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Department of Electronics Nanoelectronics 06 Atsufumi Hirohata 10:00 27/January/2014 Monday (G 001)

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Quick Review over the Last Lecture ( Rutherfords ) model : But suffers from a limitation : ( energy ) levels ( Bohrs ) model : Experimental proof : ( Balmer ) series ( Uncertainty ) principle : ( )

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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 Scholar 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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06 Schrödinger Equation Schrödinger equation Particle position Hermite operator 1D trapped particle

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Schrödinger Equation In order to express the de Broglie wave, Schrödinger equation is introduced in 1926 : E : energy eigen value and : wave function Wave function represents probability of the presence of a particle * : complex conjugate (e.g., z = x + iy and z * = x - iy ) Propagation of the probability (flow of wave packet) : Operation = observation : operator de Broglie wave observed results

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Complex Conjugates in the Schrödinger Equation In the Schrödinger equation, 1D wave propagating along x -direction is expressed as For the wave along -x -direction, At t = 0, For B = - A and = -, these waves cannot be distinguished. On the other hand, waves with complex conjugates : At t = 0, contains momentum-vector information

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Position of a Particle In the Schrödinger equation, the meaning of a particle position is : Expectation = Expected numbers Probability Standard deviation : Expectation of a momentum : Eigen value and eigen function : If a function stays the same after applying an operator, for example, Eigen value : Eigen function :

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Hermite Operator Assuming a Hamiltonian satisfies : H : Hermite operator By using this assumption, Here, Communication relation : [A,B] = AB - BA

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Bra-ket Notation Paul A. M. Dirac invented a notation based on Heisenberg-Borns matrix analysis : A wave function consists of a complex vector : Here, By assuming, a ket vector is defined as * In order to calculate a probability,

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Bra-ket Notation (Cont'd) A bra vector is defined as Therefore, an inner product is written as This bra-ket notation satisfies bra + ket = bracket

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1D Trapped Particle In 1D space, a free particle motion is expressed by the Schrödinger equation : For a time-independent case, (x) = e x can be assumed. x 0 L V (x) = 0V (x) = 0 By using, Since the particle is trapped in 0 x L, = 0 outside of this region. Boundary conditions : (0) = (L) = 0.

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1D Trapped Particle - Zero-point motion According to the Schrödinger equation, energy is defined as Even for the minimum eigen energy with n = 1, zero-point motion *

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1D Trapped Particle (Cont'd) Therefore, the wave function is written as In order to satisfy the normalisation : Finally, the wave function is obtained as * E

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D Harmonic Oscillator A crystalline lattice can be treated as a harmonic oscillator : * E Oscillation E Phonon oscillation

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