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Scienza dei Materiali -lecture 3- Vanni Lughi Università degli Studi di Trieste Dipartimento di Ingegneria e Architettura A.A. 2021-22

Scienza dei Materiali -lecture 3-

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Page 1: Scienza dei Materiali -lecture 3-

Scienza dei Materiali-lecture 3-

Vanni Lughi

Università degli Studi di TriesteDipartimento di Ingegneria e Architettura

A.A. 2021-22

Page 2: Scienza dei Materiali -lecture 3-

Quantum mechanics – basic concepts

1. Introduction of quanta: Plank (black body radiation), Einstein (Photoelectric Effect*)

2. Wave-particle dualism: Einstein (P.E. effect), De Broglie (Matter Waves*)

3. Matter waves and wavepackets

4. Uncertainty

5. Statistical approach

6. Waves and wave equation

7. Schroedinger Equation

Page 3: Scienza dei Materiali -lecture 3-

Quantum mechanicsBlack body radiation

Page 4: Scienza dei Materiali -lecture 3-

*Sidetrack: Photoelectric Effect

Einstein’s hypotheses for explaining the effect:

• Radiation is made up of photons• Photons are absorbed or emitted in descrete amounts• Photon energy is h• Photons behave like waves

Hertz Experiments

Ekin = h - W

Page 5: Scienza dei Materiali -lecture 3-

*A sidetrack: Photoelectric Effect

Application of the photoelectric effect: X-Ray Photoelectron Spectroscopy (XPS, ESCA)

295 290 285 280

0

2000

4000

6000

8000

10000

12000

Inte

sity (

a.u

.)

Binding Energy (eV)

XPS_C 1s

C - C / C = C

C - O

C = O

O - C = O

Envelope

600 400 200 0

0

20000

40000

60000

Inte

sit

y (

a.u

.)

Binding Energy (eV)

Page 6: Scienza dei Materiali -lecture 3-

En

erg

y

+ -

*A sidetrack: Photoelectric Effect

A “brother” of the PE effect: Photovoltaic Effect

Page 7: Scienza dei Materiali -lecture 3-

Quantum Mechanics – basic concepts

Light is a wave BUT can be emitted, absorbed, propagated in quanta (photons) with energy and momentum:

Matter has a partcle-like properties, BUT also wave-like properties(matter waves*, de Broglie hypothesis):

2. Wave-particle dualism: Einstein, De Broglie

Page 8: Scienza dei Materiali -lecture 3-

What is the wavelengthof an electron?

Excercises:• Calculate the wavelength of an electron (algebraic formula)• Calculate the wavelength of a “slow” electron (e.g. one accelerated by a V = 54 V )• Read about Davisson and Germer’s experiment (they find that the electrons can diffract from metal surfaces

Experimental proof of matter waves)• Calculate the wavelength of “fast” electrons (hint: account for the relativistic speed correction). Use the

formula to estimate the error when neglecting relativistic effects for electrons accelerated by V = 54 MV

Page 9: Scienza dei Materiali -lecture 3-

*Sidetrack:Matter waves and electron microscopy

Page 10: Scienza dei Materiali -lecture 3-

*Sidetrack: Matter waves and LEEDThank you Davisson and Germer!

Page 11: Scienza dei Materiali -lecture 3-

Sidetrack in a sidetrack:Fourier Analysis

Any function can be expressed as a Fourier series

Fourier transform can simplify the analysis (e.g. FT-IR spectroscopy)

Page 12: Scienza dei Materiali -lecture 3-

Quantum Mechanics – basic concepts3. Matter waves and wave packets

Single plane wave

Wave packet Example: two waves

𝑣𝑝 =𝜔

𝑘

Page 13: Scienza dei Materiali -lecture 3-

Quantum Mechanics – basic concepts3. Matter waves and wave packets

The velocity of the particle is the group velocity of the wavepacket

𝑣𝑔 = ቤ𝑑𝜔

𝑑𝑘𝑘0

= ቚ𝑣𝑝𝑘0+ 𝑘 ቤ

𝑑𝑣𝑝𝑑𝑘

𝑘0

Example: non relativistic particleHomework: relativistic particle

Page 14: Scienza dei Materiali -lecture 3-

Quantum Mechanics – basic concepts3. Matter waves and wave packets. Example: rectangular wavepacket

What is the uncertainty?

Page 15: Scienza dei Materiali -lecture 3-

Quantum Mechanics – basic concepts3. Matter waves and wave packets

Page 16: Scienza dei Materiali -lecture 3-

Quantum Mechanics – basic concepts3. Matter waves and wave packets

Dispersive Wavepackets

Page 17: Scienza dei Materiali -lecture 3-

Quantum Mechanics – basic concepts3. Matter waves and wave packets

https://demonstrations.wolfram.com/WavepacketForAFreeParticle/

https://physics.nyu.edu/~ts2/Animation/quantum.html

https://www.ncnr.nist.gov/staff/dimeo/se_sim.html

Page 18: Scienza dei Materiali -lecture 3-

Quantum mechanics – basic concepts

4. Uncertainty