# Department of Electronics

## Presentation on theme: "Department of Electronics"— Presentation transcript:

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

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

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)

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

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

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

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 :

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

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, *

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

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.

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 *

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 *

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