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Chapter 40 Introduction to Quantum Physics
Need for Quantum Physics
Problems remained from classical mechanics that relativity didn’t explain Attempts to apply the laws of classical physics to explain the behavior of matter on the atomic scale were consistently unsuccessful Problems included:
blackbody radiation
The electromagnetic radiation emitted by a heated object
photoelectric effect
Emission of electrons by an illuminated metal
Quantum Mechanics Revolution
Between 1900 and 1930, another revolution took place in physics A new theory called quantum mechanics was successful in explaining the behavior of particles of microscopic size The first explanation using quantum mechanics was introduced by Max Planck
Many other physicists were involved in other subsequent developments
Blackbody Radiation
An object at any temperature is known to emit thermal radiation
Characteristics depend on the temperature and surface properties The thermal radiation consists of a continuous distribution of wavelengths from all portions of the em spectrum
Blackbody Radiation, cont.
At room temperature, the wavelengths of the thermal radiation are mainly in the infrared region As the surface temperature increases, the wavelength changes
It will glow red and eventually white
The basic problem was in understanding the observed distribution in the radiation emitted by a black body
Classical physics didn’t adequately describe the observed distribution
Blackbody Radiation, final
A black body is an ideal system that absorbs all radiation incident on it The electromagnetic radiation emitted by a black body is called blackbody radiation
Blackbody Approximation
A good approximation of a black body is a small hole leading to the inside of a hollow object The hole acts as a perfect absorber The nature of the radiation leaving the cavity through the hole depends only on the temperature of the cavity
Blackbody Experiment Results
The total power of the emitted radiation increases with temperature
Stefan’s law (from Chapter 20): P = σAeT4
The peak of the wavelength distribution shifts to shorter wavelengths as the temperature increases
Wien’s displacement law λmaxT = 2.898 x 10-3 m.K
Stefan’s Law – Details
P = σAeT4
P is the power σ is the Stefan-Boltzmann constant
σ = 5.670 x 10-8 W / m2 . K4
Stefan’s law can be written in terms of intensity
I = P/A = σT4
For a blackbody, where e = 1
Wien’s Displacement Law
λmaxT = 2.898 x 10-3 m.K
λmax is the wavelength at which the curve peaks T is the absolute temperature
The wavelength is inversely proportional to the absolute temperature
As the temperature increases, the peak is “displaced” to shorter wavelengths
Intensity of Blackbody Radiation, Summary
The intensity increases with increasing temperature The amount of radiation emitted increases with increasing temperature
The area under the curve
The peak wavelength decreases with increasing temperature
Active Figure 40.3
(SLIDESHOW MODE ONLY)
Rayleigh-Jeans Law
An early classical attempt to explain blackbody radiation was the RayleighJeans law 2πck BT I λ,T λ4
At long wavelengths, the law matched experimental results fairly well
Rayleigh-Jeans Law, cont.
At short wavelengths, there was a major disagreement between the Rayleigh-Jeans law and experiment This mismatch became known as the ultraviolet catastrophe
You would have infinite energy as the wavelength approaches zero
Max Planck
Introduced the concept of “quantum of action” In 1918 he was awarded the Nobel Prize for the discovery of the quantized nature of energy
Planck’s Theory of Blackbody Radiation
In 1900 Planck developed a theory of blackbody radiation that leads to an equation for the intensity of the radiation This equation is in complete agreement with experimental observations He assumed the cavity radiation came from atomic oscillations in the cavity walls Planck made two assumptions about the nature of the oscillators in the cavity walls
Planck’s Assumption, 1
The energy of an oscillator can have only certain discrete values En
En = nhƒ
n is a positive integer called the quantum number h is Planck’s constant ƒ is the frequency of oscillation
This says the energy is quantized Each discrete energy value corresponds to a different quantum state
Planck’s Assumption, 2
The oscillators emit or absorb energy when making a transition from one quantum state to another
The entire energy difference between the initial and final states in the transition is emitted or absorbed as a single quantum of radiation An oscillator emits or absorbs energy only when it changes quantum states
Energy-Level Diagram
An energy-level diagram shows the quantized energy levels and allowed transitions Energy is on the vertical axis Horizontal lines represent the allowed energy levels The double-headed arrows indicate allowed transitions
More About Planck’s Model
The average energy of a wave is the average energy difference between levels of the oscillator, weighted according to the probability of the wave being emitted This weighting is described by the Boltzmann distribution law and gives the probability of a state being occupied as being proportional to E kBT where E is the energy of the state e
Planck’s Model, Graphs
Active Figure 40.7
(SLIDESHOW MODE ONLY)
Planck’s Wavelength Distribution Function
Planck generated a theoretical expression for the wavelength distribution 2πhc 2 I λ,T 5 hcλk T B λ e 1
h = 6.626 x 10-34 J.s h is a fundamental constant of nature
Planck’s Wavelength Distribution Function, cont.
At long wavelengths, Planck’s equation reduces to the Rayleigh-Jeans expression At short wavelengths, it predicts an exponential decrease in intensity with decreasing wavelength
This is in agreement with experimental results
Photoelectric Effect
The photoelectric effect occurs when light incident on certain metallic surfaces causes electrons to be emitted from those surfaces
The emitted electrons are called photoelectrons
Photoelectric Effect Apparatus
When the tube is kept in the dark, the ammeter reads zero When plate E is illuminated by light having an appropriate wavelength, a current is detected by the ammeter The current arises from photoelectrons emitted from the negative plate and collected at the positive plate
Active Figure 40.9
(SLIDESHOW MODE ONLY)
Photoelectric Effect, Results
At large values of ∆V, the current reaches a maximum value
All the electrons emitted at E are collected at C
The maximum current increases as the intensity of the incident light increases When ∆V is negative, the current drops When ∆V is equal to or more negative than ∆Vs, the current is zero
Photoelectric Effect Feature 1
Dependence of photoelectron kinetic energy on light intensity
Classical Prediction
Electrons should absorb energy continually from the electromagnetic waves As the light intensity incident on the metal is increased, the electrons should be ejected with more kinetic energy
Experimental Result
The maximum kinetic energy is independent of light intensity The maximum kinetic energy is proportional to the stopping potential (∆Vs)
Photoelectric Effect Feature 2
Time interval between incidence of light and ejection of photoelectrons
Classical Prediction
At low light intensities, a measurable time interval should pass between the instant the light is turned on and the time an electron is ejected from the metal This time interval is required for the electron to absorb the incident radiation before it acquires enough energy to escape from the metal
Experimental Result
Electrons are emitted almost instantaneously, even at very low light intensities
Photoelectric Effect Feature 3
Dependence of ejection of electrons on light frequency
Classical Prediction
Electrons should be ejected at any frequency as long as the light intensity is high enough
Experimental Result
No electrons are emitted if the incident light falls below some cutoff frequency, ƒc The cutoff frequency is characteristic of the material being illuminated No electrons are ejected below the cutoff frequency regardless of intensity
Photoelectric Effect Feature 4
Dependence of photoelectron kinetic energy on light frequency
Classical Prediction
There should be no relationship between the frequency of the light and the electric kinetic energy The kinetic energy should be related to the intensity of the light
Experimental Result
The maximum kinetic energy of the photoelectrons increases with increasing light frequency
Photoelectric Effect Features, Summary
The experimental results contradict all four classical predictions Einstein extended Planck’s concept of quantization to electromagnetic waves All electromagnetic radiation can be considered a stream of quanta, now called photons A photon of incident light gives all its energy hƒ to a single electron in the metal
Photoelectric Effect, Work Function
Electrons ejected from the surface of the metal and not making collisions with other metal atoms before escaping possess the maximum kinetic energy Kmax
Kmax = hƒ – φ
φ is called the work function The work function represents the minimum energy with which an electron is bound in the metal
Some Work Function Values
Photon Model Explanation of the Photoelectric Effect
Dependence of photoelectron kinetic energy on light intensity
Kmax is independent of light intensity K depends on the light frequency and the work function
Time interval between incidence of light and ejection of the photoelectron
Each photon can have enough energy to eject an electron immediately
Photon Model Explanation of the Photoelectric Effect, cont.
Dependence of ejection of electrons on light frequency
There is a failure to observe photoelectric effect below a certain cutoff frequency, which indicates the photon must have more energy than the work function in order to eject an electron Without enough energy, an electron cannot be ejected, regardless of the light intensity
Photon Model Explanation of the Photoelectric Effect, final
Dependence of photoelectron kinetic energy on light frequency
Since Kmax = hƒ – φ As the frequency increases, the kinetic energy will increase
Once the energy of the work function is exceeded
There is a linear relationship between the kinetic energy and the frequency
Cutoff Frequency
The lines show the linear relationship between K and ƒ The slope of each line is h The x-intercept is the cutoff frequency
This is the frequency below which no photoelectrons are emitted
Cutoff Frequency and Wavelength
The cutoff frequency is related to the work function through ƒc = φ / h The cutoff frequency corresponds to a cutoff wavelength c hc λc ƒc φ Wavelengths greater than λc incident on a material having a work function φ do not result in the emission of photoelectrons
Arthur Holly Compton
1892 - 1962 Director of the lab at the University of Chicago Discovered the Compton Effect Shared the Nobel Prize in 1927
The Compton Effect, Introduction
Compton and Debye dealt with Einstein’s idea of photon momentum The two groups of experimenters accumulated evidence of the inadequacy of the classical wave theory The classical wave theory of light failed to explain the scattering of x-rays from electrons
Compton Effect, Classical Predictions
According to the classical theory, em waves incident on electrons should:
have radiation pressure that should cause the electrons to accelerate set the electrons oscillating
Compton Effect, Observations
Compton’s experiments showed that, at any given angle, only one frequency of radiation is observed
Compton Effect, Explanation
The results could be explained by treating the photons as point-like particles having energy hƒ Assume the energy and momentum of the isolated system of the colliding photon-electron are conserved This scattering phenomena is known as the Compton effect
Compton Shift Equation
The graphs show the scattered x-ray for various angles The shifted peak, λ’ is caused by the scattering of free electrons h λ' λo 1 cos θ mec
This is called the Compton shift equation
Compton Wavelength
The unshifted wavelength, λo, is caused by x-rays scattered from the electrons that are tightly bound to the target atoms The Compton wavelength is h 0002 . 43 nm mec
Photons and Waves Revisited
Some experiments are best explained by the photon model Some are best explained by the wave model We must accept both models and admit that the true nature of light is not describable in terms of any single classical model Also, the particle model and the wave model of light complement each other
Louis de Broglie
1892 – 1987 Originally studied history Was awarded the Nobel Prize in 1929 for his prediction of the wave nature of electrons
Wave Properties of Particles
Louis de Broglie postulated that because photons have both wave and particle characteristics, perhaps all forms of matter have both properties The de Broglie wavelength of a particle is h h λ p mv
Frequency of a Particle
In an analogy with photons, de Broglie postulated that a particle would also have a frequency associated with it
E ƒ h
These equations present the dual nature of matter
particle nature, m and v wave nature, λ and ƒ
Davisson-Germer Experiment
If particles have a wave nature, then under the correct conditions, they should exhibit diffraction effects Davisson and Germer measured the wavelength of electrons This provided experimental confirmation of the matter waves proposed by de Broglie
Complementarity
The principle of complementarity states that the wave and particle models of either matter or radiation complement each other Neither model can be used exclusively to describe matter or radiation adequately
Electron Microscope
The electron microscope depends on the wave characteristics of electrons The electron microscope has a high resolving power because it has a very short wavelength Typically, the wavelengths of the electrons are about 100 times shorter than that of visible light
Quantum Particle
The quantum particle is a new model that is a result of the recognition of the dual nature Entities have both particle and wave characteristics We must choose one appropriate behavior in order to understand a particular phenomenon
Active Figure 40.20
(SLIDESHOW MODE ONLY)
Ideal Particle vs. Ideal Wave
An ideal particle has zero size
An ideal wave has a single frequency and is infinitely long
Therefore, it is localized in space
Therefore,it is unlocalized in space
A localized entity can be built from infinitely long waves
Particle as a Wave Packet
Multiple waves are superimposed so that one of its crests is at x = 0 The result is that all the waves add constructively at x = 0 There is destructive interference at every point except x = 0 The small region of constructive interference is called a wave packet
The wave packet can be identified as a particle
Wave Envelope
The blue line represents the envelope function This envelope can travel through space with a different speed than the individual waves
Active Figure 40.21
(SLIDESHOW MODE ONLY)
Speeds Associated with Wave Packet
The phase speed of a wave in a wave packet is given by
v phase ω
k
This is the rate of advance of a crest on a single wave
The group speed is given by
v g dω
dk
This is the speed of the wave packet itself
Speeds, cont.
The group speed can also be expressed in terms of energy and momentum dE d p 2 1 vg 2p u dp dp 2m 2m
This indicates that the group speed of the wave packet is identical to the speed of the particle that it is modeled to represent
Electron Diffraction, Set-Up
Electron Diffraction, Experiment
Parallel beams of mono-energetic electrons that are incident on a double slit The slit widths are small compared to the electron wavelength An electron detector is positioned far from the slits at a distance much greater than the slit separation
Electron Diffraction, cont.
If the detector collects electrons for a long enough time, a typical wave interference pattern is produced This is distinct evidence that electrons are interfering, a wave-like behavior The interference pattern becomes clearer as the number of electrons reaching the screen increases
Active Figure 40.23
(SLIDESHOW MODE ONLY)
Electron Diffraction, Equations
A minimum occurs when
dθsin
λ 2
or θsinθ
h 2 pd
This shows the dual nature of the electron
The electrons are detected as particles at a localized spot at some instant of time The probability of arrival at that spot is determined by finding the intensity of two interfering waves
Electron Diffraction, Closed Slits
If one slit is closed, the maximum is centered around the opening Closing the other slit produces another maximum centered around that opening The total effect is the blue line It is completely different from the interference pattern (red curve)
Electron Diffraction Explained
An electron interacts with both slits simultaneously If an attempt is made to determine experimentally which slit the electron goes through, the act of measuring destroys the interference pattern
It is impossible to determine which slit the electron goes through
In effect, the electron goes through both slits
The wave components of the electron are present at both slits at the same time
Werner Heisenberg
1901 – 1976 Developed matrix mechanics Many contributions include:
Uncertainty principle
Rec’d Nobel Prize in 1932
Prediction of two forms of molecular hydrogen Theoretical models of the nucleus
The Uncertainty Principle, Introduction
In classical mechanics, it is possible, in principle, to make measurements with arbitrarily small uncertainty Quantum theory predicts that it is fundamentally impossible to make simultaneous measurements of a particle’s position and momentum with infinite accuracy
Heisenberg Uncertainty Principle, Statement
The Heisenberg uncertainty principle states: if a measurement of the position of a particle is made with uncertainty ∆x and a simultaneous measurement of its x component of momentum is made with uncertainty ∆p, the product of the two uncertainties can never be smaller than h xpx
2
Heisenberg Uncertainty Principle, Explained
It is physically impossible to measure simultaneously the exact position and exact momentum of a particle The inescapable uncertainties do not arise from imperfections in practical measuring instruments The uncertainties arise from the quantum structure of matter
Heisenberg Uncertainty Principle, Another Form
Another form of the uncertainty principle can be expressed in terms of energy and time h E t 2
This suggests that energy conservation can appear to be violated by an amount ∆E as long as it is only for a short time interval ∆t
Need for Quantum Physics
Problems remained from classical mechanics that relativity didn’t explain Attempts to apply the laws of classical physics to explain the behavior of matter on the atomic scale were consistently unsuccessful Problems included:
blackbody radiation
The electromagnetic radiation emitted by a heated object
photoelectric effect
Emission of electrons by an illuminated metal
Quantum Mechanics Revolution
Between 1900 and 1930, another revolution took place in physics A new theory called quantum mechanics was successful in explaining the behavior of particles of microscopic size The first explanation using quantum mechanics was introduced by Max Planck
Many other physicists were involved in other subsequent developments
Blackbody Radiation
An object at any temperature is known to emit thermal radiation
Characteristics depend on the temperature and surface properties The thermal radiation consists of a continuous distribution of wavelengths from all portions of the em spectrum
Blackbody Radiation, cont.
At room temperature, the wavelengths of the thermal radiation are mainly in the infrared region As the surface temperature increases, the wavelength changes
It will glow red and eventually white
The basic problem was in understanding the observed distribution in the radiation emitted by a black body
Classical physics didn’t adequately describe the observed distribution
Blackbody Radiation, final
A black body is an ideal system that absorbs all radiation incident on it The electromagnetic radiation emitted by a black body is called blackbody radiation
Blackbody Approximation
A good approximation of a black body is a small hole leading to the inside of a hollow object The hole acts as a perfect absorber The nature of the radiation leaving the cavity through the hole depends only on the temperature of the cavity
Blackbody Experiment Results
The total power of the emitted radiation increases with temperature
Stefan’s law (from Chapter 20): P = σAeT4
The peak of the wavelength distribution shifts to shorter wavelengths as the temperature increases
Wien’s displacement law λmaxT = 2.898 x 10-3 m.K
Stefan’s Law – Details
P = σAeT4
P is the power σ is the Stefan-Boltzmann constant
σ = 5.670 x 10-8 W / m2 . K4
Stefan’s law can be written in terms of intensity
I = P/A = σT4
For a blackbody, where e = 1
Wien’s Displacement Law
λmaxT = 2.898 x 10-3 m.K
λmax is the wavelength at which the curve peaks T is the absolute temperature
The wavelength is inversely proportional to the absolute temperature
As the temperature increases, the peak is “displaced” to shorter wavelengths
Intensity of Blackbody Radiation, Summary
The intensity increases with increasing temperature The amount of radiation emitted increases with increasing temperature
The area under the curve
The peak wavelength decreases with increasing temperature
Active Figure 40.3
(SLIDESHOW MODE ONLY)
Rayleigh-Jeans Law
An early classical attempt to explain blackbody radiation was the RayleighJeans law 2πck BT I λ,T λ4
At long wavelengths, the law matched experimental results fairly well
Rayleigh-Jeans Law, cont.
At short wavelengths, there was a major disagreement between the Rayleigh-Jeans law and experiment This mismatch became known as the ultraviolet catastrophe
You would have infinite energy as the wavelength approaches zero
Max Planck
Introduced the concept of “quantum of action” In 1918 he was awarded the Nobel Prize for the discovery of the quantized nature of energy
Planck’s Theory of Blackbody Radiation
In 1900 Planck developed a theory of blackbody radiation that leads to an equation for the intensity of the radiation This equation is in complete agreement with experimental observations He assumed the cavity radiation came from atomic oscillations in the cavity walls Planck made two assumptions about the nature of the oscillators in the cavity walls
Planck’s Assumption, 1
The energy of an oscillator can have only certain discrete values En
En = nhƒ
n is a positive integer called the quantum number h is Planck’s constant ƒ is the frequency of oscillation
This says the energy is quantized Each discrete energy value corresponds to a different quantum state
Planck’s Assumption, 2
The oscillators emit or absorb energy when making a transition from one quantum state to another
The entire energy difference between the initial and final states in the transition is emitted or absorbed as a single quantum of radiation An oscillator emits or absorbs energy only when it changes quantum states
Energy-Level Diagram
An energy-level diagram shows the quantized energy levels and allowed transitions Energy is on the vertical axis Horizontal lines represent the allowed energy levels The double-headed arrows indicate allowed transitions
More About Planck’s Model
The average energy of a wave is the average energy difference between levels of the oscillator, weighted according to the probability of the wave being emitted This weighting is described by the Boltzmann distribution law and gives the probability of a state being occupied as being proportional to E kBT where E is the energy of the state e
Planck’s Model, Graphs
Active Figure 40.7
(SLIDESHOW MODE ONLY)
Planck’s Wavelength Distribution Function
Planck generated a theoretical expression for the wavelength distribution 2πhc 2 I λ,T 5 hcλk T B λ e 1
h = 6.626 x 10-34 J.s h is a fundamental constant of nature
Planck’s Wavelength Distribution Function, cont.
At long wavelengths, Planck’s equation reduces to the Rayleigh-Jeans expression At short wavelengths, it predicts an exponential decrease in intensity with decreasing wavelength
This is in agreement with experimental results
Photoelectric Effect
The photoelectric effect occurs when light incident on certain metallic surfaces causes electrons to be emitted from those surfaces
The emitted electrons are called photoelectrons
Photoelectric Effect Apparatus
When the tube is kept in the dark, the ammeter reads zero When plate E is illuminated by light having an appropriate wavelength, a current is detected by the ammeter The current arises from photoelectrons emitted from the negative plate and collected at the positive plate
Active Figure 40.9
(SLIDESHOW MODE ONLY)
Photoelectric Effect, Results
At large values of ∆V, the current reaches a maximum value
All the electrons emitted at E are collected at C
The maximum current increases as the intensity of the incident light increases When ∆V is negative, the current drops When ∆V is equal to or more negative than ∆Vs, the current is zero
Photoelectric Effect Feature 1
Dependence of photoelectron kinetic energy on light intensity
Classical Prediction
Electrons should absorb energy continually from the electromagnetic waves As the light intensity incident on the metal is increased, the electrons should be ejected with more kinetic energy
Experimental Result
The maximum kinetic energy is independent of light intensity The maximum kinetic energy is proportional to the stopping potential (∆Vs)
Photoelectric Effect Feature 2
Time interval between incidence of light and ejection of photoelectrons
Classical Prediction
At low light intensities, a measurable time interval should pass between the instant the light is turned on and the time an electron is ejected from the metal This time interval is required for the electron to absorb the incident radiation before it acquires enough energy to escape from the metal
Experimental Result
Electrons are emitted almost instantaneously, even at very low light intensities
Photoelectric Effect Feature 3
Dependence of ejection of electrons on light frequency
Classical Prediction
Electrons should be ejected at any frequency as long as the light intensity is high enough
Experimental Result
No electrons are emitted if the incident light falls below some cutoff frequency, ƒc The cutoff frequency is characteristic of the material being illuminated No electrons are ejected below the cutoff frequency regardless of intensity
Photoelectric Effect Feature 4
Dependence of photoelectron kinetic energy on light frequency
Classical Prediction
There should be no relationship between the frequency of the light and the electric kinetic energy The kinetic energy should be related to the intensity of the light
Experimental Result
The maximum kinetic energy of the photoelectrons increases with increasing light frequency
Photoelectric Effect Features, Summary
The experimental results contradict all four classical predictions Einstein extended Planck’s concept of quantization to electromagnetic waves All electromagnetic radiation can be considered a stream of quanta, now called photons A photon of incident light gives all its energy hƒ to a single electron in the metal
Photoelectric Effect, Work Function
Electrons ejected from the surface of the metal and not making collisions with other metal atoms before escaping possess the maximum kinetic energy Kmax
Kmax = hƒ – φ
φ is called the work function The work function represents the minimum energy with which an electron is bound in the metal
Some Work Function Values
Photon Model Explanation of the Photoelectric Effect
Dependence of photoelectron kinetic energy on light intensity
Kmax is independent of light intensity K depends on the light frequency and the work function
Time interval between incidence of light and ejection of the photoelectron
Each photon can have enough energy to eject an electron immediately
Photon Model Explanation of the Photoelectric Effect, cont.
Dependence of ejection of electrons on light frequency
There is a failure to observe photoelectric effect below a certain cutoff frequency, which indicates the photon must have more energy than the work function in order to eject an electron Without enough energy, an electron cannot be ejected, regardless of the light intensity
Photon Model Explanation of the Photoelectric Effect, final
Dependence of photoelectron kinetic energy on light frequency
Since Kmax = hƒ – φ As the frequency increases, the kinetic energy will increase
Once the energy of the work function is exceeded
There is a linear relationship between the kinetic energy and the frequency
Cutoff Frequency
The lines show the linear relationship between K and ƒ The slope of each line is h The x-intercept is the cutoff frequency
This is the frequency below which no photoelectrons are emitted
Cutoff Frequency and Wavelength
The cutoff frequency is related to the work function through ƒc = φ / h The cutoff frequency corresponds to a cutoff wavelength c hc λc ƒc φ Wavelengths greater than λc incident on a material having a work function φ do not result in the emission of photoelectrons
Arthur Holly Compton
1892 - 1962 Director of the lab at the University of Chicago Discovered the Compton Effect Shared the Nobel Prize in 1927
The Compton Effect, Introduction
Compton and Debye dealt with Einstein’s idea of photon momentum The two groups of experimenters accumulated evidence of the inadequacy of the classical wave theory The classical wave theory of light failed to explain the scattering of x-rays from electrons
Compton Effect, Classical Predictions
According to the classical theory, em waves incident on electrons should:
have radiation pressure that should cause the electrons to accelerate set the electrons oscillating
Compton Effect, Observations
Compton’s experiments showed that, at any given angle, only one frequency of radiation is observed
Compton Effect, Explanation
The results could be explained by treating the photons as point-like particles having energy hƒ Assume the energy and momentum of the isolated system of the colliding photon-electron are conserved This scattering phenomena is known as the Compton effect
Compton Shift Equation
The graphs show the scattered x-ray for various angles The shifted peak, λ’ is caused by the scattering of free electrons h λ' λo 1 cos θ mec
This is called the Compton shift equation
Compton Wavelength
The unshifted wavelength, λo, is caused by x-rays scattered from the electrons that are tightly bound to the target atoms The Compton wavelength is h 0002 . 43 nm mec
Photons and Waves Revisited
Some experiments are best explained by the photon model Some are best explained by the wave model We must accept both models and admit that the true nature of light is not describable in terms of any single classical model Also, the particle model and the wave model of light complement each other
Louis de Broglie
1892 – 1987 Originally studied history Was awarded the Nobel Prize in 1929 for his prediction of the wave nature of electrons
Wave Properties of Particles
Louis de Broglie postulated that because photons have both wave and particle characteristics, perhaps all forms of matter have both properties The de Broglie wavelength of a particle is h h λ p mv
Frequency of a Particle
In an analogy with photons, de Broglie postulated that a particle would also have a frequency associated with it
E ƒ h
These equations present the dual nature of matter
particle nature, m and v wave nature, λ and ƒ
Davisson-Germer Experiment
If particles have a wave nature, then under the correct conditions, they should exhibit diffraction effects Davisson and Germer measured the wavelength of electrons This provided experimental confirmation of the matter waves proposed by de Broglie
Complementarity
The principle of complementarity states that the wave and particle models of either matter or radiation complement each other Neither model can be used exclusively to describe matter or radiation adequately
Electron Microscope
The electron microscope depends on the wave characteristics of electrons The electron microscope has a high resolving power because it has a very short wavelength Typically, the wavelengths of the electrons are about 100 times shorter than that of visible light
Quantum Particle
The quantum particle is a new model that is a result of the recognition of the dual nature Entities have both particle and wave characteristics We must choose one appropriate behavior in order to understand a particular phenomenon
Active Figure 40.20
(SLIDESHOW MODE ONLY)
Ideal Particle vs. Ideal Wave
An ideal particle has zero size
An ideal wave has a single frequency and is infinitely long
Therefore, it is localized in space
Therefore,it is unlocalized in space
A localized entity can be built from infinitely long waves
Particle as a Wave Packet
Multiple waves are superimposed so that one of its crests is at x = 0 The result is that all the waves add constructively at x = 0 There is destructive interference at every point except x = 0 The small region of constructive interference is called a wave packet
The wave packet can be identified as a particle
Wave Envelope
The blue line represents the envelope function This envelope can travel through space with a different speed than the individual waves
Active Figure 40.21
(SLIDESHOW MODE ONLY)
Speeds Associated with Wave Packet
The phase speed of a wave in a wave packet is given by
v phase ω
k
This is the rate of advance of a crest on a single wave
The group speed is given by
v g dω
dk
This is the speed of the wave packet itself
Speeds, cont.
The group speed can also be expressed in terms of energy and momentum dE d p 2 1 vg 2p u dp dp 2m 2m
This indicates that the group speed of the wave packet is identical to the speed of the particle that it is modeled to represent
Electron Diffraction, Set-Up
Electron Diffraction, Experiment
Parallel beams of mono-energetic electrons that are incident on a double slit The slit widths are small compared to the electron wavelength An electron detector is positioned far from the slits at a distance much greater than the slit separation
Electron Diffraction, cont.
If the detector collects electrons for a long enough time, a typical wave interference pattern is produced This is distinct evidence that electrons are interfering, a wave-like behavior The interference pattern becomes clearer as the number of electrons reaching the screen increases
Active Figure 40.23
(SLIDESHOW MODE ONLY)
Electron Diffraction, Equations
A minimum occurs when
dθsin
λ 2
or θsinθ
h 2 pd
This shows the dual nature of the electron
The electrons are detected as particles at a localized spot at some instant of time The probability of arrival at that spot is determined by finding the intensity of two interfering waves
Electron Diffraction, Closed Slits
If one slit is closed, the maximum is centered around the opening Closing the other slit produces another maximum centered around that opening The total effect is the blue line It is completely different from the interference pattern (red curve)
Electron Diffraction Explained
An electron interacts with both slits simultaneously If an attempt is made to determine experimentally which slit the electron goes through, the act of measuring destroys the interference pattern
It is impossible to determine which slit the electron goes through
In effect, the electron goes through both slits
The wave components of the electron are present at both slits at the same time
Werner Heisenberg
1901 – 1976 Developed matrix mechanics Many contributions include:
Uncertainty principle
Rec’d Nobel Prize in 1932
Prediction of two forms of molecular hydrogen Theoretical models of the nucleus
The Uncertainty Principle, Introduction
In classical mechanics, it is possible, in principle, to make measurements with arbitrarily small uncertainty Quantum theory predicts that it is fundamentally impossible to make simultaneous measurements of a particle’s position and momentum with infinite accuracy
Heisenberg Uncertainty Principle, Statement
The Heisenberg uncertainty principle states: if a measurement of the position of a particle is made with uncertainty ∆x and a simultaneous measurement of its x component of momentum is made with uncertainty ∆p, the product of the two uncertainties can never be smaller than h xpx
2
Heisenberg Uncertainty Principle, Explained
It is physically impossible to measure simultaneously the exact position and exact momentum of a particle The inescapable uncertainties do not arise from imperfections in practical measuring instruments The uncertainties arise from the quantum structure of matter
Heisenberg Uncertainty Principle, Another Form
Another form of the uncertainty principle can be expressed in terms of energy and time h E t 2
This suggests that energy conservation can appear to be violated by an amount ∆E as long as it is only for a short time interval ∆t
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