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Quantum Phenomena II: Matter Matters Quantum Phenomena II: Matter Matters Chris Parkes March 2005 Hydrogen atom Quantum numbers Electron intrinsic spin Other atoms More electrons! Pauli Exclusion Principle Periodic Table Particle Physics The fundamental particles The fundamental forces Cosmology The big bang Fundamental PhysicsAtomic Structure

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2 “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

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3 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…..

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

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5 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 1.This looks like p 2 /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 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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6 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 (r, , ) = R(r) Y( , )Try (r, , ) = R(r) Y( , ) LHS(r) = RHS( , )=CSeparate out the radial parts and the angular parts, LHS(r) = RHS( , )=C Mass of electron m, Charge of proton,electron e For single electron heavy ion would have q=Ze

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7 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 p e-e- L=mvr Total angular momentum

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8Solution L are Laguerre functions They are a series with specific solutions for n and l values a 0 is a lengthknown as Bohr radius x m Similiarly can solve the angular part of eqn Specific solutions for l and m values Spherical harmonics involving another series of constants a i

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9 Find the H Energy levels Reminder lmn (r, , ) = R nl (r) Y lm ( , )Reminder lmn (r, , ) = R nl (r) Y lm ( , ) So now we have the solution! Substitute intoSubstitute 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 n = 2 n = 3 n = 4 Ionised atom E0 E0 -ve, relative to ionised atom

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10 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 m l ) fixes z component of angular momentum Integer in range –l to +l

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11 If you only learn 5 things from this….. 1. Solving Schrödinger Discrete states 2. Quantum numbers n,l,m 1. Energy, ang. mom, z cmpt L 3. Energy 1/n 2, scale is eV 4. Know the ranges n,l,m can take 5. ….Hence understand how to calculate the states

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12 Angular Momentum Quantum picture of Angular Momentum Quantum picture of Angular Momentum

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13 States and their spectroscopic notation nlm 1001s ,0,1,2s2p ,0,+1-2,-1,0,1,23s3p3d ,0,+1-2,-1,0,+1,+2-3,-2,-1,0,+1,+24s4p4d4f

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14 Angular momentum is QUANTIZED We now know Energy is quantized Familiar from seeing transition photons E.g. Balmer seriesn f =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=2L= f g EiEi EfEf Emission Photon TOTAL Angular momentum L Quantum number l

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15 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

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16 The states Hydrogen wavefunctionsHydrogen wavefunctions Where is the electron ?Where is the electron ?

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17 The first few states Can substitute into our expressions n,l,m and find out nlm (r, , ) = R nl (r) Y lm ( , )= R(r) P( ) F( ) Probability depend on wavefunction squared

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18 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/length 3 ) So P(r)dr depends on volume of shell of sphere r dr Integrating probability over and Volume is 4 r 2 dr Normalised so integral is 1 ] ? [ ] ?

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

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20 Hydrogen 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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21 Fine structure Energy levels given by quantum number n Now add a magnetic field…

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22 Adding Angular Momentum L 1 specified by l 1,m 1 L 2 specified by l 2,m 2 How would we combine them ? what is l tot, m tot ? L1L1 L2L2 L tot z m1m1 m2m2 m tot Easy (classical like) bit, adding components And obv. So for the total… Anti-parallelparallel

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23 Zeeman Effect Observe energy spectrum of H atoms Now …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 10,000 th ) Fine Structure Discrete states as Ang.mom. quantized Nature, vol February 1897, pg. 347

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24 Zeeman effect Potential energy contribution as classical 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 4p 3s

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25 How many lines on that last photo …? l=0, m l =01 line l=1, m l =-1,0,13 lines l=2, m l =-2,1,0,1,25 lines …… l=a, m l has 2a+1 lines Stern-Gerlach Experiment 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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26 “Anomalous” Zeeman Effect We need another source of ang. mom., m l is not enough m l is not enough We know we can add angular momentum EVEN numbers of splittings, something is missing…… Total orbital spin and m l =-l….+l, m s =-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

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27 Complex 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, m j =+3/2,+1/2,-1/2,-3/2 j=1/2, m j =+1/2,-1/ STATES

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28 Electron Spin 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 S and m s, just like l and m l Spin is just another standard characteristic of a particle like its mass or charge Electron spin is 1/2 up and down spins Gyromagnetic ratio g~2

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

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

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31 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

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32 Multi-Electron atoms Lowest energy configuration Start adding the electrons filling up each state H Energy levels depend only on 1/n 2 Each state contains two electrons n=1 n=2 n=3 Energy l=012 Helium Beryllium Hydrogen Lithium Boron Carbon 2p 2s 1s m l m s for filling states Full shells But order in shell? Z=2 Z=10 Z=18

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33 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

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34 Energy Order of States Screening 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

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36 Everything you ever need to know about Chemistry Closed Shell HeliumZ=21s 2 all n=1 states full NeonZ=101s 2 2s 2 2p 6 all n=1,2 states full ArgonZ=181s 2 2s 2 2p 6 3s 2 3p 6 Krypton…. Noble gases – non reactive, stable, RH column One electron more LithiumZ=3He + 2s SodiumZ=11Ne + 3s PotassiumZ=19Ar + 4s RubidiumZ=37Kr + 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)

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