L1_P02

1.021 , 3.021, 10.333, 22.00 : Introduction to Modeling and Simulation : Spring 2012 Part II – Quantum Mechanical Methods : Lecture 1

It ’ s A Quantum W orld: The Theory of Quantum Mechanics

Jeffrey C. Grossman

Department of Materials Science and Engineering

Massac husetts Institute of T ec hnology

3.021 Content O v er vi e w

I. P a r ticle a nd continuum methods

1. Atoms, molecules, chemistr y

2. Contin uum modeling a ppr oaches and solution a ppr oaches 3.Statistical mechanics

4. Molecular dynamics, Monte Carlo 5.Visualization and data anal ysis

6. Mechanical pr oper ties – a pplication: ho w things fail (and ho w to pr e v ent it) 7.Multi-scale modeling paradigm

8. Biological systems (sim ulation in bioph ysics) – ho w pr oteins w ork and ho w to model them

II. Qua ntum mecha nical methods we ar e her e

We l c o m e t o P a r t 2 !

The next 11 lectur es will co v er atomistic quantum modeling of materials.

Note: ther e will be a substitute lectur er on T uesda y , April 10 and no class on Thursda y , April 12.

Par t II T opics

1. It ’ s a Quantum W orld: The Theor y of Quantum Mechanics

2. Quantum Mechanics: Practice Mak es P erf ect

3. Fr om Man y-Body to Single-Par ticle; Quantum Modeling of Molecules

4. Application of Quantum Modeling of Molecules: Solar Thermal Fuels

5. Application of Quantum Modeling of Molecules: Hydr ogen Storage

6. Fr om Atoms to Solids

7. Quantum Modeling of Solids: Basic Pr oper ties

8. Advanced Pr op . of Materials: What else can w e do?

9. Application of Quantum Modeling of Solids: Solar Cells Par t I

10. Application of Quantum Modeling of Solids: Solar Cells Par t II

11. Application of Quantum Modeling of Solids: Nanotechnolog y

Lesson outline

• W h y quantum mechanics?

• W a v e aspect of matter

• Interpr etation

• The Schrödinger equation

• Simple examples

quantum

modeling

Multi-scale modeling

Courtesy of Elsevier, Inc., http://www.sciencedirect.com . Used with permission.

It ’ s a quantum w orld!

Motivation

If w e understand elect r ons , then w e understand e v er ything .

(almost) ...

?

electrical

pr oper ties

?

optical

pr oper ties

mec hanical

pr oper ties

?

Quantum modeling/ sim ulation

A simple i r on atom ...

Image © Greg Robson on Wikimedia Commons. License: CC-BY-SA. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/ .

W h y Quantum Mechanics?

Accurate/predictive structural/atomistic properties, when we need to span a wide range of coordinations, and bond-breaking, bond-forming takes place.

(But beware of accurate energetics with poor statistics !)

EDIP Si potential Tight-binding

Elect r onic , optical, magnetic p r oper ties

Image removed due to copyright restrictions. Please see Vangberg, T. R. Lie, and A. Ghosh. "Symmetry-Breaking Phenomena in Metalloporphyrn in Pi-Cation Radicals."

Courtesy of Elsevier, Inc., http://www.sciencedirect.com . Used with permission.

Jahn- T eller eff ect in Non-r esonant Raman in silicates (Lazzeri and Mauri) porph yrins (A. Ghosh)

Reactions

1,3-butadiene + eth ylene → cyclohex ene

Standa r d Model of Matter

• Atoms are made by MASSIVE, POIN T - LIKE NUCLEI (protons+neutrons)

• Surrounded by tightly bound, rigid shells of CORE ELECTRONS

• Bound together by a glue of V ALENCE ELECTRONS (gas vs. atomic orbitals)

It ’ s r eal!

Cu-O Bond Ti-O Bond (experiment) (theor y)

Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Zuo, J., M. Kim, et al. "Direct O bservation of d-orbital H oles and Cu-Cu

Impor tance of Solving f or this Pictu r e with a Computer

• It p r o vides us mic r oscopic understanding

• It has pr edictiv e po w er (it is “first-principles”)

• It allo ws cont r olled “gedan k en” experiments

• Challenges:

‣ Length scales

‣ Time scales

‣ Accuracy

W h y quantum mechanics?

Classical mechanics Ne wton ’ s l a ws (1687) F ⌃ =

P r oblems?

d ( m ⌃ v ) dt

W h y quantum mechanics?

P r oblems in classical p h ysics that led to quantum mechanics:

• “classical atom”

• quantization of p r oper ties

• w a v e aspect of matter

• (black-body radiation), ...

Quantum mechanists

W erner Heisenberg, Max Planck , Louis de Br oglie , Alber t Einstein , Niels Boh r , Erwin Schrödinge r , Max Born, J ohn v on Neumann,

Paul Dirac , W olfgang Pauli (1900 - 1930)

“Classical atoms”

e -

+

p r oblem:

accelerated charge causes radiation, atom is not stable!

h yd r ogen atom

Quantization of p r oper ties

_

_ _

_

photoelectric E e

eff ect

r A

Image by MIT OpenCourseWare.

E = k ( r — r A ) = h ( v — v A )

h = 2 v k = 6 . 6 · 10 — 34 W atts ec. 2

Einstein: photon E = k r

Quantization of p r oper ties

atomic spectra

Quantization of p r oper ties

Energ y

E = k r

possible energ y states

The D oubl e - S lit Experi ment

Dr Q uant um expl ai ns t he doubl e - sl i t experi m ent .

F r o m th e film : W hat t he bl eep do w e know ?

S ee Lect ure 1 vi deo f or f ul l cl i p.

"A n y one who is not shoc k ed b y quantum theor y has not understood it"

Niels Bohr

Schrödinger ’ s Cat

Courtesy of Dan Lurie on Flickr. License: CC-BY-NC-SA.

Erwin Schr ö dinger

(1887 – 1961)

"I don't li k e it, and I'm sor r y I e v er had an ything to do

EPR Parad o x

Einstein–Podolsky–Rosen

This image is in the public domain. Source: Clkrer.com .

W a v e-Par ticle Duality

• W aves have particle-like properties:

• Photoelectric effect: quanta (photons) are exchanged discretely

• Energy spectrum of an incandescent body looks like a gas of very hot particles

• Particles have wave-like properties:

• Electrons in an atom are like standing waves (harmonics) in an organ pipe

• Electrons beams can be dif fracted, and we can see the fringes

Interf e r ence Patterns

Constructi v e Interfer ence

Destructi v e Interfer ence

+

-

A 1

A 2

Resultant A 1 + A 2

Resultant A 1 - A 2

W a v e Interactions

Image by MIT OpenCourseWare.

Interf e r ence Patterns

license. For more information, see http://ocw.mit.edu/help/faq-fair- use/ .

Bucky- and soccer balls

30 Courtesy of the University of Vienna. Used with permission.

When is a par ticle li k e a w a v e?

W a v elengths: Elect r on: 10 -10 m

C60 Fulle r ene: 10 -12 m

Base ball: 10 -34 m

Human w a v elength: 10 -35 m

20 o r der s of ma gnitude smaller tha n the dia meter of the n 3 u 1 cleus of a n atom!

Classical vs. quantum

It is the mechanics of w a v es rather than classical par ticles .

This photo is in the public domain.

W a v e aspect of matter

light matter

w a v e character par ticle character

_

_ _

_

Image by MIT OpenCourseWare.

Image in public domain. See Wikimedia Commons .

Mechanics of a Par ticle

d 2 r 

r ( t ) v ( t )

m dt 2

F ( r )

The sum of the kinetic and potential energ y is conser v ed.

M

m

V

v

Image by MIT OpenCourseWare.

Description of a W a v e

The w a v e is an excitation (a vibration): W e need to kno w the amplitude of the excitation at e v er y point and at e v er y instant

   ( r , t )

Mechanics of a W a v e

Fr ee par ticle , with an assigned momentum:

 ( r , t )  A exp[ i ( k  r   t )]

W a v e aspect of matter

par ticle : E and momentum p ⇥

w a v e: fr equency and w a v e v ector ⇥ k

E = h v = k r

p → = k → k = h

Z

→ k

| → k |

h

de B r oglie: fr ee par ticle can be described as

plane w a v e

G ( → r , t ) = Ae i ( → k · → r — v t )

with Z =

mv

37

H o w do w e desc r ibe the physical behavior of pa r ticles as w a v es?

The Schrödinger equation

a w a v e equation:

second derivativ e in space first derivativ e in time

k 2

— 2 m D

2

+ V ( → r , t ) G ( → r , t ) = i k G ( → r , t )

6 t

⇥

6

2

k 2

H = — 2 m D

Hamiltonian

p 2

+ V ( → r , t ) =

p ⇤ = — i k D

= 2 m + V = T + V

In practice ...

H time independent: G ( → r , t ) = G ( → r ) · f ( t )

f ˙ ( t )

i k f ( t ) =

H G ( → r )

G ( → r )

= const. = E

—

k

G ( → r , t ) = G ( → r ) · e

i E t

H G ( → r ) = E G ( → r )

time independent Schrödinger equation stationar y Schrödinger equation

Par ticle in a b o x

boundar y conditions

Boundar y condition s caus e quantization !

41

Schrödinger equation

general solution

Par ticle in a b o x

quantization

normalization

solution

Image adapted from Wikimedia Commons, http://commons.wikimedia.org .

quantum n umber

Simple examples

?

electr on in squar e w ell

Image adapted from Wikimedia Commons, http://commons.wikimedia.org .

electr on in h ydr ogen atom

Harmonic oscillator

solv e Schrödinger equation

Harmonic oscillator

 3

 2

3

n=3 n=2

n=1

7 h 

2

5 h 

2

3 h 

2



n=0

1 h 

2

 2

 x

2

 1

 2

1

Classical limits

 0

 2

0

Graphs of the quant um harmonic oscillat or potential and wavefunct io ns.

Image by MIT OpenCourseWare.

Interp r etation of a w a v efunction

G ( → r , t ) w a v e function (complex)

| G | = G G

2 interpr etation as pr obability to find par ticle

(that is, if a measur ement is made)

Image by MIT OpenCourseWare.

⇤

∫

G G ⇥ dV = 1

— ⇤

Connection to r eality?

potential: 1/r

e -

+

h yd r ogen atom

Courtesy of Wikimedia Commons . License CC-BY-SA. This content is excluded from our Creative

Ma n y Interp r etations of

Quantum Mechanics!

Adapted from Wikipedia . This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair- use/ .

Summar y

R e vi e w

• W h y quantum mechanics?

• W a v e aspect of matter

• Interpr etation

• The Schrödinger equation

• Simple examples

Literatu r e

• Gr eine r , Quantum Mechanics: An Int r oduction

• Thalle r , Visual Quantum Mechanics

• F e ynman, The F e ynman Lectur es on P h ysics

• wikipedia, “quantum mechanics”, “Hamiltonian operator”,

“Schrödinger equation”, ...

MIT OpenCourseWare http://ocw.mit.edu

3 . 021 J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modelling and Simulation

Spring 20 1 2

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .