7. Hydrogen Atoms,4

lecture 27, October 32, 2017I crack myself up sometimes

housekeeping

I got nothin’

today

Hydrogen atom, more

WHERE WE WERE1

.

Schrodinger equation for 3 dimensional configuration

- E'

I . Er Cryer ) + resting 3fGm03o )+ rtzswzo Fan

]4Cno . on

a[-

Far4 ( no , g) = EH v. t.cl )reduced

mass

rneat.tl#EyhIIp3Ieaoti%a0ent+iEnaos4nemdr,

a,on = Rue ( r ) Ye

"

( 0.0 ) En = Effteogtfzznz"

Eigen functions"

"

eigen valves"

QUANTUM NUMBERS of the QUANTVM CENTRAL FORCE SOLUTION

n : Principle Quantum Number

n= 1,2 , 3 . - . A

l : Orbital Angular Momentum Quantum Number

l= 0, 1,2 ... ( n - I ) Of l En

\ connected to R

me: Magnetic Quantum Number

me = - l

, - ltl

, - l +2

, . ,-1,0 , I , e. . l- I

,l

- l E me < l

a few : Yen

MATHEMMCA

l=2L=A✓ #

L =

AVTLz

3k Lz = melt

25zh

gmet 2

= 0,

1,

-1,

2,

-2

A A ¢ me ,0 0•#a*mr=o- hi - hi # me= -1- 2h -2At.me: -2-3k

me =- l

,

- ltl,

-1+2, . ,-1,0 , 1 , ... l

- 1,

l

Normalization :

1 = fff 1124121 YA , a) tndvsitdddqall space

1 = lotRchtdv§ ( "

IYCA , g) Tswododq

can sort of visualize2 different probability distributions :

r21R(r)Tdr

IYIY a e) Tsinododq

Most probable ? in hydrogen ,2=1

- 2r/aoPlo = 4- me :

P÷r4 = tap [ zri% . n ( yo )E% ]

= 0 for extremum

= 4- e"%[ zv . zado ]Gp

: =) ✓ = Ao HOW COOL is THAT ?p (

most probable p10 10

0.6- v

1-

oiler. " a.

A'"

iii.... :

.

: " a.

Average"

place"

fn electron ?

( r ) = ( rpiolr ) vdr = 4g, µr3e→%°dr

line | zheitdz = n !

Change variables : z = 2%0

( v ) =Ag

( 3 ! ) Pioab

. fmost probable

< r > = } a .o , µ averageIta

Va.

WHERE IS THE ELECTRON ?

Average"

place"

fn electron ?

( r ) = ( rpiolr ) vdr = 4g, µr3e→%°dr

line | zheitdz = n !

Change variables : z = 2%0

( v ) =Ag

( 3 ! ) Pioab

. fmost probable

< r > = } a .o , µ averageIta

Va.

WHERE IS THE ELECTRON ?

Probability of electron outside the 1st Bohr orbit ?

assume : Is

x

Pcr > a.) = f P , .cn ) drao

=

Qag ( r2e→%°dv another one if more integrals .

= { f%j×d×|←£×d×= - tz(×+z×+z)[

×

×= 2%0

: zPC ✓ s a.) = 5e - 0.68⇒ 43 of tn

time ! How about clearly outside of

its atom ?

> v= looao- 83

Pcr > lwa D= 10 to

How cmnt ZS state. . .

YI → snuuvetnh .- r/2ao

Rao = ctfap ( 2 - ⇒e

-

density peaks

25 electrons spend

a significant time

in is land

Energy depends on n alone En= e-4

"

( 4. € of ¥zznzTHE BOHR ENERGY

But each n can have many ls'

& me 's

substantial degeneracy .

#stat=

- 0.85 4s 4p 4d 4f 1 + 3 +5 + 7 = 16

-1.51 35 3P 3d 1+3+5 = 9

-3.4 Zs ZP 1+3 = 4

- 13,6 ← 15 1

.

# degenerates = n2 forH

TRANSITIONS l=0 l= 1 l=z l=3

18 . n=4-1.5 . h=3

-3,4

ZPn=2

Ecevj-5

: 3.6 > ← h=1

-15 -

BOHR SCHROEDINGER

TRANSITIONS l=0 l= I l=z l=3

i8 . n=4-1.5

I Pcschen. n=3

→ µ§ Balmer

zpn=z

Eton"

}yuman: 3.6 V > ← h=1

-15 -

✓7

translate

to to in

BOHR s .E . SCHROEDINGER

TRANSITIONS - - - the almost veal way.

.

m ,

ftp..mg#seEafes"

Be$ y

in an

www.er.mn,

the t¥¥oYqY#±¥*§3a djosoinatina electricradiation or T \ P dipole4M¥he I =eF

4a( r , A 4) & 4,3 ( r , A. y ) each wave functions for" levels

"

A & B

→ for a transition, they must

"

overlap"

in the"

presence

"

ofan electric dipole operator .

< p >= |H4a*(r , 0 , g) QF 4,3 ( no , a) r2dvsi0d0d¢

a-

If integrals are zero ⇒ no transition from �1� → A

non - zero ⇒ transitions from B → A can happen

& calculation yields prob & rate

C intensities )

( p > = /rzR(rYaRCr)Bdr ffyeamact, of YMCA , a) sinododq.

complicated . --

but always control the"

allowed"

and"

forbidden"

non-zero transitions

|* which become

particular al 's

t

always Al=±1

so : p → s ? sure

s → s, pep ... ? hope

"

selection Rules"

t not quite

SELECTION RULESl=0 l= 1 l=2 l=3

18 . n=4-1.5 . n=3

-3,4 ; ← h=1

-15 -

BOHR SCHROEDINGER

→

Orbiting electric charges → a magnetic moment . & µ

Classical E. & M : current loop A ¥#.rmagnetic moment |µ|= IA ( R . H . rule )

If an eieotrr charge going in a circle I = dye = 0¥ €7 v

So L= mvr t

vr= 2¥+ A = IN

L=m(air )r

= an

II=

2M¥ →B L

µ= IA =

QgA =

Qzkmor F= zqmi

'

€H•- rere= . e- I tµZM

Strange superposition ofclassical & quantum mechanical notions :

We found . .

L= theme

a vector w/ quantized projection onto Lz axis - Lz= met

we have no preferred direction - . . jcould be anywhere

something hurt pioh a particular direction and break that symmetry

& define the"

axisof quantization

"

Magnetic fields do that .

tEmHe÷¥¥

±¥q¥€Tf¥iB #

IFEIEfItE5t@EfEaIBteidiYo.hYnathEy3aaIsniizaamJauqma0PyqIgiGEjrpgizemLz-tezImmewmaasaasnltgEeg.t

.q!eg.accordingto L→

µz=- e- time defines Bohr Magneto µB=

ezIm=9.3×152451,2 M

µz=-

µB me ⇒atomic magnetic moments can the

threat. of

as multiple , of µB .

Remember : E =f× Be → seems to align Fand Be

→ a

E = DI = Q Lx Bdt ZM

%B→

de perpendicular to plane ofB&I

=#.

. .,

, 1 - i

←

¥¥e.

←iµ, ldll

= Lsinttdo→ :L4 d¢= 1=4L5m0 o : L Sino- :£- ...E , . -+ I toll¥€s'sEIf. ' ' a =t

→ :

dld¢ d¢ = ltldtlsino

dcf = Qhm( L Bino)dt

Tino

dof = Bzlfndt ⇒ precession @ angular

frequency''

we day= QI

ZM

Lavmov Frequency

Got a torque ? through an angle ?

→you got work done

dw = Edt

dw = - µBsiQdO ( opposes B)

dw = d ( ii. B )

unh stored as potential energy: dw = - DV

an induced magnetic potential energy V = - ii. B

depends on :

→

. B

. the more aligned , the less is ✓

But... µ is quantized according to Lz µz=

-

µB me

l=l,

fn example :

' -

Y from V = - F ' Bme of L

- 1 ->

✓ is indeed quantized

The magnetic field defines a particular direction In space

→ the quantisation axis , along Be . . . of

✓ = e- E. Be = e -1131LagZM

ZM

✓ = MB meB

me ranges over - l → + l

Bottom line :

apply a magnetic field → energy levels change

degeneracy lifted

Lorentz Predicted , Zeeman discovered → bcfne Q . M .

spectra : By 111

13=0 BFO

Mmejl ←} MBB

- ⇒ I } MBBme . i

/ selection RULEA me = 0 , ± I-<me=O -✓

l=o 4=1 4=0 l= 1

"

NORMAL

"

ZEEMAN EFFECT

1 → 3 splitting . -" normal

"

because classical picture predicts

rough behavior

"

ANOMALOUS"

ZEEMAN EFFECT

1 → 4

or \ experimentally confusing , classically , disturbing1 → 6

"

Stern - Gerlach Experiment"

→ slides !

“Spin”some%mesweuse“normallanguage”

torefertoaphysicalpropertywithoutacommon-senseanalog

andsome6mes,justtobesilly

Rememberacurrentloop:where’stheBfield?

B

I

Likeabarmagnet,withthenorthpoleup...

RememberthatOerstedfoundthatacompassneedlefollowedtheBfield,sothissetupwouldalso

externalBfield

calleda“magne%cdipole”

dipolemomentcharacterizesthemagne%smofacurrentloop

Acurrentloop’smagne%smcanbecharacterized

μ

I

morecurrent,higherthemoment–theμ themoretorque

μ iscalledthe“magne%cdipolemoment”

externalBfield

μ

Bohrwasnotthewholestory

buthisquan%za%oncondi%onwasmostlycorrect

Orbitalangularmomentum-thes,p,d,f,g,h,etcoriginateswithSchroedinger’sevolu%onofBohr’soriginalidea:

L

v

L = mvr =nh

2⇡

InSchroedinger’smodel Lz = m`h

2⇡

But,theelectron’sorbitisaliMlecurrentB, μ

I

=0:s=1:p=2:detc

externalBfield

μ

So,atomicorbitsshouldtwist

rememberspectroscopicnota8on

fromChemistry?

That’swhatOMoSternandWaltherGerlachsetabouttomeasurein1922...

clevermagnetpoleshapes

N

SSo,theatomsmovetoalignthedipolewiththemagnetfield

silveratoms

theBohrModelpredictedabunchofspots

sinceSilverisacomplicatedatom:

1s22s22p63s23p63d104s24p64d105s1

IfBohrwasright

IfBohrwaswrongWhattheysaw

Separa%oninto2loca%ons,whichmadenosense.

Separa%oninto2spots-evidenceof“quan%za%on”ofsomesort…why2?

Somethingis“quan%zed”

youcan’talwaysdowhatyou’resupposedtodo

youngDutchexperimentersGeorgeUhlenbeck*andSamGoudsmithadanidea

toldbytheiradvisor“don’tpublish”

theydid.

μ

*eventuallyprofessoratUofM

Theelectronislikeaspinningcharge...

Likefororbitalangularmomentum,

Lz = m�h

2�

Electronshaveanintrinsicangularmomentum

Sz = msh

2⇡

But,the“spin”canonlytakeontwovalues:

ms = +12

ms = �12

or

Wesay“spin,plus1/2”or“spinup”and“spin,minus1/2”or“spindown”

+ 1/2

– 1/2

Stern & Gerlach got a hodge . podqe of results

Oxygen 5 spots 2s2zp4

silver 2 spots - . - 5s

Interpreted as an angular - momentum - like seletim rule w|"

l"

= Yz

Your 'e not much if you're not Dutch . . .

Govdsmit & Vhlenbeoh pound

electron has an inherent puberty : "

spin"

5=42with projections

like L : 1 staffe = aye

Yzt -'

iii.s

s / i , Isl = Dehf f

- yztn

¥↳ Sy = ± Yztn = mstns

spin

has features like angular momentum . - magnetic - like

NO CIRCULAR CURRENTS !

NO CHARGES GOING AROUND ANYTHING !

NOT A BALL OF CHARGE ROTATING1

. €¥§#SPIN IS INHERENT TO ELECTRONS

tDEFINE AN ELECTRON !

lousy name !

no - thing is" spinning

"• An excitation of the

electron field

• having mass = 0.5N new

. having electric charge = - 1.6×15 '9c

• having intrinsic spin = You

'

Magnetic Moment

→

E's = - age e- sZme

/a fudge factor . - measurable : gyromagnetic

ratio ( stay tuned )

No reason or " implementation"

of spin in Quantum Mechanics

Until 1928 ! Paul Divac → first relativistic quantum mechanics

The Divac Equation designed to deal with negative E

• 4 wave functions resulted

�2�with + energies → electron

oosterhof" { @with - energies -7 anti - electron

mathematics

Spin is an inherently Relativistic property of electrons . . .

. . .and neutrons

, protons → all" fermions

"

Also a quantum mechanics of integer- spin excitations → all

"

bosons"

S = 0 pious . kaons . Hits boson

. - -

S = 1 photon , rwo meson , W , Z - -

(that's why al = ±l

Angular momentum is conserved -8 radiates away

and tales angular momentum oftt from system

So,

Al is reduced .

ABOUTg

.

We have 2 moments now :

orbital avatar momentum " Re = - MfE ( = - guest )

spin

"

intrinsic"

angular momentum: F's = -

2M¥I ( = - go.gg

§ )

gs is strictly a relativistic quantity .

gs=2

geis sometimes included .

get

More g .

e¥e•gF×=h¥a# onits ,

"

strength gearinteraction

"

Relativistic Quantum Field Theory

ex

@

•te ¥}8 since x small , each additional× } " vertex " is smaller than predecessor .8

Full treatment of electrons requires

Aav y

+ + }+

0 ×

} } 3m

+ combinations & additional TToetc .

} §

Contributions addup

and modify

static properties if he electron

→

its (e) = ge e- sZmec

(net effect is to wrdify get

2

"

g- 2"

. . .

"

qee minus two"

...

a race fn precision

between theory ( . 103 individual Feynman diagrams )

and experiment ( heroic Penning Trap experiments )

Conventionally , what 's reported is

Ae = £-2z

- 13

experiment : AE"

= 1 1596521807.3 ( 2. 8) xlo

=

0.00115965218073±0,00000000000028

theory Ase"

= 0,00115965218178

± 0.00000000000006 ( 4 loop uncertainties )

±

0,00000000000004( 5 loop uncertainties )

HAD± 0,00000000000002 Sae±

0.00000000000076Saens

( L )

AFM - Asen = - 10.5 ( 8. 1) × 1513 1,1J difference

The most precise enunciation about naturein the history of . . everything