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Department of Electronics Nanoelectronics 05 Atsufumi Hirohata 12:00 Wednesday, 21/January/2015 (P/L 006)

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Quick Review over the Last Lecture Light quantum : E = ( h ) ( Compton ) scattering Electron ( interference ) ( De Broglie ) wave

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Contents of Nanoelectonics I. Introduction to Nanoelectronics (01) 01 Micro- or nano-electronics ? II. Electromagnetism (02 & 03) 02 Maxwell equations 03 Scalar and vector potentials III. Basics of quantum mechanics (04 ~ 06) 04 History of quantum mechanics 1 05 History of quantum mechanics 2 06 Schrödinger equation IV. Applications of quantum mechanics (07, 10, 11, 13 & 14) V. Nanodevices (08, 09, 12, 15 ~ 18)

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05 History of Quantum Mechanics 2 Rutherford’s model Bohr’s model Balmer series Uncertainty principle

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Early Models of an Atom In 1904, J. J. Thomson proposed a plum pudding model : Negatively charged “plums” (electrons) are surrounded by positively charged “pudding.” * http://www.wikipedia.org/ ** http://www.nararika.com/butsuri/kagakushi/genshi/genshiron.htm In 1904, Hantaro Nagaoka proposed a Saturn model : Negatively charged electrons rotate around positively-charged core.

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Rutherford's Model In 1909, Ernest Rutherford carried out a Au foil experiment : -ray was introduced onto a very thin Au foil. * http://www.wikipedia.org/ cannot be explained by the plum pudding model, and the Saturn model was adopted. Rutherford back scattering was observed. The size of the core is estimated to be 10 -14 m.

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Rutherford Back Scattering Powerful tool for materials analysis : * http://www.toray-research.co.jp/kinougenri/hyoumen/hyo_006.html Analyser Sample atoms Example

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Limitation of Rutherford's Model In classical electromagnetism, An electron rotating around the core loses its energy by irradiating electromagnetic wave, and falls into the positively-charged core. In 1913, Niels H. D. Bohr proposed a quantum rule : An electron can permanently rotate around the core when occupying an orbital with m e : electron mass, v : electron speed, r : orbital radius, n : quantum number and h : Planck constant stable state energy levels ( n = 1, 2, 3, …) * http://www.wikipedia.org/

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Bohr's Model Allowed transitions between the energy levels : From the energy level of E n to that of E n ’, a photon is absorbed when E n ’ - E n > 0. a photon is released when E n ’ - E n ’ < 0. * http://www.wikipedia.org/ Meaning of the quantum rule : De Broglie wave length is defined as By substituting this relationship into the quantum rule, Electron as a standing wave in an orbital.

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Proof of Bohr's Model In 1914, James Franck and Gustav L. Hertz proved discrete energy levels : An acceleration voltage is tuned to allow diluted gas to absorb the energy. * http://nobelprize.org/ ** http://www.wikipedia.org/ A proof of the quantum rule

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Balmer Series in a Hydrogen Atom In 1885, Johann Jakob Balmer proposed an empirical formula : Balmer series observation : Balmer formula : * http://www.wikipedia.org/ : wavelength, B : constant (364.56 nm), n = 2 and m : an integer ( m > n ) A proof of the discrete electron orbitals

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Rydberg Formula In 1888, Johannes R. Rydberg generalised the Balmer formula : Rydberg formula for Hydrogen : Rydberg formula for atoms : * http://www.wikipedia.org/ R H : Rydberg constant (10973731.57 m −1 ) vac : wavelength of the light emitted in a vacuum, Z : atomic number, m and n : integers

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Balmer Series in a Hydrogen Atom Other series were also found : * http://www.bigs.de/en/shop/htm/termsch01.html Lyman series (1906) Ultra violet Balmer series (1885) Visible light Brackett series (1922) Near infrared Paschen series (1908) Infrared Pfund series (1924) Far infrared Humphreys series (1953) Far infrared

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Heisenberg's Thought Experiment In 1927, Werner Karl Heisenberg proposed the uncertainty principle : Resolution of a microscope is defined as * http://www.wikipedia.org/ In order to minimise, Larger Small Here, Small large p (damage to samples) Also, larger difficult to identify paths incident wave x -direction electron In addition, incident wave can be reflected within Momentum along x is within

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Observation of the Uncertainty Principle Zero-point motion : He does not freeze near 0 K under atmospheric pressure. Spin fluctuation in an itinerant magnet : * http://www.e-one.uec.ac.jp/~kuroki/cobalt.html

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Precise Quantum Physical Definition Relationship between an observation error in position and disturbance in momentum : Heisenberg’s uncertainty principle using operators : Heisenberg’s uncertainty principle using standard deviations : Here, communication relation : If A and B are commutative operators, the right-hand side is 0. This leads the error and disturbance to be 0. However, A and B are not commutative operators, the right-hand side is not 0. This leads the error and disturbance to have a trade-off. → Precise definition

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Corrections to Heisenberg’s Uncertainty Principle Ozawa’s relationship : * Optical proof : ** * M. Ozawa, Phys. Rev. A 67, 042105 (2003); ** S.-Y. Baek et al., Sci. Rep. 3, 2221 (2013).

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Experimental Data Error and disturbance measured : ** S.-Y. Baek et al., Sci. Rep. 3, 2221 (2013).

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Ozawa’s Relationship Ozawa’s relationship : ** S.-Y. Baek et al., Sci. Rep. 3, 2221 (2013).

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