3.012 Fund of Mat Sci: Bonding – Lecture 5/6 THE HYDROGEN … · 2020-07-09 · 3.012 Fund of Mat...

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

3.012 Fund of Mat Sci: Bonding – Lecture 5/6

THE HYDROGEN ATOM

Comic strip removed for copyright reasons.


Last Time

• Metal surfaces and STM• Dirac notation• Operators, commutators, some postulates


Homework for Mon Oct 3

• Study: 18.4, 18.5, 20.1 to 20.5.• Read – before 3.014 starts next week:

22.6 (XPS and Auger)


Second Postulate

• For every physical observable there is a corresponding Hermitian operator


Hermitian Operators1. The eigenvalues of a Hermitian operator are real

2. Two eigenfunctions corresponding to different eigenvaluesare orthogonal

3. The set of eigenfunctions of a Hermitian operator is complete

4. Commuting Hermitian operators have a set of common eigenfunctions


The set of eigenfunctions of a Hermitianoperator is complete

Figure by MIT OCW.


Third Postulate

• In any single measurement of a physical quantity that corresponds to the operator A, the only values that will be measured are the eigenvalues of that operator.


Position and probability

Graph of the probability density for positions of a particle in a one-dimensional hard box removed for copyright reasons.

Graphs of the probability density for positions of a particle in a one-dimensional hard box according to classical mechanics removed for copyright reasons.

See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, p. 554, figure 15.2.

See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, p. 555, figure 15.3.


Quantum double-slit

Source: Wikipedia


Quantum double-slit

Above: Thomas Young's sketch of two-slit diffraction of light. Narrow slits at A and B act as sources, and waves interfering in various 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."


Fourth Postulate

• If a series of measurements is made of the dynamical variable A on an ensemble described by Ψ, the average (“expectation”) value is

ΨΨΨΨ

=A

Aˆ


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.


Spherical Coordinates

sin cossin sincos

x ry rz r

θ ϕθ ϕθ

===

z

θ

0

φ

P

y

r = r

x

Figure by MIT OCW.


3-d Integration

Diagram of an infinitesimal volume element in spherical polar coordinates removed for copyright reasons.

See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, p. 1006, figure B.4.

Angular Momentum

Classical Quantum

L r p= ×r r r



Commutation Relation

2 2 2 2

2 2 2

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ, , , 0

ˆ ˆ ˆ, 0

x y z

x y z

x y z

L L L L

L L L L L L

L L i L

= + +

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = =⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎡ ⎤ = ≠⎣ ⎦ h

Angular Momentum in Spherical Coordinates

22 2

2 2

ˆ

1 1ˆ sinsin sin

zL i

L

ϕ

θθ θ θ θ ϕ

∂= −

∂

⎛ ⎞∂ ∂ ∂⎛ ⎞= − +⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠

h

h



Simultaneous eigenfunctions of L2, Lz

( ) ( )( ) ( ) ( )2 2

ˆ , ,ˆ , 1 ,

m mz l l

m ml l

L Y m Y

L Y l l Y

θ ϕ θ ϕ

θ ϕ θ ϕ

=

= +

h

h

( ) ( ) ( ),m ml l mY θ ϕ θ ϕ= Θ Φ


Spherical Harmonics in Real Form

Figure by MIT OCW.


An electron in a central potential (I)2

2 2

2 22

2 2 2

2 22

2 2 2 2 2

ˆ ( ) needs to be in spherical coordinates2

1 1 1ˆ sin ( )2 sin

ˆ1ˆ ( )2

sin

e

e

e

LH r

H V rm

H r V rm r r

V rm r r r

r r

r

rϑ

ϑ ϑ ϑ ϑ ϕ

= − ∇ + ∇

⎡ ⎤

⎡ ⎤∂ ∂

∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞= − + + +⎢ ⎜ ⎟ ⎜ ⎟ ⎥∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

⎛ ⎞= − − +⎢ ⎥⎜ ⎟∂ ∂⎝ ⎠⎣ ⎦

h

h

h

h

r*


An electron in a central potential (II)2 2

22 2

ˆ1ˆ ( )2 2e e

d d LH r V rm r dr dr m r

⎛ ⎞= − + +⎜ ⎟⎝ ⎠

h

( ) ( ) ( , )r R r Yψ ϑ ϕ=r


An electron in a central potential (III)

2 22

2 2

1 ( 1) ( ) ( ) ( )2 2 nl nl nl

e e

d d l lr V r R r E R rm r dr dr m r

⎡ ⎤+⎛ ⎞− + + =⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

h h

What is the V(r) potential ?


2

1

-1

-2

1 2 3 4 5 6

Vcentripetal(r)

1010r(m)

Veff(r) VCoulomb(r)

1018

v(r)

(J)

Figure by MIT OCW.


The Radial Wavefunctions

for Coulomb V(r)

2

2

1

10

0 3 4

R10

r

r

r

r

-0.20

0

0

0.20.4

0.040.080.12

0.6

2

r

r

r

6 10

2 6 100

0

00

R20

R21

R30

R31

R32

4

4

8

8

12

12

16

16

4 8 12 16

-0.1

0.2

0.4

-0.04

0.04

0.08

0.02

0.04

Radial functions Rnl(r) and radial distribution functions r2R2nl(r) for

atomic hydrogen. The unit of length is aµ = (m/µ) a0, where a0 is thefirst Bohr radius.

Figure by MIT OCW.


The Radial Density2

2

1

10

0 3 4

R10

00

0.2

0.4

1 2 3 4

r2R220

r2R210

r2R221

r2R230

r2R231

r2R232

rr

r

r

r

r

r

r

r

-0.20

0

0

0.20.4

0.040.080.12

0.6

2

r

r

r

6 10

2 6 10

00

0.10.05

0.05

0.15

2 6 10

0

00

0

0

00

02 6 10

R20

R21

R30

R31

R32

0.1

4

4

8

8

12

12

16

16

4

4

8

8

12

12

16

16

00

4 8 12 16

r0 4 8 12 16

-0.1

0.2

0.4

0.04

0.08

-0.04

0.04

0.08

0.02

0.04

0.4

0.8

0

0.4

0.8

Radial functions Rnl(r) and radial distribution functions r2R2nl(r) for

atomic hydrogen. The unit of length is aµ = (m/µ) a0, where a0 is thefirst Bohr radius.

0.15

Z

Thickness dr

Y

X

r

Figure by MIT OCW.

Figure by MIT OCW.


Three Quantum Numbers

• Principal quantum number n (energy, accidental degeneracy)

• Angular momentum quantum number l (L2)l=0,1,…,n-1 (a.k.a. s, p, d… orbitals)

• Magnetic quantum number m (Lz )m=-l,-l+1,…,l-1,l

( ) ( )2 2 2 2

2 2 20 0

13.6058 eV 1 Ry8n

e Z Z ZEa n n nπε

= − = − = −


Emission and absorption lines

Courtesy of the Department of Physics and Astronomy at the University of Tennessee. Used with permission.


Balmer lines in a hot star

Courtesy of the Department of Physics and Astronomy at the University of Tennessee. Used with permission.


XPS in Condensed Matter

Diagram of Moon composition as seen in X-rays, removed for copyright reasons.


The Grand Table


Solutions in the central Coulomb Potential: the Alphabet Soup

Table of orbitals removed for copyright reasons.See "n and l versus m" at http://www.orbitals.com/orb/orbtable.htm.

http://www.orbitals.com/orb/orbtable.htm


Orbital levels in multi-electron atoms

Figure by MIT OCW.

Orb

ital E

nerg

y (k

J / m

ol)

0-82

-146

-328

-1313

0-82

-146

-328

-13131s 1s

4s4s

2s

2s

3s3s

4p 4p

3p 3p

2p 2p

4d 4d

3d3d

4f4f

Hydrogen Multielectron Atoms


Screening

Auf-bau


6s

5s

4s

4p

5p

6p5d

4d

4f

3d

3s

3p

2s

2p

1s

LOW

EN

ERG

YH

IGH

EN

ERG

Y

ENERGY LEVELS OF THE ELECTRONS ABOUT THEIR NUCLEI

Figure by MIT OCW.

3.012 Fund of Mat Sci: Bonding – Lecture 5/6 THE HYDROGEN … · 2020-07-09 · 3.012 Fund of Mat...

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