Reading and problem assignments, for Quiz #2 and to the end of the quarter (see also website)

Physics 9D, Section B, Fadley—Assignments for Spring Quarter, 2010: Midterm coverage in red, Quiz No. 2 coverage in blue, until end of quarter in black. Note there is a second page! Chapter Reading Questions/Problems 1—Classical physics to

1900 All None—a review of classical physics

2—Special relativity All except 2.9, plus 12.6 on radioactive decay

Probs. In Chap. 2: 3,4,13,24,25,27,32,37,40, 51, 54,55,57,68,76,80,82,86,92, plus prob. 12.26

15—General relativity 15.1-15.4, 15.5--optional (relates to newspaper article in S.F. Chronicle of April 5, 2004 see slides)

Probs. in Chap. 15: 2,4,9,11,17,24: plus: What is the frequency change due to velocity? Use equations in lecture for GPS.

3—Expts. Leading to quantum theory

All sections, plus Section 14.8 on accelerators, pp. 533-537, not including “Fixed Target Accelerators”

Probs. in Chap.3: 2,5,6,9,14,17,19,23: plus, Special part: If the emissivity of tungsten is 0.32 at this temperature, and the wire has a diameter of 0.01 cm, what length of wire would be necessary for a 60W bulb?,32,34,38,43,47,54 Probs. in Ch. 14- 31 and 34

4—Structure of the atom

All Sections Probs. in Chap. 4: 2,3,5,8,11,16,22,23,29,32,37,38,48

5—Wave properties of matter

Sections 5.1-5.3 for midterm, remaining sections after, plus supplementary reading on Fourier integrals

Probs. in Chap. 5: 1,3 10, 11, 12, 17, 19, 22, for midterm, then 26, 27, 28, 33, 40, 43, 45, 48 54, plus Special Problems: S1-With reference to the website: http://www.falstad.com/fourier/index.html, calculate the first two non-zero Fourier coeffients in the cosine+sine series representing a square wave, and show that they agree with the numbers given at this site. Why are there no cosine terms? S2-With reference to the supplementary reading on Fourier integrals, show that the final formula for g() on p. 3 and plotted in 4.20 is correct.

6—Quantum theory Sections 6.1-6.4, then all remaining sections --Quiz No. 2 coverage in blue--

Probs. in Chap. 6: S1-Show that the wave function for a free particle traveling to the right is an eigenfunction of the momentum operator p and the kinetic energy operator K , 2,5,7,8,9,11,15,20,23,26,28,32,37,40,41,47,54

7—The hydrogenic atom

All, plus supplementary reading on electronic transitions and selection rules (see link at website).

Questions: 1,2 Probs. in Chap. 7: 1,3,5,8,11,13,15,16,19,20(But do for 4p instead of 4d),29,32,35,38,46, plus Special problem: Show, using the full hydrogenic atom wave functions given in a lecture slide that, if you take the combinations 3d+2+3d-2 and 3d+2-3d-2, you get two of the real 3d orbitals, also shown in a lecture slide, and identify which ones.

8—Many-electron atoms & the periodic table

Sections 8.1 and 8.2 only, plus supplementary reading on exchange symmetry or many-particle wave functions (see

Question 5 Probs. in Chap. 8: 1,2,4,5,7, 11,12,23 (Remember the dipole selection rules!),24

10,11—Molecular and solid bonding, lasers

Sections 10.1 and 10.2 only, Section 11.1

Questions: 2,6 (But do for H2 and D2) Probs. in Chap. 10: 3,6,18,21,22 [Just make use of the result from Chap. 9 that the rms spread in emission frequency due to atomic motion relative to the observer will be given by:

0f kTfc m

.]

12—Nuclear decay and

structure All Questions: 2,12

Probs. in Chap. 12: 3,14,16,17,25,26(assigned previously),51

14—Elementary particles

14.1,14.2,14.3,14.5,14.6,14.8 (not including Fixed-Target Accelerators)

Probs. in Chap. 14: 1,3(See Ex. 14,1 & Eq. 14.4, and assume the nucleon moves at speed c),4,20,31,34,40

K

KH

Converting to newcoordinates

Polar angle = = arc cos(z/r)Azimuthal angle = = arc tan(y/x)

Quantum mechanically:

2

0 0

0

sin d d

cos | 2 4

The Hydrogenic

AtomSchroedinger

Equation:Spherical Polar Coordinates

e-

+Ze

r

V(r) =-Ze2/40

r

Classically:

L r pis conserved

p

The BohrFormula!

[ ()

f(

) & ()

g() in text ]

H

Solving for the

part = ():

The BohrFormula!

[ ()

f(

) & ()

g() in text ]

H

C

=

2

Changes for many-e-

atoms

The same for many-e-

atoms

!

E = E100

= Bohr energy for n = 1!

r

4r2dr

The atomic orbitals:

Z→Zeff,n

(r) in many-e-

atoms

mY ( , )" spherical harmonics"

But we can make them real for convenience

Angularnode for

= 90

Radialnode for

r = 6a0

Angularnodes

for cos2

=1/3

= 54.7,125.3

xy

z

x yz

node

for

= 90

And the same thing for the d orbitals:

eg

t2g3z2-r2 x2-y2

yz zx xy

x

yz

eg

t2g3z2-r2 x2-y2

yz zx xy

x

yz

See specialproblem forChap. 7

+ nodesfor cos

=1/3

= 54.7,125.3

r2(3cos2

– 1) =3z2

– r2

The atomic orbitals:

Z→Zeff,n

(r) in many-e-

atoms

mY ( , )" spherical harmonics"

But we can make them real for convenience

Angularnode for

= 90

Radialnode for

r = 6a0

Angularnodes

for cos2

=1/3

= 54.7,125.3

Radialnode for r = 2a0

The RadialParts Rn

(r):

3p: node at 6a0

2p: node at 2a0

The RadialProbabilityDensityPn

(r) =

r2

Rn

(r)2

:

n=4shell

n=3shell

n=2shell

The hydrogenic

atom-putting it all togetherin three dimensions:

Some of the atomic orbitals:including radialdependence

TotalNo e-

2

x2

x2

1s

2s

2p

3s

3p

3d

x2

x2

2

10

eg

t2g3z2-r2 x2-y2

yz zx xy

x

yz

eg

t2g3z2-r2 x2-y2

yz zx xy

x

yz

5 total

x y

z3 total

Comparison of three QM potential-well problems in 1 dimension

2

n m n mo

n m n m

n

ef

n n

n

f

EIGENFUNCTION PROPERTIES OF THE ATOMIC WAVEFUNCTIONSZe 1: Hydrogenic atom , from Bohr formula,V ( r )4 r

Many electron atom ,E from solution of radial Schroed

ˆEnergy H E E

H Einger with

ZV ( r )

2eff ,n

2 2

o

n m n m

2

n m n m

z

z

( r )e 14 r

:ˆAlways measure same value :

ˆSquare of angular momentum L 1

ˆZ component of angular momentu(L ) 0:

Always measure same value : (LL m

) 0m

Operator Eigenvalue

E.g.—n = 3,

= 2 “3d”

Quantization of space: The (randomly) precessing

vector model:

2

n m n m no

n m n m

n

eff

n

n

EIGENFUNCTION PROPERTIES OF THE ATOMIC WAVEFUNCTIONSZe 1: Hydrogenic atom , E from Bohr formula,V ( r )4 r

Many electron atom ,E from solution of radial Schroedi

ˆEnergy H E

H Enger with

ZV ( r )

2 2eff ,n

o

n m n m

2

n

2 2

z m n m

z

( r )e 14 r

:ˆAlways measure same value :

ˆSquare of angular momentum L 1

ˆZ component of angular momentu(L ) 0:

Always measure same value : (LL m

) 0m

Operator Eigenvalue

E.g.—n = 3,

= 2 “3p”“3d”

n m n m

n m

: ( r ) ( r )

and fina

llyˆParity evenness or o

( r )

ddness

( 1)

Degeneracy in the hydrogenic atom (without spin)

States Degeneracy• n = 1: 1s0 1• n = 2: 2s0 , 2p+1 ,2p0 ,2p-1 or• 2s0 , 2px ,2py ,2pz 4• n = 3: 3s0 , 3px ,3py ,3pz , or

3d3z2-r2 ,3dx2-y2 ,3dxy ,3dyz ,3dxz 9

• n = general n2

See additionalreading at website

TRANSITIONS BETWEEN LEVELS AND SELECTION RULES:

x= “dipole operator”

1.0

0.0time

a(t) b(t)

Emis-sion

Many transitions are notAllowed: selection rules important

Bremsstrahlung

e-e-

LineSpectra

+a continuum

S

N

SPIN ANGULAR MOMENTUM: A LAST PROPERTY OF THE ELECTRONTHE STERN-GERLACH EXPERIMENT

THE ELECTRON’S FINAL PROPERTY:CHARGE = 1.602 X 10-19

CMASS = 9.109 X 10-31

kg+…

e

SPIN ANGULARMOMENTUM: Quantum no. s = ½

SPIN MAGNETICMOMENT:s B

e

Be

28

e ss 2m

ewith2m

the Bohr magneton9.274x10 J / T

z

s s( s 1) 3 / 4s / 2

The atomic orbitals:

Z→Zeff,n

(r) in many-e-

atoms

mY ( , )" spherical harmonics"

But we can make them real for convenience

node for

= 90

node forr = 6a0

nodesfor cos2

=1/3

= 54.7,125.3

sn m m n mΨ ( r ,θ ,φ, spin ) Ψ ( r ,θ ,φ )x [α( ) or β( )]

With spin

Degeneracy in the hydrogenic atom with spin

States Degeneracy Degeneracywithout spin with spin

• n = 1: 1s0 Not 2• n = 2: 2s0 , 2p+1 ,2p0 ,2p-1, or 4 8

2s0 , 2px ,2py ,2pz• n = 3: 3s0 , 3px ,3py ,3pz , or

3d3z2-r2 ,3dx2-y2 ,3dxy ,3dyz ,3dxz 9 18

• n = general n2 2n2

L Be

Be

e LL2m

ewith : the Bohr magneton2m

ORBITAL ANGULAR MOMENTUMORBITAL MAGNETIC MOMENT

Classical interationof magnetic momentand B field:(Young and Freedman, Ch. 27)

E potential energy

B BcosLowest energy for

parallel to B

Quantization of space: The (randomly) precessing

vector model:

Classical interationof magnetic momentand B field:(Young and Freedman, Ch. 27)

E potential energy

B BcosLowest energy for

parallel to B

A FINAL (RELATIVISTIC) ATOMIC INTERACTION: SPIN-ORBIT COUPLING

SPIN-ORBIT SPLITTING IN HYDROGEN 2p (SMALL!):

LINE SPECTRA OF DIFFERENT SOURCES:

Atomic hydrogen

Sodium

Helium

Neon

Mercury

Molecular hydrogen = H2

WHY LINES?

Sodium-

D-line emiss.

Sodium-

D-line absorp.

The Sun: blackbody emission plus absorption

From Serway, Moses, and Moyer—”Modern Physics”

Spin-orbit splitting in spectroscopy: the sodium doublet

What properties do wave functions of overlapping(thus indistinguishable) particles have?—electrons as example:

1 1 2 2

2 21 1 2 2 2 2 1 1

1 1 2 2 2 2 1

( r ,s ; r ,s ), including spin of both electronsBut labels can' t affect any measurable quantity.E.g. probability density :

( r ,s ; r ,s ) ( r ,s ; r ,s )Therefore

( r ,s ; r ,s ) 1 ( r ,s ; r ,s

1

12 1 1 2 2

12 1 1 2 2

12

)

P ( r ,s ; r ,s )

with P permutation operator r ,s ; r ,sand eigenvalues of 1

Finally , all particles in two classes :FERMIONS : ( incl. e ' s ) : ant

1 3 5s , , .

isymmetric

P 1

BOSONS : ( incl. photons )2 2 2

:

..

12s 0,1,2,...symmetric

P 1

Probability of finding twoelectrons at the same point in space with the same spin (

or ) is zero: “the Fermi Hole”

the Exchange InteractionHund’s

1st

rule & magnetism

e1-

e3-

e2-

Antisymmetry

and the Pauli Exclusion Principle:

Try Helium, 2 electrons in ground state 1s wave functions, “1s2"

Simple normalized antisymmetric

trial wave function is

1 1 2 2 1s 1 1 1s 2 2 1s 1 1 1s 2 2

2 2 1 1 1s 2 2 1s 1 1 1s 2 2 1s 1 1

1 1

1( r ,s ; r ,s ) ( r ,s ) ( r ,s ) ( r ,s ) ( r ,s )2

int erchanging labels gives1( r ,s ; r ,s ) ( r ,s ) ( r ,s ) ( r ,s ) ( r ,s )2

( r ,s ; r

2 2,s ), as required

Can’t tell which electron is spin up--indistinguishable

Also, if we try to put both electrons in 1s with spin-up (), first termalways cancels second term, and = 0!

Therefore, we have the PauliExclusion Principle

Thus, in a many-electron system:

●Anti-symmetry of total wave function implies:Pauli Exclusion Principle:

No two electrons can have all the same quantum nos.n, , m

, msor, if spin-orbit split

n, , j, mj

●Electronic structure determined by filling n, (or n, , j ) levels from lowest to highest energy (En

from radial Schroedinger

Eqn. with Zeff

depending on n and )

●Partially filled subshells

n, (or n, , j) have their lowest energy when a maximum no. of electrons have parallel spins = highest total

spin angular momentum = (Hund’s

First

Rule), and then they couple to yield highest total

orbital angular momentum = (Hund’s

Second Rule)

S

L

K

KH

Converting to newcoordinates

Polar angle = = arc cos(z/r)Azimuthal angle = = arc tan(y/x)

Quantum mechanically:

2

0 0

0

sin d d

cos | 2 4

The Hydrogenic

AtomSchroedinger

Equation:Spherical Polar Coordinates

e-

+Ze

r

V(r) =-Ze2/40

r

Classically:

L r pis conserved

p

Intraatomic electron screening

in many-electron atoms--a self-

consistent Q.M.calculation

Plus radial one- electron functions:

Pn

(r) rRn

(r)

Intraatomic

electron screening in many-electron atoms--a simple model

kC

1/(40

)

[Zeff

]

In many-electron atoms:For a given n, s feels nuclear chargemore than p, more than d, more than f

Lifts degeneracy on

in hydrogenicatom

Ar--Evidencefor shellstructure

QUANTUM-MECHANICAL THEORYEXPERIMENT: ELECTRON DIFFRACTION

The radial probability distribution of all of the electrons in Argon, as derived from electron diffraction experiments and quantum-mechanical theory

P(r)

The shell structure of an Ar

atom, as measured and calculatedK =1s2

L=2s22p6

M=3s23p6

Bremsstrahlung

e-e-

LineSpectra

+a continuum

Hole

Hole

(2 e-)(1 e- with hole)

(2 e-)(6 e-) (8 e-)(7 e- with hole)

Why lines?

Why similar systematicsto hydrogen, but with Zeff

?

Zeff

= Z –

no. inner e-

screening

Z –

7 -2 =Z -

9

Zeff

= Z –

no. inner e-

screening

Z –

1

Understood now!

Square root of x-ray frequency

Ato

mic

num

ber =

Z

TotalNo e-

2

x2

x2

1s

2s

2p

3s

3p

3d

x2

x2

2

10

Fillingdegeneracy

MaximumOccupation =Degeneracy

2

2

6

2

6

10

+ 14 for nf

Filling theAtomicOrbitals:

8

18

SOME PERIODIC PROPERTIES:(plus see www.webelements.com)

From www.webelements.com, with much more:

3s14s1

5s1 6s17s1

Hund’s

First Rule: highest total

spin angular

momentum

-

=

antibonding 1sa

-

1sb

+ =

bonding 1sa

+ 1sb

The quantum mechanics of covalent bonding in molecules: H2

+

with one electron

=R

TotalEnergy antibonding

a.u. = -16.16 eV(Compare –

13.61 for H atom 1s)

a.u. = +7.21 eV

negative(occupied)

positive(unoccupied)

Bonding

Anti-Bondinganti

MO 1sa

-

1sa

bondingMO 1sa

+ 1sa

HaHb

36

THE ELECTRONS IN CARBON MONOXIDE:

++++ -

+

---

+ --

+++_

+_ +_