chapter 5physics.uwyo.edu/~teyu/class/chapter5.pdfโขcalculate the constructive diffraction angle,...
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Chapter 5The Wavelike Properties of Particles
The de Broglie Hypothesis (1925)
๐ =๐ธ
โ
๐ =โ
๐
Electrons could be considered as wave and the frequency and the wavelength are related to energy and momentum as;
This gives physical interpretation of Bohrโs quantization of the angular momentum of the electron in hydrogenlike atoms.
Review of E and P vs f and l
๐ธ = ๐พ๐๐2 = ๐๐2 2 + ๐๐ 2
Massive particle
Massless particle (E&M wave)
๐ธ = ๐๐ ๐ธ = โ๐ ๐ =๐ธ
๐=โ๐
๐=โ
๐
๐ =๐ธ
โ๐ =
โ
๐
๐ =โ
๐
Non-relativistic massive particle
๐ธ โ ๐ธ๐ =๐2
2๐
๐ = 2๐๐ธ๐
Example
โข Compute the de Broglie wavelength of an electron whose kinetic energy is 10 eV.
The Davisson-Germer Experiment (1927)
Interference/Diffraction with wave
d
q
๐๐ = 2๐ ๐ ๐๐๐
Crystal lattice of atoms
D
๐๐ = 2๐ ๐ ๐๐๐
d
d
d
d varies depend on the plane you are consideringD is a intrinsic property of a particular crystal, a single value
Electron Diffraction
๐๐ = 2๐ ๐ ๐๐๐
๐๐ = ๐ท ๐ ๐๐2๐ผ
d: spacing between certain set of planes
D: spacing between lattice points
Example
โข Calculate the constructive diffraction angle, ๐, of Ni (D = 0.215 nm) for this particular crystal planes (indicated by red line), when using electrons with 54 eV energy. Assuming the crystal is square/cubic.
Electron Diffraction of Al Foil (1927)
G. P. Thomson
G. P. Thomson (son of J. J. Thomson) shared the Nobel Prize in Physics in 1937 with Davisson
State-of-the-art electron diffraction instrument: Low Energy Electron Diffraction optics
400
300
200
100
0
4003002001000
Cu(001) crystal with 134 eV electron
Diffraction with Other Particles
Cu with neutronNaCl with neutron
Oxygen nuclei with proton
LiF with He
All matter has wavelike as well as particlelike properties.
Example
โข Compute the wavelength of a neutron with kinetic energy of 0.0568 eV. Cu has lattice constant (atom-atom distance) of 0.36 nm. If you use this neutron beam to do the diffraction experiment, what would be the first angle you could see the diffraction?
Wave Packets
โข The matter wave describe the probability of finding the particle
๐2๐ฆ
๐๐ฅ2=
1
๐ฃ2๐2๐ฆ
๐๐ก2Wave Equation
๐ฆ ๐ฅ, ๐ก = ๐ฆ0๐๐๐ ๐๐ฅ โ ๐๐ก
๐ =2๐
๐
๐ =2๐
๐
๐ฃ๐ = ๐๐ =๐
๐=๐
๐
Wave number
Angular(phase) frequency
Phase velocity
Wave Packet
๐ฆ ๐ฅ, ๐ก = ๐ฆ0๐๐๐ ๐1๐ฅ โ ๐1๐ก + ๐ฆ0๐๐๐ ๐2๐ฅ โ ๐2๐ก
๐ฆ ๐ฅ, ๐ก = 2๐ฆ0๐๐๐ 1
2โ๐๐ฅ โ
1
2โ๐๐ก ๐ ๐๐ ๐๐ฅ โ ๐๐ก
http://www.acs.psu.edu/drussell/Demos/superposition/beats.gif
https://upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Wave_group.gif/440px-Wave_group.gif
https://upload.wikimedia.org/wikipedia/commons/c/c7/Wave_opposite-group-phase-velocity.gif
Wave Packet (adding more than two waves)
๐ฃ๐ =๐๐
๐๐
Group velocity and phase velocity
๐ฃ๐ =๐
๐
๐ฃ๐ =๐๐
๐๐Phase velocityGroup velocity
Directly related to particle velocity
โข Non-dispersive medium: phase velocity is independent of k(sound waves in air, E&M wave in vacuum)โข Dispersive medium: phase velocity varies with k(E&M wave in medium, water wave, electron wave)
๐ฃ๐ = ๐ฃ๐ + ๐๐๐ฃ๐
๐๐
https://upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Wave_group.gif/440px-Wave_group.gif
https://upload.wikimedia.org/wikipedia/commons/c/c7/Wave_opposite-group-phase-velocity.gif
Particle Wave Packets
๐ฃ๐ =๐๐
๐๐=๐๐ธ
๐๐=
๐
๐=๐๐ฃ
๐= ๐ฃ
๐ธ = โ๐ = โ๐
๐ =โ
๐= โ๐
๐ธ = โ๐ =๐2
2๐=โ2๐2
2๐
๐ =โ๐2
2๐
๐ฃ๐ =๐
๐= ๐๐ =
๐ธ
๐=
๐
2๐=๐๐ฃ
2๐=๐ฃ
2
Probabilistic Interpretation of the Wave Function
https://www.youtube.com/watch?v=7u_UQG1La1o
Uncertainty Relation (Classical)
โ๐โ๐ฅ~1
โ๐โ๐ก~1
Example
โข The frequency of the alternating voltage produced at electric generating stations is carefully maintained at 60.00 Hz (in North America). The frequency is monitored on a digital frequency meter in the control room. For how long must the frequency be measured and how often can the display be updated if the reading is to be accurate to within 0.01 Hz?
Uncertainty Relation (Quantum)
โ๐โ๐ฅ~โ
โ๐ธโ๐ก~โ
โ๐โ๐ฅ โฅโ
2
โ๐ธโ๐ก โฅโ
2
Particle confined in a 1D box with length of L
โ๐ 2 = ๐ โ ๐ ๐๐ฃ๐2 = ๐2 โ 2๐ ๐ + ๐2 ๐๐ฃ๐ = ๐2 โ ๐2
โ๐โ๐ฅ โฅ โ โ๐ โฅโ
๐ฟ
๐ธ =๐2
2๐=
โ๐ 2
2๐โฅ
โ2
2๐๐ฟ2
Example
โข If the particle in a one-dimensional box of length L = 0.1 nm (about the diameter of an atom) is an electron, what will be its zero-point energy?
Lifetime of an energy state
โ๐ธโ๐ก~โUncertainty Principle for energy and time
If for a particular atomic spectrum transition, the lifetime, t, is in the order of 10 ns (10โ8๐ ), there is an energy uncertainty as:
โ๐ธ~โ
๐โ 10โ7๐๐
โ๐ =โ๐ธ
โ~โ
๐
1
โ=
1
2๐๐โ 107๐ป๐ง
Thus, the uncertainty in frequency of the emitted light is:
ฮ๐
๐โ
ฮ๐ธ
๐ธ โ ๐ธ0Then, the uncertainty in wavelength is:
Example
โข When you measure a spectrum emitted by He atom, you found out there is a peak width in intensity vs energy spectrum is around 0.1 meV. Assuming this peak width originates from the energy uncertainty of the excited state of He atom, find out the lifetime of the excited state of He atom.