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NE 409 9-21-00

NE 409 9-21-00. Quantum Mechanics, the second pillar of modern physics. Atomic Radiation. Light emission from an atom is due to a transition from one energy level to another. E 2. Increasing Energy. photon. E photon = E 2 -E 1 =h n. E 1. Black Body Radiation.

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NE 409 9-21-00

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  1. NE 409 9-21-00 • Quantum Mechanics, the second pillar of modern physics.

  2. Atomic Radiation • Light emission from an atom is due to a transition from one energy level to another. E2 Increasing Energy photon Ephoton = E2-E1=hn E1

  3. Black Body Radiation • A body in thermodynamic equilibrium with it’s surroundings emits black body radiation. Temperature T Volume V Black Body Radiation I(l) dl

  4. Measured Emission Spectra Rayleigh-Jeans Law

  5. Rayleigh-Jeans Law • In late 1800’s, Maxwell’s equations were known—photons behave like waves • Assuming that the black body emitter, can calculate mode structure of waves in resonator. • Number of resonant frequencies in a cube of volume V with sides a is: • (wn/c)2=(mp/a)2+(np/a)2+(qp/a)2

  6. Solving for n • n=(c/2na)[m2+p2+q2]1/2 • In a volume element of dm x dp x dq there is one mode. So the number of modes between n=0 and n can be found by taking the 1/8 the volume of a sphere with a radius R=2nan/c • Vs=(1/8)x(4pR3/3) • Each of the modes in Vs must be multiplied by two since there can be a Transverse Electric (TE) and Transverse Magnetic (TM) component.

  7. Number of Modes • N is the number of modes • N=2x(1/8)x(4p/3)R3 • R=2nan/c • N=(8pn3n3/3c3)a3 • The mode density is defined as p(n)dn which represents the small frequency interval dn around n per unit volume.

  8. Mode density • p(n)dn=(1/V)(dN/dn)dn • p(n)dn=(8pn2ng/c3)n2dn • Where ng is the group index • ng=n+n(dn/dn)=n-l(dn/dl) • The average energy per mode is kT from the Maxwell Boltzmann’s equation. • Classical physics would calculate the energy density r(n)dn by multiplying number of modes times average energy of mode—Rayleigh-Jeans law: • r(n)dn=(8pn3n2dn/c3)kT

  9. At short wavelengths Rayleigh-Jeans law goes to infinity • Ultraviolet catastrophe

  10. Plank’s Approach • Energy density=mode density x average energy • The error must come from average energy density • Assume that the average energy goes as DE=hn

  11. Calculation of Average Energy • Using Boltzmann statistics • e1, e2, e3 …. Are the allowed energy states • The probability that an energy state ej occurs is: • Pi=exp(-ei/kT) • Average energy is: • <e>=(hnexp(-hn/kT)+2hnexp(-2hn/kT) +…)/(exp(-hn/kT)+exp(-2hn/kT)+…)

  12. Average Energy • <e>=ånhnexp(-nhn/kT)/ åexp(-nhn/kT) • Where n is summed from 1 to infinity • Taking the limits of the summation • <e>=hn/ (exp(hn/kT)-1)

  13. Energy Density • Energy density=mode density x average energy • r(n)dn=(8pn3n2dn/c3) x hn/ (exp(hn/kT)-1) • Plank’s formula explained the ultraviolet catastrophe. Each photon is emitted in distinct quanta, DE=hn, not in a continuum

  14. Einstein A and B coefficients • Atoms interact with and emit photons by three processes • Spontaneous emission • Absorption • Stimulated emission • Relationship between A and B coefficients is • A21/B21=(8phn3/c3)

  15. Important events in quantum physics • Photoelectric Effect, Hertz 1887 • Light is able to knock electrons off of a material • Observations • If light intensity increases, so does the number of ejected electrons. The kinetic energy of the ejected electrons does not increase. • If the frequency of the light is reduced, then below a critical frequency, no electrons are ejected

  16. Einstein’s Photoelectric Theory • Light is made up of photons with a finite number of energy quanta, which are localized, which cannot be subdivided, and which are absorbed and emitted only as specific energy units: • E=hn

  17. X-rays and Bragg Diffraction • A. H. Compton proved that photons carry both energy and momentum. Thus light has a wave-particle duality. • Bragg used crystals to scatter x-rays • Bragg’s law • 2dsinq=nl

  18. Duane-Hunt Law • Classical Bremsstrahlung emission should vary smoothly in an x-ray tube. It does not. Certain isolated “characteristic” frequencies exist for different materials. These x-rays are called characteristic x-rays. • The Kinetic Energy of an electron striking the material is KE=Voe

  19. X-ray production • No x-ray can have an energy greater than the energy of the electron hitting the material • Duane-Hunt Law • nmax = Voe/h

  20. The Compton Effect 1923 • E2=(pc)2 + (mc2)2 • A photon has zero mass, but has momentum • E=pc • E=hn=pc • p=hn/c=h/l • Photon-Electron collision is like any classical collision since a photon has particle-like behavior

  21. Questions?

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