( ) # velocity. Wavelengths of massive objects. From Last Time. Wavelength of electron. Wavelength of 1 ev electron. A little complicated ( ) " = h mv
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1 From Last Time Wavelengths of massive objects Light shows both particle and wavelike properties Matter shows both particle and wavelike properties. How can we make sense of this? debroglie wavelength = " = h p p=mv for a nonrelativistic (v<<c) particle with mass. " = h mv Mar. 23, 2007 Phy107 Lecture 25 1 Mar. 23, 2007 Phy107 Lecture 25 2 Wavelength of electron Macroscopic objects don t show effects of quantum mechanics. Saw this previously in pendulum: Energy levels are quantized, but discreteness is too small to be detected. Wave properties also too small to be detected Need less massive object to show wave effects Electron is a very light particle Mass of electron = 9.1x10-31 kg " = h p = h mv = 6 #10 $3 J $ s 9 #10 $31 kg ( ) # velocity ( ) Wavelength depends on mass and velocity Larger velocity, shorter wavelength Mar. 23, 2007 Phy107 Lecture 25 3 Mar. 23, 2007 Phy107 Lecture 25 Wavelength of 1 ev electron Fundamental relation is wavelength = " = h p Need to find momentum in terms of kinetic energy. p = mv, so E kinetic = p2 p = 2mE kinetic 2m " = h p = h 2mE kinetic = hc 2mc 2 E kinetic A little complicated But look at this without calculating it " = h p = hc 2 mc 2 E kinetic rest energy Same as before kinetic energy Mar. 23, 2007 Phy107 Lecture 25 5 Mar. 23, 2007 Phy107 Lecture
2 Why use rest energy? Particles important in quantum mechanics are characterized by their rest energy In relativity all observers measure same rest energy. electron: mc 2 ~ 0.5 MeV proton: mc 2 ~ 90 MeV neutron: mc 2 ~ 90 MeV Different for different particles General trends Wavelength decreases as rest energy (mass) increases Wavelength decreases as kinetic energy (energy of motion) increases 1 MeV = 1 million electron-volts Mar. 23, 2007 Phy107 Lecture 25 7 Mar. 23, 2007 Phy107 Lecture 25 8 Matter wave question A neutron has almost 2000 times the rest mass of an electron. Suppose they both have 1 ev of energy. How do their wavelengths compare? A. both same B. neutron wavelength < electron wavelength C. neutron wavelength > electron wavelength D. depends on energy Wavelength depends on momentum, as h/p. Same momentum -> same wavelength. Momentum = 2mE, depends on energy AND mass Wavelength of 1 ev electron For an electron, " = rest energy 1 ev electron, 120 ev # nm 2 $ MeV 10 ev electron 100 ev electron 1 E kinetic λ=1.23 nm λ=0.39 nm λ=0.12 nm = 1.23 ev 1/ 2 # nm E kinetic kinetic energy Mar. 23, 2007 Phy107 Lecture 25 9 Mar. 23, 2007 Phy107 Lecture Question A 10 ev electron has a wavelength of ~ 0. nm. What is the wavelength of a 0 ev electron? Can this be correct? If electrons are waves, they should demonstrate wave-like effects e.g. Interference, diffraction A. 0.2 nm B. 0. nm C. 0.8 nm A 25 ev electron has wavelength 0.25 nm, similar to atomic spacings in crystals Mar. 23, 2007 Phy107 Lecture Mar. 23, 2007 Phy107 Lecture
3 Crystals: regular arrays of atoms Layered planes of atoms Wave reflection from crystal Reflection from next plane Reflection from top plane Table salt (NaCl = Sodium Chloride) Very common cubic structure. Na and Cl atoms alternate in a regular pattern Typical spacings ~ 0.3 nm. Mar. 23, 2007 Phy107 Lecture side view Interference of waves reflecting from different atomic layers in the crystal. Difference in path length ~ spacing between atoms Mar. 23, 2007 Phy107 Lecture 25 1 & Destructive Interference Interference arises when waves change their phase relationship. Can vary phase relationship of two waves by changing physical location of speaker. in-phase 1/2 λ phase diff Destructive Mar. 23, 2007 Phy107 Lecture Mar. 23, 2007 Phy107 Lecture Patterns on the slide X-ray diffraction Molecular structure Diffraction spot arrangement indicates atomic arrangement Used to determine atomic arrangements of complex molecules. e.g. DNA X-ray diffraction pattern Mar. 23, 2007 Phy107 Lecture Mar. 23, 2007 Phy107 Lecture
4 Davisson-Germer experiment Diffraction of electrons from a nickel single crystal. Established that electrons are waves Bright spot: constructive Davisson: Nobel Prize 1937 Particle-wave duality Like light, particles also have a dual nature Can show particle-like properties (collisions, etc) Can show wavelike properties (). Like light, they are neither particle nor wave, but some new object. 5 ev electrons (λ=0.17nm) Can describe them using particle language or wave language whichever is most useful Mar. 23, 2007 Phy107 Lecture Mar. 23, 2007 Phy107 Lecture Suppose an electron is a wave P.A.M. Dirac (early 20th century): each photon interferes with itself. Interference between different photons never occurs. We now can have coherent photons in a laser, (Light Amplification by Stimulated Emission of Radiation) invented 0 years ago. These photons can in fact interfere. Here is a wave: λ where is the electron? " = h p Wave extends infinitely far in +x and -x direction x Mar. 23, 2007 Phy107 Lecture Mar. 23, 2007 Phy107 Lecture Analogy with sound Sound wave also has the same characteristics But we can often locate sound waves E.g. echoes bounce from walls. Can make a sound pulse Beat frequency: spatial localization What does a sound particle look like? One example is a beat frequency between two notes Two sound waves of almost same wavelength added. Example: Hand clap: duration ~ 0.01 seconds Speed of sound = 30 m/s Spatial extent of sound pulse = 3. meters. 3. meter long hand clap travels past you at 30 m/s Large Destructive Small Large Mar. 23, 2007 Phy107 Lecture Mar. 23, 2007 Phy107 Lecture 25 2
5 Making a particle out of waves 39 Hz 39 Hz + 38 Hz 39 Hz + 38 Hz + 37 Hz + 36 Hz Mar. 23, 2007 Phy107 Lecture Adding many sound waves Six sound waves with different wavelength added together λ 1 =λ λ 2 = λ/1.05 λ 3 = λ/1.10 λ = λ/1.15 λ 5 = λ/1.20 λ 6 = λ/1.25 Wave now resembles a particle, but what is the wavelength? Sound pulse is comprised of several wavelength The exact wavelength is indeterminate Δx J Mar. 23, 2007 Phy107 Lecture Spatial extent of localized sound wave Δx = spatial spread of wave packet Spatial extent decreases as the spread in included wavelengths increases. Mar. 23, 2007 Phy107 Lecture Δx J Same occurs for a matter wave Construct a localized particle by adding together waves with slightly different wavelengths. Since de Broglie says λ = h /p, each of these components has slightly different momentum. We say that there is some uncertainty in the momentum And still don t know exact location of the particle! Wave still is spread over Δx ( uncertainty in position) Can reduce Δx, but at the cost of increasing the spread in wavelength (giving a spread in momentum). Mar. 23, 2007 Phy107 Lecture Heisenberg Uncertainty Principle Using Δx = position uncertainty Δp = momentum uncertainty Heisenberg showed that the product Planck s ( Δx ) ( Δp ) is always greater than ( h / π ) The exact value of the product depends on the problem (pendulum, hydrogen atom, etc) Uncertainty principle question Suppose an electron is inside a box 1 nm in width. There is some uncertainty in the momentum of the electron. We then squeeze the box to make it 0.5 nm. What happens to the momentum? A. Momentum becomes more uncertain B. Momentum becomes less uncertain C. Momentum uncertainty unchanged Mar. 23, 2007 Phy107 Lecture Mar. 23, 2007 Phy107 Lecture
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