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Quantum Qualities Quantum Qualities frequency, wavelength and momentum of photons and electrons 1 Quantum Qualities Quick Trip Way Down Memory Lane In 1833, Michael Faraday observed evidence of electrons when he observed a gas discharge glow in an evacuated container. The container was a sealed glass tube with two electrodes. In 1858, Julius Plucker proved the glow was charged. The glow cast a shadow and responded to a magnetic field. Casting a shadow meant the glow was comprised of rays, and responding to a magnet meant the glow was charged. In 1876, Eugen Goldstein named the glow “cathode rays”. 2 Quantum Qualities Quick Trip Way Down Memory Lane In 1883, Thomas Alva Edison observed that a sealed light bulb deposited carbon inside the bulb and modestly called it the Edison Effect. In 1885, Johann Balmer, a Swiss schoolteacher developed an equation that describes the photographs of emission light spectra of the elements (rainbow light patterns with dark spaces among the colors). In 1896, Jean-Baptist Perrin captured cathode rays and measured their charge, proving that the rays carry charge. In 1890’s, Max Planck used spectroscopic data to create a blackbody radiation law. 3 Quantum Qualities Quick Trip Way Down Memory Lane In 1897, J. J. Thompson proved that: • cathode rays were particles. Hear J. J. Thompson describe the relative size of an electron. • the charged particles emitted by a heated cathode were the same as cathode rays. • cathode rays acted as charged particles in both electric and magnetic fields. • Later he measured the charge to mass ratio of the charged particles, later determined to be electrons. From the soundtrack of the film, Atomic Physics Copyright J. Arthur Rank Organization, Ltd., 1948, 4 Quantum Qualities The 19th Century Version of a Diffraction Experiment It was harder back then to get Edison’s bulbs to glow and certainly harder to keep them from burning out. But, it was still possible to direct a light beam to a prism and then see the diffracted light on a screen on the other side of the grating. 5 Quantum Qualities Today’s Version of a 19th Century Diffraction Experiment It was harder back then to get Edison’s bulbs (discharge tubes) to glow and certainly harder to keep them from burning out. -1000 0 +1000 On Voltage Off l1 Current Voltage Generator Diffraction Grating Discharge Tube Filter l3 Screen But, it was still possible to direct a light beam (from a discharge tube) to a prism (grating), and then see the diffracted light on a screen on the other side of the grating. 6 Quantum Qualities Developing a Model: Matching Observations to Theory Johann Balmer found a mathematical series that predicts the wavelengths of the spectra lines when hydrogen is the gas in the glowing bulb. k l 364.56 2 nm k 4 2 657 nm 486 nm 410 nm 434 nm 375 nm He put different integers in the “k” position of the equation and then noticed that his answers matched the wavelengths of the spectra lines. 7 Unfortunately, Balmer did not have any idea why the equation worked or why there were colored lines in the first place. Quantum Qualities Blackbody Radiation Max Planck performed his “blackbody radiation” experiment to figure out why the spectra lines occurred. Light Source Any object that absorbs all the light that strikes it is called a blackbody. 8 Quantum Qualities Blackbody Radiation l1 l3 Screen A blackbody will emit (radiate) light as it heats up Max Planck used this spectroscopic data to create a blackbody radiation law. 9 Quantum Qualities Blackbody Radiation l1 l3 Screen • Planck believed that the separation of light into discrete lines meant that light energy was quantified as discrete energy levels. • If an unbroken spectrum had been observed, it would have meant a continuous energy distribution. • Planck started with Balmer’s result and “quantized” the photon emissions from various blackbody radiators. 10 Quantum Qualities Blackbody Radiation Plank’s experiments produced a description of energy emitted from the blackbody as a function discrete frequency values. Frequency can be calculated from the wavelength, l: f = c/l = (speed of light wave)/(wavelength of light) Planck’s Radiation Hypothesis states that: The emitted or absorbed energy was quantized with the smallest possible value being when n =1 and that: En = n x h x f All other permitted values of the energy are integral multiples of h f. 11 Quantum Qualities Blackbody Radiation Remember that: hf is h multiplied by frequency and has units of energy (Joules). En = is the dependent variable assigned to the energy absorbed or emitted n = an integer -34 x h = 6.63 10 Joule·seconds f = frequency of light associated with the heat energy En = n x h x f 12 Quantum Qualities Quick Trip a Shorter Distance Down Memory Lane In 1905, Albert Einstein extended Planck’s concept to light and used it to explain the photoelectric effect. In 1911, Robert Millikan measured the charge of an electron. In 1913, Niels Bohr proposed a model for the hydrogen atom that combined ideas about the nucleus and electrons that explained the spectra colors and the Balmer mathematical series as well. 13 Quantum Qualities Einstein and the Photoelectric Effect (1905) (Shine a light on a metal and electrons will escape from the metal surface.) E escape = hf - hfw Photoelectric Effect Light Source Photons E escape is the energy an electron has when it leaves the surface of a metal. hf is the energy of the light shining on the metal hfw is the energy used to move the electron away from the metal atom. (The work done to free the electron from the metal surface.) Einstein said that when hf is greater than hfw an electron will leave the metal surface. 14 Quantum Qualities Millikan Oil Drop Experiment (1911) (Put a charged oil drop in two force fields and balance the forces so the oil drop does not move.) He selected the gravitational field and the electric field. Gravity will pull oil drop down. Fg = mg Electrode Electric field can pull oil drop up. Fe = -qE -1000 0 +1000 On Voltage Off Current Voltage Generator Remember that: Fg is the gravity force pulling the charged oil drop down. Fe is the electric force pulling the charged oil drop up. m is the mass of the the charged oil drop up. q is the amount of charge on the oil drop up. g and E are the gravity and electric field strengths, respectively. 15 Quantum Qualities Millikan Oil Drop Experiment (1911) (Put a charged oil drop in two force fields and balance the forces so the oil drop does not move.) mg qelectron E 1.602x10 An oil drop with a surface charge -19 Coulombs -1000 0 +1000 On Electrodes Voltage Off Current Voltage Generator • Electric field between electrodes can be adjusted to suspend the drop. • The mass of the oil drop is known. • The oil drop floats because the force on the drop in the electric field exactly counteracted the force on the drop in the gravity field. 16 Quantum Qualities Photon E=hxf Bohr Hydrogen Atom Model (1913) ΔE = Einitial - Efinal = h × f h = Planck’s constant f = frequency of the photon • Electrons reside in orbits and can absorb specific quantities of energy to move to another orbit. • If an electron returns to its original orbit, the specific amount of absorbed energy could be returned to the world as a photon of light. • The wavelength and frequency of that light could be calculated using the equation Balmer developed in 1885. • The energy of that light could be calculated using the equation Planck developed in 1900. 17 Quantum Qualities Particle Wave Duality • Photons, packets of light with energy = hf, are an example of mass less particles. • The particle wave duality of light can be shown by combining Planck’s Law and Einstein’s hypothesis. • Planck stated that an atom can only emit and absorb energy of specific values. E = Einitial - Efinal = h f • Einstein stated that the frequency times the wavelength of a photon is equal to the speed of light. l f=c c Putting the two equations together yields E = h . l 18 Quantum Qualities Particle Wave Duality The particle-wave nature of h c photons can be described using E =p c photon = h f = l the work of Planck and Einstein. This equation models the energy of a photon that is released when an electron returns from an outer orbit to its original orbit (closer to the nucleus). It can be used to relate the energy of a photon to that photon’s: •Frequency: Ephoton = h x f •Wavelength:Ephoton = (h x c)/l •Momentum: Ephoton = p x c 19 Quantum Qualities Quick Trip not so Long Ago Your Great Grandfather’s Time By 1920, engineers and scientists understood that: • photons existed • if photons ever stopped moving, they would have no mass (a photon’s rest mass equals zero). Since photons do move: • their mass is related to the energy needed for movement as defined by Einstein’s famous equation! • their mass is “relativistic” mass, m using E = mc2 • they have momentum, p , like all moving objects with mass using Pr = mrc 20 Quantum Qualities Quick Trip not so Long Ago Your Great Grandfather’s Time In 1923, Louis De Broglie understood the properties of photons: • all photons obey Planck’s quantum energy equation: E Photon = n(h)(v) • all photons move and have a relative mass connected to the energy, Ephoton, they possess. • They, like all moving objects with mass, have momentum, pr pr = (mr ) c • Some photons can be seen by humans because they have a wave frequency (color) that is detected by human eyes and/or human instruments. 21 Quantum Qualities Quick Trip not so Long Ago Your Great Grandfather’s Time In 1923, Louis De Broglie understood the properties of photons: He also believed that: • unlike photons, most things have a rest mass. • all things including photons have wave properties. Momentum and frequency values with corresponding wavelengths. Thus, a baseball’s mass is not zero when it is not being thrown but it does have a wave frequency and a wavelength associated with it when it is in motion. h Planck’s constant l any object = (m object) (vobject velocity ) 22 Quantum Qualities Quick Trip not so Long Ago Your Great Grandfather’s Time In 1923, Louis De Broglie understood the properties of photons: He also believed that: • since the mass of a baseball (0.15 kg) is so large, the wavelength associated with a moving baseball is too short for humans to detect. • since the mass of an electron (9.11 x10-31 kg) is so small, the wavelength associated with a moving electron is long enough for human instruments to detect. h Planck’s constant l any object = (m object) (vobject velocity ) 23 Quantum Qualities Quick Trip not so Long Ago Your Great Grandfather’s Time In 1926, Erwin Schrodinger: • wondered why De Broglie’s relationship only quantitatively worked when the moving particles where in a force-free environment. • developed a general equation that: • described electrons moving with a wave motion because they were under the influence of a force environment generated by the positive charge of the nucleus. • gives the same momentum and frequency (wavelength) values that would be obtained by De Broglie if the electron was moving without the influence of the atom’s nucleus. Schrodinger’s equation is simply called the “wave equation”. Unfortunately, it only works really well for the hydrogen atom. 24 Quantum Qualities The Mechanics of Electron/Atom Behavior When De Broglie’s ideas were first applied to electron behavior, the topic was called quantum mechanics. When Schrodinger’s ideas were applied to electron behavior, the topic was called wave mechanics. 25 Quantum Qualities The Mechanics of Electron/Atom Behavior In 1927, Werner Heisenberg suggested that: • the difficulties with quantum mechanics and/or wave mechanics is that there is no way to know both the position and speed of an electron at the same time. • a probability perspective turns out to be a better way to describe electron behavior within an atom. Heisenberg’s approach is known today as both: statistical quantum mechanics and/or statistical wave mechanics. 26 Quantum Qualities Quantum Qualities The major step forward in the understanding of light, electrons and atoms was the steady development of the ideas that: • energy comes into and goes out of atoms but it does so in packets or quanta. • energy packets, quanta, differ from each other by multiple integer values of Planck’s constant. • each packet of energy can be associated with a frequency (wavelength). • each packet of energy can interact with an electron within an atom. • an electron will change orbits or even leave the atom if it interacts with the right packet of energy. 27 Quantum Qualities Equation Summary of our Historical Trip In 1885, Johann Balmer: An equation that described the photographs of light spectra (rainbow light patterns with dark spaces among the colors). In 1900, Max Planck: The quantum equation that describes the relationship between the energy of a wave and the frequency of that wave. In 1905, Albert Einstein: l 364.56 28 2 k 4 2 nm Ephoton = h f E An equation that connects light energy to electrons leaving an atom. k = h f - hf w escape Quantum Qualities Equation Summary of our Historical Trip In 1911, Robet Millikan: An equation that related the charge of an electron to BOTH its mass and the electric field its suspended in. qelectron = -mg E In 1913, Niels Bohr: An equation that explained the electron orbit quantum nature of hydrogen atom as seen from the hydrogen absorption spectrum and predicted the wavelength of the lines in that spectrum. E = Einitial - E final = h f In 1923, Louis De Broglie: An equation that connects the wavelength of a moving object to its mass and velocity. 29 l object h (mobject) (vvelocity) Quantum Qualities Two Example Problems Objectives: 1. to show the use of quantum equations 2. to illustrate unit manipulations 1.) Calculate the wavelength and the frequency of one photon of light that has an energy of 3.64 x 10-19 Joules. 2.) For a photon with a wavelength of 700 nm, calculate the a) frequency, b) the energy, and c) the momentum of the photon. 30 Quantum Qualities 1.) Calculate the wavelength and the frequency of one photon of light that has an energy of 3.64 x 10-19 Joules. Knowns: Energy = 3.64 10-19 J Planck’s constant = h = 6.63 10-34 J·s 8 m 3.0 10 Speed of light = c = s Unknowns: Wavelength = l = nm Frequency = f = s -1 Equations: Ephoton = h f = h c l 31 =p c Quantum Qualities 1.) Calculate the wavelength and the frequency of one photon of light that has an energy of 3.64 x 10-19 Joules. Unknowns: Wavelength = l = nm Frequency = f = s -1 h c Equations: E =p c photon = h f = l Ephoton = h c l h c l= Ephoton Ephoton = h f f= Ephoton h 32 Quantum Qualities Wavelength Value Calculation for Example #1 Knowns: Equations: Solution: Energy = 3.64 10-19 J Planck’s constant = h = 6.63 10-34 J·s 8 m 3.0 10 Speed of light = c = s h c l= Ephoton f= Ephoton h 6.63 10 -34 J s 3.0 10 8 m/s l= 3.64 10 -19 J The wavelength of light that has 3.64 x 10-19 J of energy is l = 5.464 33 10 -7 m Quantum Qualities Wavelength Unit Conversion for Example #1 Solution: l = 5.464 10 -7 m Wavelength is typically reported in units of nanometers. Units: Remember: 9 or 1 m = 1 10 nm l = (5.464 10 -7 m) (1 10 9 1 109 nm 1m nm 546 nmnm ) = 546.4 m Answer: A photon with an energy of 3.64 10 Joules has a wavelength of 546 nm 34 -19 Quantum Qualities Frequency Calculation for Example #1 Knowns: Equations: Solution: Energy = 3.64 10-19 J Planck’s constant = h = 6.63 10-34 J·s 8 m Speed of light = c = 3.0 10 s h c l= Ephoton f= Ephoton h 3.64 10-19 J f= 6.63 10-34 J s The frequency of light that has 3.64 x 14 1 -19 10 J of energy is f = 5.49 10 s 35 Quantum Qualities Frequency Unit Conversion for Example #1 Solution: Units: f = 5.49 10 14 1 s Frequency is typically given in units of Hertz. 1 Hertz = Hz = second f = 5.49 10 14 1 = 5.49 10 14 Hz s Answer: A photon with an energy of 3.64 10-19 Joules has a frequency of 5.49 1014 Hz. 36 Quantum Qualities Summary for Problem #1 1.) Calculate the wavelength and the frequency of one photon of light that has an energy of 3.64 10-19 Joules. Answer: A photon with an energy of 3.64 10-19 Joules has a wavelength of 546 nm and a frequency of 5.49 1014 Hz. ? Is this wavelength of light in the visible part of the spectrum? If so, what color is it? Yes, a 546 nm light wave is in the visible spectrum and will be green. 37 Quantum Qualities 2.) For a photon with a wavelength of 700 nm calculate the energy, frequency, and the momentum. Known: Wavelength = l = 700 nm Planck's constant = h = 6.63 10-34 J s m Speed of light = c = 3.0 108 s Unknown: Equations: Energy = Ephoton in J Frequency = f in s –1 Momentum = p in kg-m/s Ephoton = h f = h c l 38 =p c Quantum Qualities 2.) For a photon with a wavelength of 700 nm calculate the energy, frequency, and the momentum. Unknown: Energy = Ephoton in J Frequency = f in s –1 Momentum = p in kg-m/s h c Equations: E = h f = =p c photon l Ephoton = h c l Ephoton = h f h c l =p c p f= Ephoton h 39 h c l c p= h l Quantum Qualities Wavelength Unit Conversion Known: Unknown: Wavelength = l = 700 nm 8 m Speed of light = c = 3.0 10 s Energy = Ephoton in J Frequency = f Momentum = p Units: in s –1 in kg-m/s Wavelength = l = 700 nm Wavelength is usually reported in nm and the speed of light is given in m/s. (For this problem the wavelength will be converted to meters from nanometers so the velocity can be used in meters/second.) 1 10 -9 m 700 nm = 700 nm = 700 10 -9 m 1 nm 40 Quantum Qualities Energy Calculation Known: Wavelength = l = 700 nm m Speed of light = c = 3.0 10 s Planck's constant = h = 6.63 10-34 J s 8 Unknown: Equations: Energy = Ephoton in Joules Ephoton = h c l Ephoton = f= Ephoton h c l 41 h p= h l Quantum Qualities Energy Calculation Known: Wavelength = l = 700 nm m Speed of light = c = 3.0 10 s Planck's constant = h = 6.63 10-34 J s 8 Equations: Solution: Ephoton = h c l m 6.63 10 J s 3.0 10 s Ephoton = 7 10 -7 m -34 8 Ephoton = 2.8 10 -19 Joules 42 Quantum Qualities Energy Calculation Units: Energy is typically reported in units of Joules. Answer: A photon with a wavelength of 700 nm has an energy of 3 10-19 Joules. ? Is this wavelength of light in the visible part of the spectrum? If so, what color is it? Yes, a 700 nm light wave is in the visible spectrum and will be red. 43 Quantum Qualities Frequency Calculation Known: Wavelength = l = 700 nm Planck's constant = h = 6.63 10-34 J s 8 m Speed of light = c = 3.0 10 s Unknown: Frequency = f in s –1 Momentum = p in kg-m/s Energy Equations: Ephoton = =Ephoton in J h c l f= Ephoton h 44 p= h l Quantum Qualities Frequency Calculation Known: Wavelength = l = 700 nm Planck's constant = h = 6.63 10-34 J s 8 m Speed of light = c = 3.0 10 s Equations: Solution: f= f= c p= l 3.0 10 8 7 10 -7 f = 4.3 10 14 m s m 1 s 45 h l Quantum Qualities Frequency Calculation Solution: Units: f = 4.3 10 1 s Frequency is typically reported in units of Hertz. 1 Hertz = Hz = second f = 4.3 10 14 Answer: 14 1 = 4.3 10 14 Hz s A photon with a wavelength of 700 nm 14 has a frequency of 4 10 Hz. 46 Quantum Qualities Momentum Calculation Known: Wavelength = l = 700 nm Planck's constant = h = 6.63 10-34 J s 8 m Speed of light = c = 3.0 10 s Unknown: Momentum = p Energy in kg-m/s =Ephoton in J Frequency = f Equations: Ephoton = h c l in s –1 f= Ephoton h 47 p= h l Quantum Qualities Momentum Calculation h Equations: p= Solution: 6.63 10 -34 J s p= 7 10 -7 m l p = 9.5 10 note: 1 Joule of energy = -28 This momentum unit is also the same as the units of energy divided by velocity. Js m (1 kg) (1 m) 2 2 (1 second) 48 = 1 kg m 2 s 2 Quantum Qualities Momentum Calculation Units: There are several popular choices for momentum units, but the usual units are (mass)(velocity). 1 Js =1 kg m2 s m s2 m For metric calculations these units would be kg m/s. 1 kg m2 s s 9.5 10 Answer: 2 -28 m Js m = 1 kg m s = 9.5 10 -28 kg m s A photon with an wavelength of 700 nm has a momentum of 1 10-27 kg m/s. 49 Quantum Qualities Summary for Example Problems 1 and 2 1.) Calculate the wavelength and the frequency of one photon of light that has an energy of 3.64 10-19 Joules. Answer: A photon with an energy of 3.64 10-19 Joules has a wavelength of 546 nm and a frequency of 5.49 1014 Hz. 2.) For a photon with a wavelength of 700 nm calculate the energy, frequency, energy and the momentum. Answer: A photon with a wavelength of 700 nm has an energy of 3 10-19 Joules, and a frequency of 4 1014 Hz, and a momentum of 1 10-27 kg m/s. 50 Quantum Qualities Two De Broglie Wave Equation Examples Objectives: 1. to show the use of quantum equations 2. to illustrate unit manipulations De Broglie’s equation connects the wavelength of a moving object to the mass and velocity of that moving object. l object = h (mobject )(vvelocity ) 3a.) What is the wavelength value associated with a 0.15 kg baseball moving with a velocity of 30 meters/second? -31 3b.) What is the wavelength associated with a 9.11 x10 kg electron moving with a velocity of 1.47 x 10 7 meters/second? 51 Quantum Qualities Wavelength Calculation Knowns: Planck's constant = h = 6.63 10-34 J s Mass of baseball = m = 1.5 10-1 kg Mass of electron = m = 9.11 10-31 kg m Speed of baseball = v = 3.0 101 s 7 m Speed of electron = v = 1.4 10 s Unknowns: Wavelength of baseball = l b in nm Wavelength of electron = l e in nm Equations: h l= (mobject)(vvelocity) 52 Quantum Qualities Wavelength Calculation Knowns: h = 6.63 10-34 J s 10-1 kg 1 m vball = 3.0 10 s h Equations: l = (mball)(vball) melectron = 9.11 10-31 kg 7 m velectron = 1.4 10 s h l electron = (m e )(v e) mball = 1.5 b lb le 6.64 x 10 (1.5 x 10 1 34 kg) (3.0 x 10 m/s ) 1 6.64 x 10 (9.11 x 10 31 Js 34 Js kg) (1.4 x 10 m/s ) 7 53 meters meters Quantum Qualities Wavelength Calculation Equations: lb le lb 6.64 x 10 (1.5 x 10 -1 -31 Js 1 kg) (3.0 x 10 m/s) 6.64 x 10 (9.11 x 10 -34 -34 Js kg) (1.4 x 10 7 m/s) -34 1.5 x 10 meters le 6.64 x 10 (12.8 x 10 -34 -24 Js kg m/s) le 54 6.64 x 10 12.8 -10 meters Quantum Qualities Wavelength Calculation lb 1.5 x 10 meters -11 = l e= 5.19 x 10 meters -34 baseball wavelength = electron wavelength 3a Answer: A baseball does travel on a wave path like water but going 30 meters per second the ball goes through one complete wave after it has gone 1.5 x10-34 meters. 3b Answer: An electron does travel on a wave path like water but going 1.4 x 107 meters per second the electron goes through one complete wave after it has gone 5.19 x10-11 meters. 55 Quantum Qualities 56