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The Wave – Particle Duality OR Light Waves Until about 1900, the classical wave theory of light described most observed phenomenon. Light waves: Characterized by: Amplitude (A) Frequency (n) Wavelength (l) Energy of wave a A2 YOUNG’S DOUBLE SLIT EXPERIMENT An artists impression Previous to the photoelectric effect, Young demonstrated the wave nature of light (1801) YOU CAN SEE ALL THIS USING A JAVA APPLET: http://micro.magnet.fsu.edu/primer/java/interference/doubleslit And then there was a problem… In the early 20th century, several effects were observed which could not be understood using the wave theory of light. Two of the more influential observations were: 1) Blackbody radiation 2) The photoelectric effect 3) The Compton scattering Studies of the blackbody radiation spectrum Predicting the existence of the photon What is a blackbody? • A black body is a theoretical object that absorbs all electromagnetic radiation that falls on it. No electromagnetic radiation passes through it and none is reflected. Because no light (visible electromagnetic radiation) is reflected or transmitted, the object appears black when it is cold. • If the black body is hot, these properties make it an ideal source of thermal radiation. If a perfect black body at a certain temperature is surrounded by other objects in thermal equilibrium at the same temperature, it will on average emit exactly as much as it absorbs, at every wavelength. Since the absorption is easy to understand—every ray that hits the body is absorbed—the emission is just as easy to understand. • A black body at temperature T emits exactly the same wavelengths and intensities which would be present in an environment at equilibrium at temperature T, and which would be absorbed by the body. Since the radiation in such an environment has a spectrum that depends only on temperature, the temperature of the object is directly related to the wavelengths of the light that it emits. • At room temperature, black bodies emit mostly infrared light, but as the temperature increases past a few hundred degrees Celsius, black bodies start to emit visible wavelengths, from red, through orange, yellow, and white before ending up at blue, beyond which the emission includes increasing amounts of ultraviolet. • The term "black body" was introduced by Gustav Kirchhoff in 1860. The light emitted by a black body is called black-body radiation. In the laboratory, black-body radiation is approximated by the radiation from a small hole entrance to a large cavity, a hohlraum. (this technique leads to the alternative term cavity radiation) Any light entering the hole would have to reflect off the walls of the cavity multiple times before it escaped, in which process it is nearly certain to be absorbed. This occurs regardless of the wavelength of the radiation entering (as long as it is small compared to the hole). The hole, then, is a close approximation of a theoretical black body and, if the cavity is heated, the spectrum of the hole's radiation (i.e., the amount of light emitted from the hole at each wavelength) will be continuous, and will not depend on the material in the cavity, but only on the temperature of the cavity wall. Studying the blackbody radiaton was a major challenge in theoretical physics during the late nineteenth century, that ultimately led to Quantum Theory. Starting from Stefan Boltzmann’s Law, we will move to the studies that led to the quantisation of the electromagnetic field by Max Planck: Stefan and Boltzman stated that the total I T 4 Stefan - Boltzman constant intensity (power per unit area) over all wavelengths 5.67039910-8W / m2 K 4 for a blackbody radiator is proportional to the fourth power of the absolute temperature. For hot objects other than ideal radiators, the law is expressed in the form: I eT 4 where e is the emissivity of the object (e = 1 for ideal radiator). If the hot object is radiating energy to its cooler surroundings at temperature Tc, the net radiation loss rate takes the form I e T 4 - To 4 Note: The unusually rapid increase in radiation with temperature is a consequence of the massless nature of photons, the carriers of electromagnetic energy (the same kind of behavior governs phonons, the quanta of acoustic energy in solids and liquids). Although this relationship describes the total energy emitted, it does not predict the spectral distribution. Wien’s Displacement Law Wien’s displacement law: Wien's Law tells us that objects of different temperature emit spectra that peak at different wavelengths. Hotter objects emit most of their radiation at shorter wavelengths; hence they will appear to be bluer . Cooler objects emit most of their radiation at longer wavelengths; hence they will appear to be redder. Furthermore, at any wavelength, a hotter object radiates more (is more luminous) than a cooler one. 1 l peak T o We define the proportionality constant t be : l peakT 2.898 103 m K Wien’s displacement Law: The wavelength at which the intensity reaches its maximum (peak) value decreases as the temperature is increased, in inverse proportion to the temperature. RAYLEIGH AND JEANS: AIMED TO PROVIDE A THEORETICAL EXPRESSION TO FIT WIEN’S EXPERIMENTAL EVIDENCE PROVIDED BY WIEN.FOR THAT THEY USED CLASSICAL CONCEPTS OF ELECTROMAGNETISM AND STATISTICAL MECHANICS Although the derivation of this formula is beyond the scope of this course, the form can be easily explained. In classical statistical mechanics, every possible degree of freedom should acquire an average energy of 1/2 kT. For example, a monatomic gas molecule has energy 3/2 kT because it can move in three independent directions. The total intensity at a given frequency is the product of the number of modes available and the average energy in each. As the frequency f becomes large, the predicted intensity increases without limit, even for objects at modest temperature. This is called "the ultraviolet catastrophe". On the other hand, at low frequencies, the formula gave accurate predictions of experimental results, indicating that at least some aspects were basically correct. Lets take a closer look • Step 1: Calculate the amount of radiation (number of waves/modes) at each wavelength. • Step 2: The contribution of each wave to the total energy in the box. • Step 3: The radiancy corresponding to that energy. STEP 1: Cavity modes Rayleigh and Jeans showed that the number of modes N per unit volume, per unit wavelength was inverse proportional to the 4th power of wavelength. dN 8 4 dl l STEP 2: Energy density (energy per unit volume) U per unit wavelength Modes per unit Average energy volume per unit CLASSICAL wavelength per mode TREATMENT dU 8 4 kT dN 8 dl l 4 kT dl l STEP 3: dI Rl dl Rayleigh-Jeans Law 8 Rl radiancy (intensity per unit length interval) Rl 4 kT , c λv c l 4 Classical theory: Rayleigh Jeans Law R as v or l 0 Experimental Evidence R 0 as v or l 0 Planck suggested a formula to fit the experimental data. CLASSICAL THEORY LED TO THE ULTRAVIOLET CATASTROPHE Quantum Treatment: proposed by Max Planck CLASSICAL Modes per unit Average energy TREATMENT volume per unit per mode wavelength dN 8 kT 4 dNl 8 d l 4 QUANTUM dl l dN 8 hc 1 TREATMENT 4 dl l l hc e lkT 1 He postulated that electromagnetic energy was emitted in discrete units or quanta, each with energy given by hf where h is the Planck constant, 6.6256 x 10-34 joule-second. For photon energies with hf>kT, it would no longer be possible to populate each mode with kT average energy since a fraction of hf is no longer allowed. The consequence is an additional factor that reduces the spectral distribution, approaches unity for small f, preserving the long wavelength Rayleigh-Jeans behavior but squelches the ultra-violet divergence. Plancks idea lead to the idea of quantisation of the electromagnetic radiation. The basic quantity of energy hv was associated with the notion of the photon, the quantum of light. E= hv , h=Planck’s constant Based on the above result he recalculated the radiancy to be: 8 c 1 Rl 4 hv hv l 4 e 1 KT which agreed with Wien’s experimental evidence!!!!! An experiment demonstrating the quantum nature of light Experimental verification of Planck’s ideas What is a photon according to Planck’s hypothesis ?(1912 - 1915) A photon is a quantum of the electromagnetic field with energy E=hv. What is photoelectric emission? The process of using light to eject electrons from a metal surface. Red light incident on a clean sodium plate in vacuum SODIUM PLATE Surface at 0 potential Shining Red Light on my metal plate I get no current on my ammeter. How can I Ammeter registers current I change that? Photoelectric Effect (I) “Classical” Method What if we try this ? Increase energy by Vary wavelength, fixed amplitude increasing amplitude electrons electrons emitted ? emitted ? No No No Yes, with low KE No Yes, with No high KE No electrons were emitted until the frequency of the light exceeded a critical frequency- threshold frequency, at which point electrons were emitted from the surface! (Recall: small l large n) CHANGING THE WAVELENGTH Metal X-rays UV Blue Red Light Light Zinc Yes Yes No No Sodium Yes Yes Yes No Even though there was no current present when illuminating the sodium plate with red light, that changed when the wavelength was changed to blue or UV. Thus the ability of light to kick out electrons, depends on the frequency of the light, which is associated with their KE. There is a minimum frequency required to eject electrons: threshold frequency (a characteristic for different metals) Photoelectric Effect - Explanation Electrons are attracted to the (positively charged) nucleus by the electrical force In metals, the outermost electrons are not tightly bound, and can be easily “liberated” from the shackles of its atom. It just takes sufficient energy… Classically, we increase the energy of an EM wave by increasing the intensity (e.g. brightness) Energy a A2 But this doesn’t work ?? An alternate view is that light is acting like a particle The light particle must have sufficient energy to “free” the electron from the atom. Increasing the Amplitude is simply increasing the number of light particles, but its NOT increasing the energy of each one! Increasing the Amplitude does nothing However, if the energy of these “light particle” is related to their frequency, this would explain why higher frequency light can knock the electrons out of their atoms, but low frequency light cannot… In this “quantum-mechanical” picture, the energy of the light particle (photon) must overcome the binding energy of the electron to the nucleus. If the energy of the photon exceeds the binding energy, the electron is emitted with a KE = Ephoton – Ebinding. The energy of the photon is given by E=hn, where the constant h = 6.6x10-34 [J s] is Planck’s constant. “Light particle” For example: Electron 1 at the surface requires the least possible energy to be liberated, so it escapes with the most possible kinetic energy. Electron 2 deep in the material has lost too much kinetic energy by the time it reaches the Surface and is attracted back. Electron 3 escapes, but has less kinetic energy than electron 1 Electron 4 gained enough energy to escape, but was moving in the wrong direction and was absorbed by the metal. 3 1 2 Photon Metal ion 4 Electron Einstein’s photoelectric equation • Einstein suggested that when a photon causes photoemission from a metal surface, some of the photon energy is used to overcome the work function, while the remainder appears at the kinetic energy of the ejected electron. He expressed that in his famous photoelectric equation: Kmax = hv – Φ Einstein was awarded the Nobel Prize for explaining the photoelectric effect in 1921 Work function (Ф)= Binding Energy • Is the work necessary to remove an electrons from the surface of the material. • Different metals have different work functions, but the work function of any metal is a characteristic of the metal it self, and does not change for different frequencies of light. Stopping potential • The photoelectric equation corresponds to a metal surface at zero potential. • Let the surface be held at a positive potential. If we gradually increase the positive potential, we come to a point where no electrons can escape any more. prevention of photoem ission K m ax hv eV hv eVs 0 m axim umenergyreducedto zero h h h Vs v v vo for vo e e e e h STOPPING POTENTIAL Vs Gradient=h/e vo v -Φ/e Plotting the stopping potential against frequency • Wave theory of light: 1. Any frequency of light would eject an electron. 2. A higher light intensity, would cause an increase in current. 3. The procedure of ejecting the electrons would not be immediate. • Particle theory of light: 1. Minimum frequency of light required to eject electrons: threshold frequency. 2. If the frequency is sufficient to eject electrons, increasing the intensity of light would increase the current. If the frequency is not sufficient, then any increase in the light intensity would have no effect in producing a current. Changing the frequency, would cause a change in the electron KE and stopping potential. 3. The procedure of ejecting the electrons is immediate. Photons Quantum theory describes light as a particle called a photon According to quantum theory, a photon has an energy given by E = hn = hc/l h = 6.6x10-34 [J s] Planck’s constant, after the scientist Max Planck. The energy of the light is proportional to the frequency (inversely proportional to the wavelength) ! The higher the frequency (lower wavelength) the higher the energy of the photon. 10 photons have an energy equal to ten times a single photon. Quantum theory describes experiments to astonishing precision, whereas the classical wave description cannot. The Electromagnetic Spectrum Shortest wavelengths (Most energetic photons) E = hn = hc/l h = 6.6x10-34 [J*sec] (Planck’s constant) Longest wavelengths (Least energetic photons) Can a particle exist that has no rest mass, but which nevertheless exhibits such particle-like properties as energy and momentum? According to classical theory Special Relativity nevertheless the answer is no allows for it. • Photons are particles that are never found at rest. That means that their rest mass is zero. • Photons move with the speed of light c. • Using the relationship between momentum and energy from Special Relativity, we find: E E c p mo c 2 2 o 2 2 2 2 c p c p 2 2 2 2 E pc The Compton Effect In 1924, A. H. Compton performed an experiment where X-rays impinged on matter, and he measured the scattered radiation. Incident X-ray wavelength M A l1 T Scattered X-ray T E wavelength l2 > l1 R l2 e Application of Planck’s ideas Electron comes flying out Problem: According to the wave picture of light, the incident X-ray should give up some of its energy to the electron, and emerge with a lower energy (i.e., the amplitude is lower), but should have l2l1. It was found that the scattered X-ray did not have the same wavelength ? Quantum Picture to the Rescue Electron Scattered X-ray Incident X-ray initially at E2 = hc / l2 rest (almost) E1 = hc / l1 e l2 > l1 e Ee Compton found that if you treat the photons as if they were particles of zero mass, with energy E=hc/l and momentum p=h/l The collision behaves just as if it were 2 billiard balls colliding ! Photon behaves like a particle with energy & momentum as given above! BEFORE THE COLLISION AFTER THE COLLISION Scattered photon INCIDENT PHOTON P’=hf’/c ELECTRON AT REST hf Pe=mu P=hf/c Scattered electron Kinetic energy of the scattered electron: Ke=mc2-moc2 Conservation of linear momentum and energy hf hf X : pbefore pafter cos pe cos c c hf Y : pbefore pafter 0 sin pe sin c Ebefore Eafter hf hf K e Squaring the following equations and adding them together, we eliminate pe c cos hf hf cos pe c 2 hf 2hf hf cos hf 2 2 2 pe c sin hf sin Then weequate the following two expressions for the total energy of the particle. E KE mo c 2 KE mo c 2 2 mo c 4 pe c 2 pe c 2 KE 2 2mo c 2 KE 2 2 2 E mo c 4 pe c 2 2 2 But : KE hf hf pe c 2 KE 2 2mc 2 KE hf 2hf hf hf 2mc 2 hf hf 2 2 2 on But from the conservati of momentum we have : pe c 2 hf 2hf hf cos hf 2 2 2 2mc 2 hf hf 2hf hf 1 cos The above relationship is simpler when expressedin terms of the wavelength l. Dividing by 2h 2 c 2 we have : 2mo c 2 hf hf 2hf hf 1 cos mo c f f f f 1 cos h c c c c f 1 f 1 Since and we have : c l c l mo c 1 1 1 - cos l -l h 1 cos h l l ll mo c l - l lC 1 cos , lC h mo c The above equation gives the change in wavelength expected for a photon that is scattered through the angle by a particle of rest mass m. This change is independent of the wavelength λ of the incident photon. The quantity λC is called the Compton wavelength of the scattering particle. Compton effect-Summary Compton was able to explain all he was seeing by using the photon theory of light. As incident photons collided with the electrons, they transferred some of their energy to them. Compton applied the conservation of momentum and energy to the experiment, and found the results agreed. Photons conformed with the laws of conservation of momentum and energy. This provided support that electromagnetic radiation is quantised and has momentum and energy associated with it. Summary of Photons Photons can be treated as “packets of light” which behave as a particle. To describe interactions of light with matter, one generally has to appeal to the particle (quantum) description of light. A single photon has an energy given by E = hc/l, where h = Planck’s constant = 6.6x10-34 [J s] and, c = speed of light = 3x108 [m/s] l = wavelength of the light (in [m]) Photons also carry momentum. The momentum is related to the energy by: p = E / c = h/l So is light a wave or a particle ? On macroscopic scales, we can treat a large number of photons as a wave. When dealing with subatomic phenomenon, we are often dealing with a single photon, or a few. In this case, you cannot use the wave description of light. It doesn’t work ! Matter Waves: The De Broglie hypothesis • If there is a particle of momentum p, its motion is associated with a wave of wavelength: h h l DE BROGLIE WAVELENGTH p mu MATTER WAVES The De Broglie wavelength of a moving cow would be trillions times smaller than atomic dimensions and far too small to be detected. The De Broglie wavelength of a C60 carbon molecule (buckyball) travelling at a few meters per second is approximately the size of the molecule itself (about 1nm). The De Broglie wavelength of an electron travelling at a few meters per second has a de Broglie wavelength equivalent to the width of a human hair (a fraction of a millimetre. That is large enough for the quantum wave to actually show up in experiments. Complementarity • Light exhibits both wave and particle bahaviour, and using different experimental set ups and we can either detect one or the other. e.g. the photoelectric effect demonstrates the particle nature of light (photons), while Young’s double slit experiment demonstrates the wave nature of light. Both the particle and wave picture of light are valid and under no circumstances cancel each other. In the contrary they are complementary. This is the principle of complementarity in quantum mechanics and we will see it exhibiting itself in more examples as we move across the subject The principle of complementarity also holds for particles. Both the matter wave description of a moving atom, or it’s description as a localised particle are equally valid.

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