Particle Wave Duality What is a particle? What is a wave?
Problems with Classical Physics Nature of Light? Discrete Spectra? Blackbody Radiation? Photoelectric Effect? Compton Effect? Model of Atom?
Thomas Young 1804 Double Slit is VERY IMPORTANT because it is evidence of waves. Only waves interfere like this. dsin m
REVIEW! Derive Fringe Equations For bright fringes λl ybright m ( m 0, 1, 2 ) d For dark fringes λl 1 ydark m ( m 0, 1, 2 ) d 2
Double Slit for Electrons shows Wave Interference! Key to Quantum Theory!
James Clerk Maxwell 1860s Light is an electromagnetic wave. The medium is the Ether. 1 8 c 3.0x10 m / s 0 o
The Electromagnetic Spectrum
Michelson-Morely Experiment 1887 The speed of light is independent of the motion and is always c. The speed of the Ether wind is zero. OR. Lorentz Contraction The apparatus shrinks by a factor : 1 v / c 2 2
Special Relativity 1905 2 2 E mc E p mu 0 mc u c 2 2 pc muc m c ( mc ) E u c 2 2 p E ( pc) E E mc u u E mc E mc E E E E E E u c c 1 2 c 2 2 2 2 2 2 2 2 2 2 2 2 2 2 (1 ) ( ) 2 2 2 0 0 2 0 If m = 0 (photon) photon momentum: E ( pc) ( m c ) ( pc) 2 2 2 2 2 0 p E / c
Why Continuous vs Discrete? This is a continuous spectrum of colors: all colors are present. This is a discrete spectrum of colors: only a few are present.
Kirkoff s Rules
Kirkoff s Rules for Spectra: 1859 German physicist who developed the spectroscope and the science of emission spectroscopy with Bunsen. Kirkoff Bunsen * Rule 1 : A hot and opaque solid, liquid or highly compressed gas emits a continuous spectrum. * Rule 2 : A hot, transparent gas produces an emission spectrum with bright lines. * Rule 3 : If a continuous spectrum passes through a gas at a lower temperature, the transparent cooler gas generates dark absorption lines.
Compare absorption lines in a source with emission lines found in the laboratory! Kirchhoff deduced that elements were present in the atmosphere of the Sun and were absorbing their characteristic wavelengths, producing the absorption lines in the solar spectrum. He published in 1861 the first atlas of the solar spectrum, obtained with a prism ; however, these wavelengths were not very precise : the dispersion of the prism was not linear at all.
Anders Jonas Ångström 1869 Ångström measured the wavelengths on the four visible lines of the hydrogen spectrum, obtained with a diffraction grating, whose dispersion is linear, and replaced Kirchhoff's arbitrary scale by the wavelengths, expressed in the metric system, using a small unit (10-10 m) with which his name was to be associated. Line color red blue-green violet violet Wavelength 6562.852 Å 4861.33 Å 4340.47 Å 4101.74 Å
Balmer Series: 1885 Johann Balmer found an empirical equation that correctly predicted the four visible emission lines of hydrogen Johannes Robert Rydberg generalized it in 1888 for all transitions: 1 R 1 1 λ 2 n H 2 2 R H is the Rydberg constant R H = 1.097 373 2 x 10 7 m -1 n is an integer, n = 3, 4, 5, The spectral lines correspond to different values of n H α is red, λ = 656.3 nm H β is green, λ = 486.1 nm H γ is blue, λ = 434.1 nm H δ is violet, λ = 410.2 nm
Why this shape? Why the drop?
All objects radiate energy continuously in the form of electromagnetic waves due to thermal vibrations of their molecules. When an object it heated it will glow first in the infrared, then the visible. Most solid materials break down before they emit UV and higher frequency EM waves. Long Frequency ~ Temperature Short
A good absorber reflects little and appears Black A good absorber is also a good emitter.
Blackbody Radiation 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 c
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
Stefan s Law: 1879 Rate of radiation of a Black Body P = σaet 4 P is the rate of energy transfer, in Watts σ = 5.6696 x 10-8 W/m 2. K 4 Jožef Stefan A is the surface area of the object (1835 1893) e is a constant called the emissivity e varies from 0 to 1 The emissivity is also equal to the absorptivity T is the temperature in Kelvins With his law Stefan determined the temperature of the Sun s surface and he calculated a value of 5430C. This was the first sensible value for the temperature of the Sun. Boltzmann was his student and derived Stefan s Law from Thermodynamics in 1884 and extended it to grey bodies.
Maxwell-Boltzmann Distribution: 1877 The observed speed distribution of gas molecules in thermal equilibrium is shown at right N V is called the Maxwell- Boltzmann speed distribution function The distribution of speeds in N gas molecules is The probability of finding the molecule in a particular energy state varies exponentially as the negative of the energy divided by k B T n V (E ) = n o e E /k B T m 3 / 2 o 2 mv NV 4 N v e 2 kt B Ludwig Boltzmann 1844 1906 2 /2k T B
Temperature ~ Ave KE of each particle Particles have different speeds Gas Particles are in constant RANDOM motion Equipartition of Energy: Average KE of each particle is: 3/2 kt Pressure is due to momentum transfer Speed Distribution at CONSTANT Temperature is given by the Maxwell Speed Distribution 2 3/ 2kT KE 1/ 2mvrms k =1.38 x 10-23 J/K Boltzmann s Constant
P 4 e T A Radiant heat makes it impossible to stand close to a hot lava flow. Calculate the rate of heat loss by radiation from 1.00 m 2 of 1200C fresh lava into 30.0C surroundings, assuming lava s emissivity is 1. The net heat transfer by radiation is: 4 4 2 1 P e A( T T ) 4 4 2 1 P e A( T T ) 8 4 2 4 4 1(5.67 x10 J / smk )1 m ((303.15 K) (1473.15 K) ) P 266kW
Blackbody Experiment Results The total power of the radiation emitted from the surface increases with temperature Stefan s law: P = AeT 4 P is the power and is the Stefan-Boltzmann constant: = 5.670 x 10-8 W / m 2. K 4 (0<e < 1, for a blackbody, e = 1) The peak of the wavelength distribution shifts to shorter wavelengths as the temperature increases Wien s displacement law (T must be in kelvin):
Finding peak wavelengths
Finding peak wavelengths
The heating effect of a medium such as glass or the Earth s atmosphere that is transparent to short wavelengths but opaque to longer wavelengths: Short get in, longer are trapped!
Intensity of Blackbody Radiation I = P/A = T 4 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 Combining gives the Rayleigh- Jeans law: 1 I λt, ~ λ 4
Problems with the Wein s World 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 1 I λt, ~ λ 4
Max Planck: Father of Quantum Introduced the concept of quantum of action in 1900 to solve the black body mystery In 1918 he was awarded the Nobel Prize for the discovery of the quantized nature of energy
Strings are Quantized The possible frequencies and energy states of a wave on a string are quantized. f v n 2 l
Planck s Two Assumptions The energy of an oscillator can have only certain discrete values E n = nhƒ This says the energy is quantized Each discrete energy value corresponds to a different quantum state 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
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 B e E k T
Planck s Wavelength Distribution Function Planck generated a theoretical expression for the wavelength distribution I λt, 2πhc 2 5 hc λkbt λ e 1 h = 6.626 x 10-34 J. s h is a fundamental constant of nature
Planck s Model, Graphs
Intensity of Blackbody Radiation P40.61 The total power per unit area radiated by a black body at a temperature T is the area under the I(λ, T)-versus-λ curve, as shown in Figure 40.3. (a) Show that this power per unit area is I λt, 2πhc 2 5 hc λkbt λ e 1 0 I 4 λ, T dλ T where I(λ, T) is given by Planck s radiation law and σ is a constant independent of T. This result is Stefan s law. To carry out the integration, you should make the change of variable x = hc/λkt and use the fact that 0 x e x 3 dx 1 4 15
Atomic Energy is quantized. It comes in chunks of Planck s constant, h. Max Planck NEVER liked the idea of quantized energy states. E nhf, n= 0,1,2,3,... 34 h 6.626x10 Js
Why doesn t Planck like Quantum????? In classical physics particles have continuous energy states.to say they have discrete energy states would mean that you can only drive at 10mph and 20mph but not at 15mph or at any speed in between 10 and 20 mph! In classical physics only special bound states have discrete or quantum energy states.
In 1886 Hertz noticed, in the course of his investigations, that a negatively charged electroscope could be discharged by shining ultraviolet light on it. In 1899, Thomson showed that the emitted charges were electrons. The emission of electrons from a substance due to light striking its surface came to be called the photoelectric effect. The emitted electrons are often called photoelectrons to indicate their origin, but they are identical in every respect to all other electrons. The Photoelectric Effect
The PROBLEM with the Photoelectric Effect The Problem with Waves: Increasing the intensity of a low frequency light beam doesn t eject electrons. This didn t agree with wave picture of light which predicts that the energy of waves add so that if you increase the intensity of low frequency light (bright red light) eventually electrons would be ejected but they don t! There is a cut off frequency, below which no electrons will be ejected no matter how bright the beam! Also there is no time delay in the ejection of electrons as the waves build up!
The Photoelectric Effect Proof that Light is a Particle The Problem with Waves: Increasing the intensity of a low frequency light beam doesn t eject electrons. This didn t agree with wave picture of light which predicts that the energy of waves add so that if you increase the intensity of low frequency light (bright red light) eventually electrons would be ejected but they don t! There is a cut off frequency, below which no electrons will be ejected no matter how bright the beam! Also there is no time delay in the ejection of electrons as the waves build up!
Characteristics of the Photoelectric Effect 1. The current I is directly proportional to the light intensity. 2. Photoelectrons are emitted only if the light frequency f exceeds a threshold frequency f 0. 3. The value of the threshold frequency f 0 depends on the type of metal from which the cathode is made. 4. If the potential difference ΔV is positive, the current does not change as ΔV is increased. If ΔV is made negative, the current decreases until, at ΔV = V stop the current reaches zero. The value of V stop is called the stopping potential. 5. The value of V stop is the same for both weak light and intense light. A more intense light causes a larger current, but in both cases the current ceases when ΔV = V stop.
The Photoelectric Effect
Photoelectric Effect Problem 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 (DV s )
Photoelectric Effect Problem 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 Problem 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, ƒ c The cutoff frequency is characteristic of the material being illuminated No electrons are ejected below the cutoff frequency regardless of intensity
Photoelectric Effect Problem 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
Einstein s Postulates: Light Quanta Einstein framed three postulates about light quanta and their interaction with matter: 1. Light of frequency f consists of discrete quanta, each of energy E = hf, where h is Planck s constant h = 6.63 10 34 J s. Each photon travels at the speed of light c = 3.00 10 8 m/s. 2. Light quanta are emitted or absorbed on an all-ornothing basis. A substance can emit 1 or 2 or 3 quanta, but not 1.5. Similarly, an electron in a metal can absorb only an integer number of quanta. 3. A light quantum, when absorbed by a metal, delivers its entire energy to one electron.
E hf Light is quantized in chunks of Planck s constant. Electrons will not be ejected in the Photoelectric Effect unless every photon has the right energy. One photon is completely absorbed by each electron ejected from the metal. As you increase the intensity of the beam, more electrons are ejected, but their energy stays the same. Photons
EX 39.2 The energy of a light quantum
Einstein s Explanation of the Photoelectric Effect An electron that has just absorbed a quantum of light energy has E elec = hf. (The electron s thermal energy at room temperature is so much less than that we can neglect it.) This electron can escape from the metal, becoming a photoelectron, if In other words, there is a threshold frequency for the ejection of photoelectrons because each light quantum delivers all of its energy to one electron.
Einstein s Explanation of the Photoelectric Effect A more intense light delivers a larger number of light quanta to the surface. These quanta eject a larger number of photoelectrons and cause a larger current. There is a distribution of kinetic energies, because different photoelectrons require different amounts of energy to escape, but the maximum kinetic energy is The stopping potential V stop is directly proportional to K max. Einstein s theory predicts that the stopping potential is related to the light frequency by
hf KE E max 0 Work to eject (Work Function) or Binding Energy Photon Energy Max KE of ejected electron
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 The cutoff frequency is related to the work function through ƒ c = φ / h The cutoff frequency corresponds to a cutoff wavelength λ c c ƒ c hc φ
Work Function
The Photoelectric Effect What is the maximum velocity of electrons ejected from a material by 80nm photons, if they are bound to the material by 4.73eV? Ignore relatavistic effects. KMax hf BE hc BE 34 8 6.6310 J s 3.0010 m s 19 1.6010 J 9 4.73 ev 80.010 m 1 ev K 18 1.7295 10 J 2 1.72910 18 J 12 1 2 2KE 6 mv v 1.9510 m s 31 2 m 9.1110 kg 2 (SR: K ( 1) mc )
Maximum photoelectron speed
Compton Effect, Classical 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 Predictions
Compton Effect, Observations Compton s experiments showed that, at any given angle, only one frequency of radiation is observed
Compton Effect, Explianed 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 photonelectron 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' λ 1 cos o θ mc e This is called the Compton shift equation
Arthur Holly Compton 1892-1962 Director of the lab at the University of Chicago Discovered the Compton Effect, 1923 Shared the Nobel Prize in 1927
The phenomenon in which an X-ray photon is scattered from an electron, the scattered photon having a smaller frequency than the incident photon is called The Compton Effect. 2 E mc p mv Divide: hc E hf pc p E h pc
The photon transfers momentum, acts like a particle. p p p incident scattered electron h λ' λ o 1 cosθ mc e The Compton wavelength of a particle is equivalent to the wavelength of a photon whose energy is the same as the rest mass of the particle. It gives the limits of measuring the position of a particle using traditional QM and not QED. The compton wavelength of the elctron is: h mc e 2. 43x10 12 m
Compton Wavelength
, The incident X-ray photon has an energy of 3.98 kev and is scattered by an angle of 140.0 degrees. a) What is the wavelength of incident X-ray? b) What is the wavelength of the scattered X-ray? c) What is the energy of the scattered X-ray? d) What is the kinetic energy of the recoil electron? e) What is the de Broglie wavelength of the recoil electron? 8 c 2.998x10 m/ s 31 me 9.109x10 kg 34 h 6.626x10 Js 19 1.602x10 / J ev
Joseph John Thomson Plum Pudding Model 1904 Received Nobel Prize in 1906 Usually considered the discoverer of the electron Worked with the deflection of cathode rays in an electric field His model of the atom A volume of positive charge Electrons embedded throughout the volume
1911: Rutherford s Planetary Model of the Atom A beam of positively charged alpha particles hit and are scattered from a thin foil target. Large deflections could not be explained by Thomson s pudding model. (Couldn t explain the stability or spectra of atoms.)
1911: Rutherford s Planetary Model of the Atom A beam of positively charged alpha particles hit and are scattered from a thin foil target. Large deflections could not be explained by Thomson s model. (Couldn t explain the stability or spectra of atoms.)
Classical Physics at the Limit WHY IS MATTER (ATOMS) STABLE?
Electrons exist in quantized orbitals with energies given by multiples of Planck s constant. Light is emitted or absorbed when an electron makes a transition between energy levels. The energy of the photon is equal to the difference in the energy levels: E nhf, n= 0,1,2,3,... 34 h 6.626x10 Js E E E hf i f
Light Absorption & Emission E E E hf i f
Bohr s Assumptions 1. Electrons in an atom can occupy only certain discrete quantized states or orbits. 2. Electrons are in stationary states: they don t accelerate and they don t radiate. 3. Electrons radiate only when making a transition from one orbital to another, either emitting or absorbing a photon. Postulate: The angular momentum of an electron is always quantized and cannot be zero: L n h 2 ( n 1,2,3,...)
Bohr s Derivation of the Energy for Hydrogen: Conservation of E: E K U F is centripetal: (1) Sub back into E: From Angular Momentum: (2) h From : L n = mvr ( n 1,2,3,...) 2 Sub r back into (1): Sub into (2): v 2 2 2 2kq 2q q e e e nh 4 0hn 2 0hn Why is it negative?
Bohr Line Spectra of Hydrogen Bohr s Theory derived the spectra equations that Balmer, Lyman and Paschen had previously found experimentally! 1 1 1 ( ) 2 RZ n 2 2 f n i 7 1 R 1.097 x10 m Balmer: Visible Lyman: UV Paschen: IR
1. Bohr model does not explain why electrons don t radiate in orbit. 2. Bohr model does not explain splitting of spectral lines. 3. Bohr model does not explain multi-electron atoms. 4. Bohr model does not explain ionization energies of elements. E Z ev 2 0 13.6
1. Bohr model does not explain angular momentum postulate: The angular momentum of an electron is always quantized and cannot be zero*: L n h 2 ( n 1,2,3,...) *If L=0 then the electron travels linearly and does not orbit. But if it is orbiting then it should radiate and the atom would be unstable eek gads! WHAT A MESS!
If photons can be particles, then why can t electrons be waves? p E hf h c c debroglie Wavelength: e h p Electrons are STANDING WAVES in atomic orbitals.
2r n n 1924: de Broglie Waves Explains Bohr s postulate of angular h p momentum quantization: 2r n n h h 2 r n n n p e m v m vr e n h n 2 h L mevrn n 2 ( n 1,2,3,...)
1924: de Broglie Waves Electrons are STANDING WAVES in atomic orbitals. h p 2r n n
De Broglie Wavelength h mv 34 h 6.626x10 J s
Lynda s De Broglie Wavelength 6.626x10 34 J s (75 kg)2 m / s 4.4x10 36 m Too small to notice or to interact with anything!
Particle-Wave: Light A gamma ray photon has a momentum of 8.00x10-21 kg m/s. What is its wavelength? What is its energy in Mev? h p 663. 10 800. 10 21 34 J s kg m s 829. 10 14 m E 21 8 8.00 10 kg m s 3.00 10 m s pc 1 MeV 1.6010 J 12 2.40 10 J 15.0 MeV 13
Electron De Broglie Wavelength for electron v =.1c h / mv 6.626x10 34 J s 31 7 (9.1x10 kg)(3x10 m / s) 2.4x10 11 m
Limits of Vision Electron Waves 11 e 2.4x10 m
Electron Microscope Electron microscope picture of a fly. The resolving power of an optical lens depends on the wavelength of the light used. An electron-microscope exploits the wave-like properties of particles to reveal details that would be impossible to see with visible light.
Electron Microscope Salmonella Bacteria Stem Cells The fossilized shell of a microscopic ocean animal is magnified 392 times its actual size.
Double Slit for Electrons shows Wave Interference
Double Slit for Electrons A modified oscilloscope is used to perform an electron interference experiment. Electrons are incident on a pair of narrow slits 0.060 0 μm apart. The bright bands in the interference pattern are separated by 0.400 mm on a screen 20.0 cm from the slits. Determine the potential difference through which the electrons were accelerated to give this pattern.
Particle-Wave Duality Interference pattern builds one electron at a time. Electrons act like waves going through the slits but arrive at the detector like a particle.
Trying to see what slit an electron goes through destroys the interference pattern.
Feynman version of the Uncertainty Principle It is impossible to design an apparatus to determine which hole the electron passes through, that will not at the same time disturb the electrons enough to destroy the interference pattern. -Richard Feynman
Next Time. Electron waves are probability waves in the ocean of uncertainty. - Richard Feynman: