Quantum Phenomena II: Matter Matters

Quantum Phenomena II:Matter Matters • Hydrogen atom • Quantum numbers • Electron intrinsic spin • Other atoms • More electrons! • Pauli Exclusion Principle • Periodic Table Atomic Structure Fundamental Physics • Particle Physics • The fundamental particles • The fundamental forces • Cosmology • The big bang http://ppewww.ph.gla.ac.uk/~parkes/teaching/QP/QP.html Chris Parkes March 2005

“Curiouser and curiouser” cried Alice • 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 series • Some consequences of QM • Applications • Emphasis on awareness not mathematical rigour • First few lectures – Young & Freedman 41.1->41.4 • Lectures main points same, but more complex treatment • Last few lectures – Y&F 44

Understanding atoms • Key to all the elements & chemistry • Non-relativistic QM – the schrödinger equation • What are atoms made of ? • Nucleus (p,n) ,e • What are the nucleons made of • quarks • Why are p,n clamped together in the middle? • Strong nuclear force • Second part of this course….. • How do we analyse atoms • The first part of this course…..

Find the energy levels for a Hydrogen atom • Find the wavefunction for the hydrogen atom

Schrödinger Equation :solving H atom • Wavefunction • Probability to find a particle at • P(x,y,z) dx dy dz = | (x,y,z)|2 dx dy dz • This looks like p2/2m + U = E in classical mechanics • n,l,m are quantum numbers • E depends on n only for H (also l for multi electron atoms) • BUT now we have a wavefunction (x,y,z) BIG Difference from classical physics. No longer know where a particle is Just how likely it will be at x,y,z dy dz dx

Spherical co-ordinates • Potential energy of one electron in orbit around one proton • Spherical symmetry, so use spherical polars • rewrite schrödinger in r, ,  • Rather than x,y,z • Mass of electron m, Charge of proton,electron e • For single electron heavy ion would have q=Ze • Try (r,,) = R(r) Y(,) • Separate out the radial parts and the angular parts, LHS(r) = RHS(,)=C

Radial Equation • BUT this looks a lot like schrödinger eqn • With rR(r)=(r) • And with an extra term • What is the extra term ? • Think classically • Potential U + K.E. term Or rearranged as L=mvr e- p Total angular momentum

Solution • 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 eqn • Specific solutions for l and m values Spherical harmonicsinvolving another series of constants ai

Find the H Energy levels • Reminder lmn(r,,) = Rnl(r) Ylm(,) • So now we have the solution! • Substitute into • And find an expression for E • The energy only depends on n • n is Principle quantum number • Not on l,m for coulomb potential U Ionised atom n = 4 n = 3 n = 2 E0 -ve, relative to ionised atom

Quantum Numbers • Atom can only be in a discrete set of states n,l,m • Diff. From classical picture with any orbit • Principle n fixes energy - quantized • Integer >=1 • l fixes angular momentum L • Integer in range 0 to n-1 • m (or ml ) fixes z component of angular momentum • Integer in range –l to +l

If you only learn 5 things from this….. • Solving Schrödinger  Discrete states • Quantum numbers n,l,m • Energy, ang. mom, z cmpt L • Energy  1/n2 , scale is eV • Know the ranges n,l,m can take • ….Hence understand how to calculate the states

Angular Momentum • Quantum picture of Angular Momentum

States and their spectroscopic notation

Angular momentum is QUANTIZED • We now know Energy is quantized • Familiar from seeing transition photons • E.g. Balmer series nf=2 • BUT we have also learnt • and l takes discrete values • s state is l=0 L= • p state is l=1 L= • d state is l=2 L= • f • g Ei Photon Ef Emission TOTAL Angular momentum L Quantum number l

m - z component of l - magnetic quantum number • choice of z axis purely a convention • Important for interactions of atom with magnetic field along z (later) Cartoon of components for l=2, p state • 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

The states • Hydrogen wavefunctions • Where is the electron ?

The first few states • Can substitute into our expressions n,l,m and find out nlm(r,,) = Rnl(r) Ylm(,)= R(r) P() F() Probability depend on wavefunction squared

Visualising the states(1) • States with zero angular momentum are isotropic • Indep.of  and  • n00(r,,)= nlm(r) • P(x,y,z) dx dy dz = | (x,y,z)|2 dx 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 [] ? Integrating probability over  and  Volume is 4r2dr r Normalised so integral is 1 dr

Visualising the states(2) 1s state n=1,l=m=0 • 2s, 3s states wavefunction PDF P(r) in units of a

Hydrogen Atom PDFs Scale increases with increasing n l=0 spherically symmetric m=0 no z cmpt of ang.momentum z x

Fine structure • Energy levels given by quantum number n • Now add a magnetic field…

Adding Angular Momentum • L1 specified by l1,m1 • L2 specified by l2,m2 • How would we combine them ? • what is ltot, mtot ? z Ltot m2 L2 mtot m1 Easy (classical like) bit, adding components L1 And obv. So for the total… Anti-parallel parallel

Zeeman Effect Nature, vol. 5511 February 1897, pg. 347 • Observe energy spectrum of H atoms • Now …add magnetic field • Atoms have moving charges, hence magnetic interaction • Spectral lines split (Pieter Zeman, 1896) Discrete states as Ang.mom. quantized Angular momentum has made small contribution to energy (order 10,000th ) Fine Structure

Zeeman effect • Potential energy contribution as classical Magnetic dipole Sodium 4p3s Potential energy in magnetic field Now, put magnetic field along z axis Bohr magneton B Orbital Magnetic Interaction energy equation So, for example, p state l=1, with possible m=-1,0,+1, splits into 3 Energy levels according to Zeeman effect

How many lines on that last photo …? Stern-Gerlach Experiment • l=0, ml=0 1 line • l=1, ml=-1,0,1 3 lines • l=2, ml=-2,1,0,1,2 5 lines • …… • l=a, ml has 2a+1 lines Experiment with silver atoms, 1921, saw some EVEN numbers of lines Non-uniform B field, need a force not just a twist Odd no.

“Anomalous” Zeeman Effect EVEN numbers of splittings, something is missing…… • We need another source of ang. mom., ml is not enough • We know we can add angular momentum and Total orbital spin 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 Energy splitting as before but with an extra factor of g=2 Due to relativistic effects

Complex Example: Sodium p state STATES 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

Electron Spin Gyromagnetic ratio g~2 • Like a spinning top! • But not really…point-like particle as far as we know • Orbital and Intrinsic spin is familiar • Earth spinning on axis while orbiting the sun Electron spin is 1/2 S and ms, just like l and ml up and down spins Spin is just another standard characteristic of a particle like its mass or charge

Total Angular momentum: A Top 5… • Orbital angular momentum L, e orbiting nucleus • L2=l(l+1)h • Quantum number l • notation l=spdfg…., l=0,1,2,3,4… • lhas z-component ml, (-l….+l) • Interacts with magnetic field, U=mlBB • Zeeman effect gives splitting of states • Spin s=1/2, intrinsic property of electron • Has ms =-1/2, +1/2 • So splits an l state into two • Total Angular Momentum J • Sum of orbital and spin • Anomalous Zeeman effect / Stern-Gerlach Expt

Multi-Electron Atoms • Everything that isn’t hydrogen!

Pauli Exclusion Principle • No two electrons can occupy the same quantum mechanical state • Actually true for all fermions (1/2 integer spin) • Nothing to do with Electrostatic repulsion • Also true for neutrons • Deeply imbedded principle in QM • If all electrons were in the n=1 state all atoms would behave like hydrogen ground state • No chemistry – same properties

Multi-Electron atoms Hydrogen Helium Lithium Beryllium Boron Carbon • Lowest energy configuration • Start adding the electrons filling up each state l=0 1 2 Energy Z=18 n=3 Z=10 n=2 Z=2 n=1 ml ms for filling states Full shells But order in shell? 1s • H Energy levels depend only on 1/n2 • Each state contains two electrons 2s 2p

Central field approximation • We have neglected any interaction of electrons • BUT we no longer have a coulomb potential • U now depends on electrons we have already added • Approximation - Electron moving in averaged out field due to all others • Screening 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 electrons nucleus p,s states extra peaks at low r, more time close to nucleus, less screened

Energy Order of States • Screening shifts the states • f above d above p above s • but also 3d is above 4s E Gap number Z=18 Z=10 Z=2

Everything you ever need to know about Chemistry • Closed 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 column • One 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, easily get ions • Similarly Be, Mg, Ca form 2+ ions (alkaline earth metals) • And F,Cl,Br form 1- ions to get closed shell (halogens)

Quantum Phenomena II: Matter Matters