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**Department of Electronics**

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

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**Quick Review over the Last Lecture**

( Rutherford’s ) model : ( Uncertainty ) principle : ( ) But suffers from a limitation : ( energy ) levels ( Bohr’s ) model : Experimental proof : ( Balmer ) series

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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 : de Broglie wave observed results operator

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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-Born’s 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 : V (x) = 0 For a time-independent case, (x) = e x can be assumed. x L 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 : E Finally, the wave function is obtained as *

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１D Harmonic Oscillator A crystalline lattice can be treated as a harmonic oscillator : Oscillation E Oscillation E Phonon oscillation *

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