Quantum Quantum Phenomena Phenomena

II:II:Matter Matter MattersMatters

Chris Parkes March 2005

Hydrogen atomHydrogen atom Quantum numbers Electron intrinsic spin

Other atomsOther atoms More electrons! Pauli Exclusion Principle Periodic Table

Particle PhysicsParticle Physics The fundamental particles The fundamental forces

CosmologyCosmology The big bang

Fundamental PhysicsAtomic Structure

http://ppewww.ph.gla.ac.uk/~parkes/teaching/QP/QP.html

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““Curiouser and curiouser” cried Alice Curiouser and curiouser” cried Alice

Christine Davies’ first part Christine Davies’ first part Basics intro: Rutherford’s atom, blackbody radiation, photo-

electric effect, wave particle duality, uncertainty principle, schrödingers equation, intro. to H atom.

This lecture seriesThis lecture series Some consequences of QM Applications Emphasis on awareness not mathematical rigour

First few lectures – Young & Freedman 41.1->41.4First few lectures – Young & Freedman 41.1->41.4 Lectures main points same, but more complex treatment

Last few lectures – Y&F 44Last few lectures – Y&F 44

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Understanding atomsUnderstanding atoms

Key to all the elements & chemistryKey to all the elements & chemistry Non-relativistic QM – the schrödinger equation

What are atoms made of ?What are atoms made of ? Nucleus (p,n) ,e

What are the nucleons made ofWhat are the nucleons made of quarks

Why are p,n clamped together in the middle?Why are p,n clamped together in the middle? Strong nuclear force

Second part of this course…..Second part of this course…..

How do we analyse atomsHow do we analyse atoms The first part of this course…..The first part of this course…..

Find the energy levels for a Find the energy levels for a Hydrogen atomHydrogen atom

• Find the wavefunction for the hydrogen atom

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SchrSchröödinger Equation :dinger Equation :solving H atomsolving H atom

WavefunctionWavefunction Probability to find a particle at

P(x,y,z) dx dy dz = | (x,y,z)|2 dx dy dz

nlmnnlmnlm EUzyxm

2

2

2

2

2

22

2

1. This looks like p2/2m + U = E in classical mechanics

2. n,l,m are quantum numbers

3. E depends on n only for H (also l for multi electron atoms)

4. BUT now we have a wavefunction (x,y,z)

dx

dy

dz

)ionnormalisat( 1ddd,,2

zyxzyx

BIG Difference from classical physics.

No longer know where a particle is

Just how likely it will be at x,y,z

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Spherical co-ordinatesSpherical co-ordinates Potential energy of one electron in orbit around one protonPotential energy of one electron in orbit around one proton

Spherical symmetry, so use spherical polarsSpherical symmetry, so use spherical polars rewrite schrödinger in r, , Rather than x,y,z

rker

e

r

qqrU 2

2

0

21

0 4

1

4

1)(

nlmnnlm EUrr

rrrm

2

2

2222

22

sin

1)(sin

sin

1)(

1

2

•Try (r,(r,,,) = R(r) Y() = R(r) Y(,,))

•Separate out the radial parts and the angular parts, LHS(r) = RHS(LHS(r) = RHS(,,)=C)=C

• Mass of electron m, Charge of proton,electron eFor single electron heavy ion would have q=Ze

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Radial EquationRadial Equation

BUT this looks a lot like schrBUT this looks a lot like schröödinger eqndinger eqn With rR(r)=(r) And with an extra term

What is the extra term ?What is the extra term ? Think classically Potential U + K.E. term

)1()]([2

)(2

2

2

2

llCrUEmr

rRdr

d

R

r

Or rearranged as )())](1(2

)([)(2 2

2

2

22

rRErRllmr

rUrRdr

d

m

p

e-

L=mvr

)1(

2

)1(

22

)(

2

1

22

2

2

2

2

2

22

llL

mr

ll

mr

L

mr

mvrmv

Total angular momentum

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SolutionSolution

L are Laguerre functions

They are a series with specific solutions for n and l values a0 is a length known as Bohr radius

0.529 x10-10 m

Similiarly can solve the angular part of eqnSimiliarly can solve the angular part of eqn

Specific solutions for l and m values

0,)( na

r

lnl

nl eLrrR

1

0 01 )22)(1(

12:,

ln

iji

ii b

lji

nli

nabwhererb

2/300 2 ab

a0

402

me2

||

0

|| cossin),(ml

i

ii

mimlmlm aeNY

Spherical harmonics involving another series of constants ai

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Find the H Energy levelsFind the H Energy levels

• Reminder Reminder lmnlmn(r,(r,,,) = R) = Rnlnl(r) Y(r) Ylmlm((,,))• So now we have the solution!

• Substitute intoSubstitute into

And find an expression for EAnd find an expression for E

The energy only depends on nThe energy only depends on n n is Principle quantum number Not on l,m for coulomb potential U

nlmnnlmnlm EUzyxm

2

2

2

2

2

22

2

eVnn

meE

22220

2

4 6.13

32

n = 2

n = 3

n = 4

Ionised atom

E0

-ve, relative to ionised atom

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Quantum NumbersQuantum Numbers

Atom can only be in a discrete set of states n,l,mAtom can only be in a discrete set of states n,l,m Diff. From classical picture with any orbit

Principle n fixes energy - quantizedPrinciple n fixes energy - quantized Integer >=1

l fixes angular momentum Ll fixes angular momentum L Integer in range 0 to n-1

m (or mm (or ml l ) fixes z component of angular ) fixes z component of angular momentummomentum Integer in range –l to +l

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If you only learn 5 things from If you only learn 5 things from this…..this…..

1.1. Solving SchrSolving Schröödinger dinger Discrete states Discrete states

2.2. Quantum numbers n,l,mQuantum numbers n,l,m1. Energy, ang. mom, z cmpt L

3.3. Energy Energy 1/n 1/n2 2 , scale is eV, scale is eV

4.4. Know the ranges n,l,m can takeKnow the ranges n,l,m can take

5.5. …….Hence understand how to calculate the states.Hence understand how to calculate the states

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Angular MomentumAngular Momentum

• Quantum picture of Angular MomentumQuantum picture of Angular Momentum

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States and their States and their spectroscopic notationspectroscopic notation

nn ll mm

11 00 00 1s1s

22 00

11

00

-1,0,1,-1,0,1,

2s2s

2p2p

33

00

11

22

00

-1,0,+1-1,0,+1

-2,-1,0,1,2-2,-1,0,1,2

3s3s

3p3p

3d3d

44

00

11

22

33

00

-1,0,+1-1,0,+1

-2,-1,0,+1,+2-2,-1,0,+1,+2

-3,-2,-1,0,+1,+2-3,-2,-1,0,+1,+2

4s4s

4p4p

4d4d

4f4f

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Angular momentum is Angular momentum is QUANTIZEDQUANTIZED

We now know Energy is quantizedWe now know Energy is quantized Familiar from seeing transition photons E.g. Balmer series nf=2

BUT we have also learnt BUT we have also learnt

and l takes discrete values

s state is l=0 s state is l=0 L=L= p state is l=1 p state is l=1 L=L= d state is l=2d state is l=2 L=L= ff gg

1

E0hc

1ni

2 1

n f2

Ei

Ef

Emission

Photon

)1(22 llL

632

2

0

TOTAL Angular momentum L

Quantum number l

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m m - z component of l - z component of l

- magnetic quantum - magnetic quantum numbernumber

choice of z axis purely a convention choice of z axis purely a convention Important for interactions of atom with magnetic field along z Important for interactions of atom with magnetic field along z

(later)(later)

Cartoon of components for l=2, p state

2

0

2

z

z

z

z

z

L

L

L

L

L

6|| L

c.f. Classical behaviour

•state has angular mometum and this has a component along z axis

But quantum

•States are quantized

•Ang. momentum can be zero

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The statesThe states

• Hydrogen wavefunctionsHydrogen wavefunctions• Where is the electron ?Where is the electron ?

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The first few statesThe first few states Can substitute into our expressions n,l,m and find out Can substitute into our expressions n,l,m and find out

nlmnlm(r,(r,,,) = R) = Rnlnl(r) Y(r) Ylmlm((,,)= R(r) P()= R(r) P() F() F())

Probability depend on wavefunction squared

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Visualising the states(1)Visualising the states(1)

States with zero angular momentum are isotropicStates with zero angular momentum are isotropic Indep.of and n00(r,,)= nlm(r)

P(x,y,z) dx dy dz = | P(x,y,z) dx dy dz = | (x,y,z)|(x,y,z)|2 2 dx dy dzdx dy dz

i.e. probability in cube of vol dV is P dV Probability density fn PDF (dim. 1/length3) So P(r)dr depends on volume of shell of sphere

r

dr

Integrating probability over and

Volume is 4r2dr

drrr

drrPrdrrP22

2

)(4

),,(4)(

Normalised so integral is 1

[] ?] ?

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Visualising the states(2)Visualising the states(2)

2s, 3s states2s, 3s states

wavefunction PDF P(r)

1s state n=1,l=m=0

in units of a

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Hydrogen Atom PDFsHydrogen Atom PDFs

Scale increases with increasing n

l=0 spherically symmetric

m=0 no z cmpt of ang.momentum

z

x

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Fine structureFine structure

• Energy levels given by quantum number n

• Now add a magnetic field…

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Adding Angular Adding Angular MomentumMomentum

LL11 specified by l specified by l11,m,m11

LL22 specified by l specified by l22,m,m22

How would we combine them ?How would we combine them ?

what is lwhat is ltottot, m, mtottot ? ?21 LLLTotal

212121 ,....,1, llllllltot

21 mmmtot

|| tottot lm

L1

L2

Ltot

z

m1

m2mtot

Easy (classical like) bit, adding components

And obv.

So for the total…

Anti-parallel parallel

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Zeeman EffectZeeman Effect

Observe energy spectrum of H atomsObserve energy spectrum of H atoms Now …add magnetic fieldNow …add magnetic field

Atoms have moving charges, hence magnetic interaction Spectral lines split (Pieter Zeman, 1896)

Angular momentum has made small contribution to energy (order 10,000th )

Fine Structure

Discrete Discrete statesstates

as as

Ang.mom. Ang.mom. quantizedquantized

Nature, vol. 5511 February 1897,

pg. 347

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Zeeman effectZeeman effect

Potential energy contribution as classicalPotential energy contribution as classical

BmU

Bm

emBL

m

eBU

BU

Lm

eAI

Bl

lzz

22

2

Magnetic dipole

Potential energy in magnetic field

Now, put magnetic field along z axis

Orbital Magnetic Interaction energy equation

Bohr magneton B

So, for example, p state l=1, with possible m=-1,0,+1, splits into 3 Energy levels according to Zeeman effect

Sodium 4p3s

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How many lines on that last photo …? How many lines on that last photo …?

l=0, ml=0, mll=0=0 1 line1 line l=1, ml=1, mll=-1,0,1=-1,0,1 3 lines3 lines l=2, ml=2, mll=-2,1,0,1,2=-2,1,0,1,2 5 lines5 lines ………… l=a, l=a, mmll has has 2a+12a+1 lines lines

Stern-Gerlach ExperimentStern-Gerlach Experiment

lz mL

Odd no.

Experiment with silver atoms, 1921, saw some EVEN numbers of lines

Non-uniform B field, need a force not just a twist

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““Anomalous” Zeeman EffectAnomalous” Zeeman Effect

We need another source of ang. mom.,We need another source of ang. mom.,

mmll is not enough is not enough We know we can add angular momentumWe know we can add angular momentum

EVEN numbers of splittings, something is missing……

SLJ

Total orbital spin zzz SLJ and

ml=-l….+l, ms=-1/2, +1/2 Intrinsic property of electron

So, every previous state we can split into two

(careful though for total as 1+1/2 = 2-1/2!!)

Using +1/2 or –1/2 electron spin states

BgmU BsEnergy splitting as before but with an extra factor of g=2Due to relativistic effects

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Complex Example: Sodium p stateComplex Example: Sodium p state

l=1 hence j=1+1/2 or j=1-1/2

j=3/2 or j=1/2, now 2 states

j=3/2, mj=+3/2,+1/2,-1/2,-3/2

j=1/2, mj=+1/2,-1/2

4

2

STATES

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Electron SpinElectron Spin

Like a spinning top!Like a spinning top! But not really…point-like particle as far as we know

Orbital and Intrinsic spin is familiarOrbital and Intrinsic spin is familiar Earth spinning on axis while orbiting the sun

S and ms, just like l and ml

Spin is just another standard characteristic of a particle like its mass or charge

Electron spin is 1/2

2222

4

3)1

2

1(

2

1)1( ssS

2

1,

2

1sm up and down spins

BgmU BsGyromagnetic ratio g~2

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Total Angular momentum: A Top 5…Total Angular momentum: A Top 5…

1.1. Orbital angular momentum L, Orbital angular momentum L, e orbiting nucleuse orbiting nucleus L2=l(l+1)h

2.2. Quantum number lQuantum number l notation l=spdfg…., l=0,1,2,3,4…

3.3. ll has z-component mhas z-component mll,, (-l….+l) (-l….+l) Interacts with magnetic field, U=mlBB Zeeman effect gives splitting of states

4.4. Spin s=1/2Spin s=1/2, intrinsic property of electron, intrinsic property of electron Has ms =-1/2, +1/2 So splits an l state into two

5.5. Total Angular Momentum JTotal Angular Momentum J Sum of orbital and spin Anomalous Zeeman effect / Stern-Gerlach Expt

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Multi-Electron AtomsMulti-Electron Atoms

Everything that isn’t hydrogen!Everything that isn’t hydrogen!

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Pauli Exclusion PrinciplePauli Exclusion Principle

No two electrons can occupy the same quantum No two electrons can occupy the same quantum mechanical statemechanical state Actually true for all fermions (1/2 integer spin)

Nothing to do with Electrostatic repulsionNothing to do with Electrostatic repulsion Also true for neutrons

Deeply imbedded principle in QMDeeply imbedded principle in QM

If all electrons were in the n=1 state all atoms If all electrons were in the n=1 state all atoms would behave like hydrogen ground statewould behave like hydrogen ground state No chemistry – same properties

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Multi-Electron atomsMulti-Electron atoms

Lowest energy configurationLowest energy configuration Start adding the electrons filling up each state

•H Energy levels depend only on 1/n2

•Each state contains two electrons

n=1

n=2

n=3

Energy l=0 1 2

Helium

Beryllium

Hydrogen

Lithium

BoronCarbon

2p

2s

1sml ms for filling states

Full shells

But order in shell?

Z=2

Z=10

Z=18

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Central field approximationCentral field approximation

We have neglected any interaction of electronsWe have neglected any interaction of electrons BUT we no longer have a coulomb potentialBUT we no longer have a coulomb potential

U now depends on electrons we have already added

Approximation - Electron moving in averaged out Approximation - Electron moving in averaged out field due to all othersfield due to all others

Screening EffectScreening Effect Higher n, l means more screening See less charge (Gauss’ law) Radial solutions of schrödinger have

changed E now depends on l not just n

r

ZerU

2

04

1)(

electrons

nucleus

p,s states extra peaks at low r, more time close to nucleus, less screened

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Energy Order of States Energy Order of States Screening shifts the statesScreening shifts the states

f above d above p above s but also 3d is above 4s

Z=2

Z=10

Z=18

E Gap number

35

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Everything you ever need to know Everything you ever need to know about Chemistryabout Chemistry

Closed ShellClosed Shell Helium Z=2 1s2 all n=1 states

full Neon Z=10 1s22s22p6 all n=1,2

states full Argon Z=18 1s22s22p63s23p6

Krypton….

Noble gases – non reactive, stable, RH columnNoble gases – non reactive, stable, RH column One electron moreOne electron more

Lithium Z=3 He + 2s Sodium Z=11 Ne + 3s Potassium Z=19 Ar + 4s Rubidium Z=37 Kr + 5s

Alkali metals, effective screening, weak binding, Alkali metals, effective screening, weak binding, easily get ionseasily get ions

Similarly Be, Mg, Ca form 2+ ions (alkaline earth Similarly Be, Mg, Ca form 2+ ions (alkaline earth metals)metals)

And F,Cl,Br form 1- ions to get closed shell And F,Cl,Br form 1- ions to get closed shell (halogens)(halogens)