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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
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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
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 4r2dr 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 4p3s 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=mlBB • 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)