation to a blackbodylife approximation to a blackbody
Blackbody Radiation GraphBlackbody Radiation Graph
Experimental data for Experimental data for distribution of energy in distribution of energy in blackbody radiationblackbody radiation
As the temperature increases, As the temperature increases, the total amount of energy the total amount of energy increasesincreases
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increasesincreases�� Shown by the area under Shown by the area under
the curvethe curve
As the temperature increases, As the temperature increases, the peak of the distribution the peak of the distribution shifts to shorter wavelengthsshifts to shorter wavelengths
Wien’s Displacement LawWien’s Displacement Law
The wavelength of the peak of the blackbody distribution The wavelength of the peak of the blackbody distribution was found to follow was found to follow Wein’s Displacement LawWein’s Displacement Law
λλmaxmax T = 0.2898 x 10T = 0.2898 x 10--22 m • Km • K
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λλmaxmax is the wavelength at the curve’s peakis the wavelength at the curve’s peakT is the absolute temperature of the object emitting the radiationT is the absolute temperature of the object emitting the radiation
The Ultraviolet CatastropheThe Ultraviolet Catastrophe
Classical theory did not match Classical theory did not match the experimental datathe experimental dataAt At longlong wavelengths, the wavelengths, the match is good match is good �� RayleighRayleigh--JeansJeans lawlaw
2 ckTπ
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At At shortshort wavelengths, classical wavelengths, classical theory predicted theory predicted infinite energyinfinite energyAt short wavelengths, At short wavelengths, experiment showed no energyexperiment showed no energyThis contradiction is called the This contradiction is called the ultraviolet catastropheultraviolet catastrophe
4
2 ckTP
πλ
=
Planck’s ResolutionPlanck’s Resolution
Planck hypothesized that the Planck hypothesized that the blackbody radiation was produced by blackbody radiation was produced by resonatorsresonators�� Resonators were submicroscopic charged oscillatorsResonators were submicroscopic charged oscillators
The resonators could only have The resonators could only have discrete energiesdiscrete energies
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EEnn = n h ƒ= n h ƒ
n is called the n is called the quantum numberquantum numberƒ is the frequency of vibrationƒ is the frequency of vibrationh is h is Planck’s constantPlanck’s constant, , h=6.626 x 10h=6.626 x 10--3434 J sJ s
Key point is Key point is quantized energy statesquantized energy states
QUICK QUIZ
A photon (quantum of light) is reflected from a mirror. A photon (quantum of light) is reflected from a mirror. True or falseTrue or false: :
(a) Because a photon has a zero mass, it does not exert a force on (a) Because a photon has a zero mass, it does not exert a force on the mirror. the mirror.
(b) Although the photon has energy, it cannot transfer any energy to (b) Although the photon has energy, it cannot transfer any energy to the surface because it has zero mass. the surface because it has zero mass.
(c) The photon carries momentum, and when it reflects off the mirror, (c) The photon carries momentum, and when it reflects off the mirror, it undergoes a change in momentum and exerts a force on it undergoes a change in momentum and exerts a force on
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it undergoes a change in momentum and exerts a force on it undergoes a change in momentum and exerts a force on the mirror. the mirror.
(d) Although the photon carries momentum, its change in momentum (d) Although the photon carries momentum, its change in momentum is zero when it reflects from the mirror, so it cannot exert a is zero when it reflects from the mirror, so it cannot exert a force on the mirror.force on the mirror.
(a)(a) False False (b)(b) False False (c)(c) True True (d)(d) FalseFalse
p Ft∆ =
27.2 Photoelectric Effect27.2 Photoelectric Effect
When light is incident on certain metallic surfaces, electrons are When light is incident on certain metallic surfaces, electrons are emitted from the surfaceemitted from the surface�� 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
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The effect was first discovered by HertzThe effect was first discovered by HertzThe successful explanation of the effect was given by Einstein in The successful explanation of the effect was given by Einstein in 19051905
�� Received Nobel Prize in 1921 for paper on electromagnetic radiation, of Received Nobel Prize in 1921 for paper on electromagnetic radiation, of which the photoelectric effect was a partwhich the photoelectric effect was a part
Photoelectric Effect SchematicPhotoelectric Effect Schematic
When light strikes E, When light strikes E, photoelectrons are emittedphotoelectrons are emitted
Electrons collected at C and Electrons collected at C and passing through the ammeter passing through the ammeter are a current in the circuitare a current in the circuit
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are a current in the circuitare a current in the circuit
C is maintained at a positive C is maintained at a positive potential by the power supplypotential by the power supply
Photoelectric Current/Voltage GraphPhotoelectric Current/Voltage Graph
The current increases with The current increases with intensity, but reaches a intensity, but reaches a saturation level for large saturation level for large ∆V’s∆V’s
No current flows for voltages No current flows for voltages less than or equal to less than or equal to ––∆V∆V , the , the
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less than or equal to less than or equal to ––∆V∆Vss, the , the stopping potentialstopping potential
�� The stopping potential is The stopping potential is independent of the radiation independent of the radiation intensityintensity
Features Not Explained by Classical Features Not Explained by Classical Physics/Wave TheoryPhysics/Wave Theory
No electrons are emitted if the incident light frequency is No electrons are emitted if the incident light frequency is below some below some cutoff frequencycutoff frequency that is characteristic of the that is characteristic of the material being illuminatedmaterial being illuminatedThe maximum kinetic energy of the photoelectrons is The maximum kinetic energy of the photoelectrons is independent of the light intensityindependent of the light intensity
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independent of the light intensityindependent of the light intensityThe maximum kinetic energy of the photoelectrons The maximum kinetic energy of the photoelectrons increases with increasing light frequencyincreases with increasing light frequencyElectrons are emitted from the surface almost Electrons are emitted from the surface almost instantaneously, even at low intensitiesinstantaneously, even 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 , would be emitted when a quantized oscillator jumped from one energy level to the when a quantized oscillator jumped from one energy level to the next lower onenext lower one�� Extended Planck’s idea of quantization to electromagnetic Extended Planck’s idea of quantization to electromagnetic
radiationradiation
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The photon’s energy would be The photon’s energy would be E = hE = hƒƒEach photon can give all its energy to an electron in the metalEach photon can give all its energy to an electron in the metalThe maximum kinetic energy of the liberated photoelectron is The maximum kinetic energy of the liberated photoelectron is
KE = hKE = hƒ ƒ –– ΦΦ
ΦΦ is called the is called the work functionwork function of the metalof the metal
Explanation of Classical “Problems”Explanation of Classical “Problems”
The effect is not observed below a certain cutoff The effect is not observed below a certain cutoff frequency frequency since the photon energy must be greater than since the photon energy must be greater than or equal to the work functionor equal to the work function�� Without this, electrons are not emitted, regardless of the intensity Without this, electrons are not emitted, regardless of the intensity
of the lightof the light
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The maximum KE dependsThe maximum KE depends only on the only on the frequencyfrequency and and the work functionthe work function, not on the intensity, not on the intensityThe The maximum KE increasesmaximum KE increases with with increasing frequencyincreasing frequencyThe effect is The effect is instantaneousinstantaneous since there is a onesince there is a one--toto--one one interaction between the photon and the electroninteraction between the photon and the electron
Verification of Einstein’s TheoryVerification of Einstein’s Theory
Experimental observations of a Experimental observations of a linear relationship between KE linear relationship between KE and frequency confirm and frequency confirm Einstein’s theoryEinstein’s theory
The xThe x--intercept is the intercept is the cutoff cutoff
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The xThe x--intercept is the intercept is the cutoff cutoff frequencyfrequency
cf h
Φ=
27.3 Application: Photocells27.3 Application: Photocells
Photocells are an application of the photoelectric effectPhotocells are an application of the photoelectric effectWhen light of sufficiently high frequency falls on the cell, When light of sufficiently high frequency falls on the cell, a current is produceda current is producedExamplesExamples�� Streetlights, garage door openers, elevatorsStreetlights, garage door openers, elevators
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Streetlights, garage door openers, elevatorsStreetlights, garage door openers, elevators
27.4 The Compton Effect27.4 The Compton Effect
Compton directed a beam of xCompton directed a beam of x--rays toward a block of graphiterays toward a block of graphiteHe found that the scattered xHe found that the scattered x--rays had a slightly longer wavelength rays had a slightly longer wavelength that the incident xthat the incident x--raysrays�� This means they also had less energyThis means they also had less energy
The amount of energy reduction depended on the angle at which the The amount of energy reduction depended on the angle at which the xx--rays were scatteredrays were scattered
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xx--rays were scatteredrays were scatteredThe change in wavelength is called the The change in wavelength is called the Compton shiftCompton shift
Compton ScatteringCompton Scattering
Compton assumed the Compton assumed the photons acted like other photons acted like other particles in collisionsparticles in collisions
Energy and momentum Energy and momentum
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Energy and momentum Energy and momentum were conservedwere conserved
The shift in wavelength isThe shift in wavelength is )cos1(cm
h
eo θ−=λ−λ=λ∆
Compton wavelength
Compton ScatteringCompton Scattering
The quantity The quantity h/mh/meecc is called the is called the Compton wavelengthCompton wavelength�� 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 anglescattering angle and and not not on on the the wavelengthwavelength
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the the wavelengthwavelengthExperiments confirm the results of Compton scattering and Experiments confirm the results of Compton scattering and strongly support the photon conceptstrongly support the photon concept
Problem: Compton scatteringProblem: Compton scattering
A beam of 0.68A beam of 0.68--nm photons undergoes Compton scattering from free nm photons undergoes Compton scattering from free electrons. What are the energy and momentum of the photons that electrons. What are the energy and momentum of the photons that emerge at a 45emerge at a 45°°angle with respect to the incident beam? angle with respect to the incident beam?
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QUICK QUIZ 1
An 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.
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(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 2
A 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.
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(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 2
A 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.
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(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.
27.8 Photons and Electromagnetic Waves27.8 Photons and Electromagnetic Waves
Light has a dual nature.Light has a dual nature. It exhibits both wave and particle It exhibits both wave and particle characteristicscharacteristics�� Applies to all electromagnetic radiationApplies to all electromagnetic radiation
The The photoelectric effectphotoelectric effect and and Compton scatteringCompton scattering offer evidence for offer evidence for the the particle nature of lightparticle nature of light
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�� When light and matter interact, light behaves as if it were composed of When light and matter interact, light behaves as if it were composed of particlesparticles
InterferenceInterference and and diffractiondiffraction offer evidence of the offer evidence of the wave nature of lightwave nature of light
28.9 Wave Properties of Particles28.9 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 photons have wave and particle characteristics, perhaps all forms of matter have both propertiesall forms of matter have both properties
Furthermore, the frequency and wavelength of matter Furthermore, the frequency and wavelength of matter
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Furthermore, the frequency and wavelength of matter Furthermore, the frequency and wavelength of matter waves can be determinedwaves can be determinedThe The de Broglie wavelengthde Broglie wavelength of a particle isof a particle is
The frequency of matter waves isThe frequency of matter waves ismvh=λ
hE
ƒ =
The DavissonThe Davisson--Germer ExperimentGermer Experiment
They scattered lowThey scattered low--energy electrons from a nickel targetenergy electrons from a nickel target
They followed this with extensive They followed this with extensive diffraction measurementsdiffraction measurements from from various materialsvarious materials
The wavelength of the electrons calculated from the diffraction data The wavelength of the electrons calculated from the diffraction data
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The wavelength of the electrons calculated from the diffraction data The wavelength of the electrons calculated from the diffraction data agreed with the expected de Broglie wavelengthagreed with the expected de Broglie wavelength
This confirmed the wave nature of electronsThis confirmed the wave nature of electrons
Other experimenters have confirmed the wave nature of other Other experimenters have confirmed the wave nature of other particlesparticles
Review problem: the wavelength of a protonReview problem: the wavelength of a proton
Calculate the de Broglie wavelength for a proton (mCalculate the de Broglie wavelength for a proton (mpp=1.67x10=1.67x10--2727 kg ) kg ) moving with a speed of 1.00 x 10moving with a speed of 1.00 x 1077 m/s.m/s.
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Calculate the de Broglie wavelength for a proton (mCalculate the de Broglie wavelength for a proton (mpp=1.67x10=1.67x10--2727 kg ) moving with a kg ) moving with a speed of 1.00 x 10speed of 1.00 x 1077 m/s.m/s.
Given:
v = 1.0 x 107m/s
Given the velocity and a mass of the proton we can compute its wavelength
pp
h
m vλ =
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Find:
λp = ?
Or numerically,
( )( )( )
34
14
31 7
6.63 103.97 10
1.67 10 1.00 10ps
J sm
kg m sλ
−−
−
× ⋅= = ×
× ×
QUICK QUIZ 3
A 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?
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(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 2
A 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?
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(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.
The Electron MicroscopeThe Electron Microscope
The electron microscope depends The electron microscope depends on the wave characteristics of on the wave characteristics of electronselectronsMicroscopes can only resolve details Microscopes can only resolve details that are slightly smaller than the that are slightly smaller than the wavelength of the radiation used to wavelength of the radiation used to
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wavelength of the radiation used to wavelength of the radiation used to illuminate the objectilluminate the objectThe electrons can be accelerated to The electrons can be accelerated to high energies and have small high energies and have small wavelengthswavelengths
27.10 The Wave Function27.10 The Wave Function
In 1926 SchrIn 1926 Schrödinger proposed a ödinger proposed a wave equationwave equation that that describes the manner in which matter waves change in describes the manner in which matter waves change in space and timespace and timeSchrSchrödinger’s wave equationödinger’s wave equation is a key element in is a key element in quantum mechanicsquantum mechanics
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quantum mechanicsquantum mechanics
SchrSchrödinger’s wave equation is generally solved for the ödinger’s wave equation is generally solved for the wave functionwave function, Ψ, Ψ
i Ht
∆Ψ = Ψ∆
The Wave FunctionThe Wave Function
The wave function depends on the particle’s position and The wave function depends on the particle’s position and the timethe time
The The value of |Ψ|value of |Ψ|22 at some location at a given time is at some location at a given time is proportional to the probability of finding the particle at proportional to the probability of finding the particle at
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proportional to the probability of finding the particle at proportional to the probability of finding the particle at that location at that timethat location at that time
27.11 The Uncertainty Principle27.11 The Uncertainty Principle
When measurements are made, the experimenter is When measurements are made, the experimenter is always faced with experimental uncertainties in the always faced with experimental uncertainties in the measurementsmeasurements
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�� Classical mechanics offers no fundamental barrier to Classical mechanics offers no fundamental barrier to ultimate refinements in measurementsultimate refinements in measurements
�� Classical mechanics would allow for measurements with Classical mechanics would allow for measurements with arbitrarily small uncertaintiesarbitrarily small uncertainties
The Uncertainty PrincipleThe Uncertainty Principle
Quantum mechanics predicts that a barrier to measurements Quantum mechanics predicts that a barrier to measurements with ultimately small uncertainties does existwith ultimately small uncertainties does exist
In 1927 Heisenberg introduced the In 1927 Heisenberg introduced the uncertainty principleuncertainty principle
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�� If a measurement of position of a particle is made with precision If a measurement of position of a particle is made with precision ∆x ∆x and a simultaneous measurement of linear momentum is made with and a simultaneous measurement of linear momentum is made with precision ∆p, then the product of the two uncertainties can never be precision ∆p, then the product of the two uncertainties can never be smaller than h/4smaller than h/4ππ
The Uncertainty PrincipleThe Uncertainty Principle
Mathematically,Mathematically,
It is physically impossible to measure simultaneously the It is physically impossible to measure simultaneously the exact position and the exact linear momentum of a exact position and the exact linear momentum of a
π≥∆∆
4h
px x
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exact position and the exact linear momentum of a exact position and the exact linear momentum of a particleparticle
Another form of the principle deals with energy and time: Another form of the principle deals with energy and time:
π≥∆∆
4h
tE
Thought Experiment Thought Experiment –– the Uncertainty the Uncertainty PrinciplePrinciple
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A thought experiment for viewing an electron with a powerful A thought experiment for viewing an electron with a powerful microscopemicroscopeIn 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 itDuring this interaction, momentum is transferred from the photon to During this interaction, momentum is transferred from the photon to the electronthe electronTherefore, the light that allows you to accurately locate the electron Therefore, the light that allows you to accurately locate the electron changes the momentum of the electronchanges the momentum of the electron
Problem: macroscopic uncertaintyProblem: macroscopic uncertainty
A 50.0A 50.0--g ball moves at 30.0 m/s. If its speed is measured to an g ball moves at 30.0 m/s. If its speed is measured to an accuracy of 0.10%, what is the minimum uncertainty in its accuracy of 0.10%, what is the minimum uncertainty in its position?position?
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A 50.0A 50.0--g ball moves at 30.0 m/s. If its speed is measured to an accuracy of 0.10%, g ball moves at 30.0 m/s. If its speed is measured to an accuracy of 0.10%, what is the minimum uncertainty in its position?what is the minimum uncertainty in its position?
Given:
v = 30 m/sδv = 0.10%m = 50.0 g
Notice that the ball is non-relativistic. Thus, p = mv, and uncertainty in measuring momentum is
( ) ( )( )( )2 3 250.0 10 1.0 10 30 1.5 10
p m v m v v
kg m s kg m s
δ− − −
∆ = ∆ = ⋅
= × × ⋅ = × ⋅
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m = 50.0 g
Find:
δx = ?
( )( )50.0 10 1.0 10 30 1.5 10kg m s kg m s= × × ⋅ = × ⋅
Thus, uncertainty relation implies
( ) ( )24
32
3
6.63 103.5 10
4 4 1.5 10
h J sx m
p kg m sπ π
−−
−
× ⋅∆ ≥ = = ×∆ × ⋅
Problem: Macroscopic measurementProblem: Macroscopic measurement
A 0.50-kg block rests on the icy surface of a frozen pond, which we can assume to be frictionless. If the location of the block is measured to a precision of 0.50 cm, what speed must the block acquire because
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to a precision of 0.50 cm, what speed must the block acquire because of the measurement process?
