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Material since exam 3

Material since exam 3. De Broglie wavelength, wavefunctions, probabilities Uncertainty principle Particle in a box Wavefunctions, energy, uncertainty relation 1D, 2D, and 3D box, wavefunctions, energy 3D hydrogen atom Quantum #’s, physical meaning of quantum #’s Energies and wavefunctions

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Material since exam 3

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  1. Material since exam 3 • De Broglie wavelength, wavefunctions, probabilities • Uncertainty principle • Particle in a box • Wavefunctions, energy, uncertainty relation • 1D, 2D, and 3D box, wavefunctions, energy • 3D hydrogen atom • Quantum #’s, physical meaning of quantum #’s • Energies and wavefunctions • Orbital magnetic dipole moment, electron spin • Multielectron atoms • State energies, electron configuration, periodic table • Lasers • Nuclear physics • Isotopes, nuclear binding energy • Radioactive decay • Decay rates, activity, radiation damage • Types of decay, half-life, radioactive dating.

  2. Matter waves • If light waves have particle-like properties, maybe matter has wave properties? • de Broglie postulated that the wavelength of matter is related to momentum as • This is called the de Broglie wavelength. Nobel prize, 1929

  3. Matter Waves • deBroglie postulated that matter has wavelike properties. • deBroglie wavelength Example: Wavelength of electron with 10 eV of energy: Kinetic energy

  4. Heisenberg Uncertainty Principle • Using • x = position uncertainty • p = momentum uncertainty • Heisenberg showed that the product (x )  (p ) is always greater than ( h / 4 ) Often write this as where is pronounced ‘h-bar’ Planck’sconstant

  5. The wavefunction 2  • Particle has a wavefunction (x) 2(x) x x dx Very small x-range = probability to find particle in infinitesimal range dx about x Larger x-range = probability to find particle between x1 and x2 particle must be somewhere Entire x-range

  6. 2 Question 0.5nm-1 What is probability that particle is found in 0.01nm wide region about -0.2nm? x 0.001 0.005 0.01 0.05 0.1 0 -0.8nm -0.2nm About what is probability that particle is in the region -1.0nm<x<0.0nm? 0.1 0.4 0.5 1.5 3.0

  7. L Particle in 1D box n=1 n Wavefunction Probability n=3 n=2

  8. n=5 Energy n=4 n=3 n=2 n=1 Particle in box energy levels • Quantized momentum • Energy = kinetic • Or Quantized Energy n=quantum number

  9. 3-D particle in box: summary • Three quantum numbers (nx,ny,nz)label each state • nx,y,z=1, 2, 3 … (integers starting at 1) • Each state has different motion in x, y, z • Quantum numbersdetermine • Momentum in each direction: e.g. • Energy: • Some quantum states have same energy

  10. Question How many 3-D particle in box spatial quantum states have energy E=18Eo? A. 1 B. 2 C. 3 D. 5 E. 6

  11. Q: ask what state is this? (121) (112) (211) All these states have the same energy, but different probabilities

  12. Spin magnetic quantum number 2 +1/2 or -1/2 3D hydrogen atom • n describes energy of orbit : • ℓ describes the magnitude of orbital angular momentum • mℓ describes the angle of the orbital angular momentum • ms describes the angle of the spin angular moment For hydrogen atom:

  13. Other elements: Li has 3 electrons n=2 states, 8 total, 1 occupied n=1 states, 2 total, 2 occupiedone spin up, one spin down

  14. Question Inert gas atoms are ones that have just enough electrons to finish filling a p-shell (except for He). How many electrons do next two inert gas atoms after helium ( neon (Ne) and argon (Ar) ) have. In this range of atomic number the subshells fill in order of increasing angular momentum. 10 & 18 4 & 8 8 & 16 12 & 20 6 & 10

  15. Multi-electron atoms • Electrons interact with nucleus (like hydrogen) • Also with other electrons • Causes energy to depend on ℓ States fill in order of energy: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d Energy depends only on n Energy depends on n and ℓ

  16. H1s1 He1s2 Li2s1 Be2s2 B2p1 C2p2 N2p3 O2p4 F2p5 Ne 2p6 Na3s1 Mg3s2 Al3p1 Si3p2 P3p3 S3p4 Cl3p5 Ar3p6 K K4s1 Ca4s2 Ca Sc3d1 Y3d2 Ga4p1 Ge4p2 As4p3 Se4p4 Br4p5 Kr4p6 8 moretransition metals The periodic table • Atoms in same column have ‘similar’ chemical properties. • Quantum mechanical explanation: similar ‘outer’ electron configurations. Na3s1

  17. Electron Configurations Atom Configuration H 1s1 He 1s2 1s shell filled (n=1 shell filled - noble gas) Li 1s22s1 Be 1s22s2 2s shell filled B 1s22s22p1 etc (n=2 shell filled - noble gas) Ne 1s22s22p6 2p shell filled

  18. Relaxation to metastable state(no photon emission) Transition by stimulated emission of photon Ruby laser operation PUMP 3 eV 2 eV Metastable state 1 eV Ground state

  19. 12 C 6 Isotopes # protons Total # nucleons Another form of carbon has 6 protons, 8 neutrons in the nucleus. This is 14C. • Carbon has 6 protons, 6 electrons (Z=6):this is what makes it carbon. • Most common form of carbon has 6 neutrons in the nucleus. Called 12C This is a different ‘isotope’ of carbonIsotopes: same # protons, different # neutrons

  20. Nuclear matter • Any particle in nucleus, neutron or proton, is called a nucleon. • “A” is atomic mass number • A=total number of nucleons in nucleus. • Experimental result • All nuclei have ~ same (incredibly high!) density of 2.3x1017kg/m3 • Volume A = number of nucleons • Radius  A1/3

  21. Mass of Hydrogen atom(1.0078 u) Mass of atom withZ protons, N neutrons Binding energy • Calculate binding energy from masses Atomic masses well-known-> easier to use

  22. Biological effects of radiation • Radiation damage depends on • Energy deposited / tissue mass (1 Gy (gray) = 1J/kg) • Damaging effect of particle (RBE, relative biological effectiveness) Radiation type RBE X-rays 1Gamma rays 1Beta particles 1-2Alpha particles 10-20 • Dose equivalent = (Energy deposited / tissue mass) x RBE • Units of Sv (sieverts) [older unit = rem, 1 rem=0.01 Sv] • Common units mSv (10-3Sv), mrem (10-3rem) • Common ‘safe’ limit = 500 mrem/yr (5 mSv/yr)

  23. Exposure from 60Co source • 60Co source has an activity of 1 µCurie • Each decay: 1.3 MeV photon emitted • Hold in your fist for one hour • all particles absorbed by a 1 kg section of your body for 1 hour • Energy absorbed in 1 kg = 0.5 rem 0.3 rem 0.1 rem 0.05 rem 0.003 rem What dose do you receive?

  24. Quantifying radioactivity • Decay rate r (Units of s-1) • Prob( nucleus decays in time t ) = r t • Activity R (Units of becquerel (1 Bq=1 s-1) or curie (1 Ci=3.7x1010 s-1) • Mean # decays / s = rN, N=# nuclei in sample • Half-life t1/2(Units of s) • time for half of nuclei to decay = t1/2

  25. Activity of Radon • 222Rn has a half-life of 3.83 days. • Suppose your basement has 4.0 x 108 such nuclei in the air. What is the activity? We are trying to find number of decays/sec. So we have to know decay constant to get R=rN

  26. Decay summary • Alpha decay • Nucleus emits He nucleus (2 protons, 2 neutrons) • Nucleus loses 2 protons, 2 neutrons • Beta- decay • Nucleus emits electron • Neutron changes to proton in nucleus • Beta+ decay • Nucleus emits positron • Proton changes to neutron in nucleus • Gamma decay • Nucleus emits photon as it drops from excited state

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