The Schrodinger Equation

Dae Yong JEONG

Inha University

2nd week

Review

Electron device (electrical energy traducing)

Understand the electron properties

energy point of view

To know the energy of electron… Wave eq. Electron (Particle – Wave Duality)

From wave viewpoint phase velocity

From wavelike particle viewpoint group velocity

What can we do with wave equation?

Energy Operator onto Wave Eq. Solve Schrodinger Eq.

Possible energy level which electron can possess.

ddxdydz **

The probability of finding an electron in a volume element dτ

The time-independent Schrodinger Eq.

Let’s derive the Wave-Eq for electron in materials.

The condition The property of the surroundings of the electron does not change with time.

Separate the “Spatial” and “temporal” part in the wave eq.

Don’t need to consider the “temporal” part

iwtezyxtzyx ),,(),,,(

2

0

2

2

2

0

kejkx

jkejkx

e

jkx

jkx

jkx

o

D-3for 1 2

2

k

Then, we can get the following relation

t

jtj

E

jejt

e

tj

tj

For time dependant

(cont) The time-independent Schrodinger Eq.

02

2

1

22

1

22

1

2

2

2

2

2

22222

VEm

Vmx

E

Vm

kV

m

hV

m

pVmvE

EEE potkintot

Operator (Refer to the Quantum mechanics):

From the classical mechanics

2

2

1

k

To calculate the “E” of electron in material.

‘V” should be defined!!

How can we define the potential energy “V”?

Ch. 4 The Schrodinger Equation

Review and Contents

Eq. for Electron Wave with energy operatior (Schrodinger Eq.)

To solve the Schrodinger Eq., “potential energy, V” should be

defined.

We will solve the Schrodinger Eq. for

Extreme case for easy calculation and understanding

Free electron (without potential)

Bound electron (trapped in potential box)

But sometimes, electron escapes by tunneling effect

More realistic case (periodic potential in Crystal)

Fig 3.20 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

The electron in the hydrogenic

atom is attracted by a central force

that is always directed toward the

positive Nucleus.

Spherical coordinates centered at

the nucleus are used to describe

the position of the electron.

The PE of the electron depends

only on r.

Electron in a Potential (electron in H atom)

From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

Electron wavefunctions and the electron energy are obtained by solving the

Schrödinger equation

Electron’s PE V(r) in hydrogenic atom is used in the Schrödinger equation

r

ZerV

o4)(

2

From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

Electron energy is quantized

Electron energy in the hydrogenic atom is quantized.

n is a quantum number, 1,2,3,…

222

24

8 nh

ZmeE

o

n

eV13.6 J1018.28

18

22

4

h

meE

o

I

Ionization energy of hydrogen: energy required to remove the electron

from the ground state in the H-atom

From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

Fig 3.22 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

(a) The polar plots of Yn,(, ) for 1s and 2p states.

(b) The angular dependence of the probability distribution, which is proportional to

| Yn,(, )|2.

Fig 3.23 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

The energy of the electron in the hydrogen

atom (Z = 1).

Fig 4.1 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

Formation of molecular

orbitals, bonding, and

antibonding ( and

*) when two H atoms

approach each other.

The two electrons pair

their spins and occupy

the bonding orbital .

Electron in H2 molecule (two atoms)

From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

Linear Combination of Atomic Orbitals

Two identical atomic orbitals 1s on atoms A and B can be combined linearly in two different ways to generate two

separate molecular orbitals and *

and * generated from a

linear combination of atomic orbitals (LCAO)

)()( 11 BsAs rr

)()( 11* BsAs rr

Wavefunction around B Wavefunction around A

Fig 4.2 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

(a) Electron probability distributions for bonding and antibonding orbitals, and *.

(b) Lines representing contours of constant probability (darker lines represent greater

relative probability).

Fig 4.3 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

(a) Energy of and * vs. the interatomic separation R.

(b) Schematic diagram showing the changes in the electron energy as two isolated H

atoms, far left and far right, come together to form a hydrogen molecule.

Fig 3.17 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

Tunneling (Application)

Fig 3.18 From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

Scanning Tunneling Microscopy (STM) image of a graphite surface where

contours represent electron concentrations within the surface, and carbon

rings are clearly visible. Two Angstrom scan. |SOURCE: Courtesy of Veeco

Instruments, Metrology Division, Santa Barbara, CA.

From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

STM image of Ni (100) surface

SOURCE: Courtesy of IBM

STM image of Pt (111) surface

SOURCE: Courtesy of IBM