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3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

3.012 Fund of Mat Sci: Bonding Lecture 4

CURIOSITY KILLED THE CAT

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Specific Heat of Graphite (Dulong and Petit)

00

500

500

1000

1000

1500

1500

2000

2000

2500

2500

Temperature (K)

C

P(J.K

-1.kg-1)

Figure by MIT OCW.

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Last Time

Expectation values of the energy in aninfinite well (particle-in-a-box)

Absorption lines (linear conjugated

molecules)

Particles in 3-dim box (quantum dots,

Farbe defects)

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4/243.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Homework for Fri 23

Study: 14.1, 14.2, 14.3 Read: 14.4

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Metal Surfaces (I)

2 2

2( ) ( ) ( )

2

dV x x E x

m dx + =

h

0 x

V(x)

V0

Figure by MIT OCW.

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Metal Surfaces (II)

y5

(y)

0

-0.5

0.5

1

-1

-5-10-15

Figure by MIT OCW.

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Scanning Tunnelling Microscopy

Piezodrive

Feedback

Loop

Sample

Probe Tip

Constant-Current

Contour

+

Sample

s

Ef

I/V

K = = 1.1 A-12m

h2( (

e-2Ks

= density of states

1/2

Ef - eV

Tip

.

Figure by MIT OCW.

Figure by MIT OCW.

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Wavepacket tunnelling through a

nanotube

Images removed for copyright reasons. See http://www.mfa.kfki.hu/int/nano/online/kirchberg2001/index.html
http://www.mfa.kfki.hu/int/nano/online/kirchberg2001/index.htmlhttp://www.mfa.kfki.hu/int/nano/online/kirchberg2001/index.html

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Dirac Notation

Eigenvalue equation:

Expectation values:

iiiiiii ErdrrVm

rHH =

+==

rrrhr)()(

2)( 2

2*

ijjiiii aA ==

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Operators and operator algebra

Examples: derivative, multiplicative

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Product of operators, and commutators

1, =

dx

dx

BA

,

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Linear and Hermitian

[ ] AAA

+=+

AA =

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Examples: (d/dx) and i(d/dx)

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First postulate All information ofan ensemble of identical physical

systems is contained in the wavefunction (x,y,z,t),which is complex, continuous, finite, and single-

valued; square-integrable. finiteisi.e.2

rdr

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Second Postulate

For every physical observable there is acorresponding Hermitian operator

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Hermitian Operators1. The eigenvalues of a Hermitian operator are real

2. Two eigenfunctions corresponding to different eigenvalues

are orthogonal

3. The set of eigenfunctions of a Hermitian operator is

complete

4. Commuting Hermitian operators have a set of common

eigenfunctions

Th f i f i f

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The set of eigenfunctions of a

Hermitian operator is complete

Figure by MIT OCW.

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Position and probability

Graphs of the probability density for positions of a particle in a

one-dimensional hard box according to classical mechanics removed for

copyright reasons.

Graph of the probability density for positions of a particle in a

one-dimensional hard box removed for copyright reasons.

See Mortimer, R. G.Physical Chemistry. 2nd ed.

San Diego, CA: Elsevier, 2000, p. 555, figure 15.3.

See Mortimer, R. G.Physical Chemistry. 2nd ed.

San Diego, CA: Elsevier, 2000, p. 554, figure 15.2.

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Commuting Hermitian operators have a

set of common eigenfunctions

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Fourth Postulate

If a series of measurements is made of thedynamical variable A on an ensemble

described by , the average

(expectation) value is

= AA

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Quantum double-slit

Source: Wikipedia

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Quantum double-slit

Above: Thomas Young's sketch of two-slitdiffraction of light. Narrow slits at A and B

act as sources, and waves interfering invarious phases are shown at C, D, E, and F.Source: Wikipedia

Image of the double-slit experiment removed for copyright reasons.

See the simulation at http://www.kfunigraz.ac.at/imawww/vqm/movies.html:

"Samples from Visual Quantum Mechanics": "Double-slit Experiment."

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Deterministic vs. stochastic

Classical, macroscopic objects: we have well-

defined values for all dynamical variables at every

instant (position, momentum, kinetic energy)

Quantum objects: we have well-defined

probabilities of measuring a certain value for a

dynamical variable, when a large number of

identical, independent, identically prepared

physical systems are subject to a measurement.

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Top Three List

Albert Einstein: Gott wurfelt nicht! [God does

not play dice!]

Werner HeisenbergI myself . . . only came to

believe in the uncertainty relations after manypangs of conscience. . .

Erwin Schrdinger: Had I known that we were

not going to get rid of this damned quantum

jumping, I never would have involved myself in

this business!

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