1. Department of Electronics
Nanoelectronics 06
Atsufumi Hirohata
Department of Electronics
09:00 Tuesday, 29/January/ (P/T 006)

2. 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

3. 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)

4. 06 Schrödinger Equation Schrödinger equation Particle position
Hermite operator
1D trapped particle

5. 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

6. 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

7. 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 :

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

9. 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, *

10. 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

11. 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.

12. 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 *

13. 1D Trapped Particle (Cont'd)
Therefore, the wave function is written as
=𝐶sin 𝑛𝜋 𝐿 𝑥
In order to satisfy the normalisation : E
0 𝐿 𝜓 ∗ 𝜓𝑑𝑥 = 0 𝐿 𝐶 2 sin 2 𝑛𝜋 𝐿 𝑥 𝑑𝑥
= 𝐶 2 ∙ 𝐿 2
∴ 𝐶 2 ∙ 𝐿 2 =1
∴𝐶= 2 𝐿
Finally, the wave function is obtained as
𝜓 𝑛 𝑥 = 2 𝐿 sin 𝑛𝜋 𝐿 𝑥 *

14. １D Harmonic Oscillator
A crystalline lattice can be treated as a harmonic oscillator :
Oscillation E
Oscillation E
 Phonon oscillation *