Agenda Administrative Matters Atomic Physics Molecules

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1 Fromm Institute for Lifeong Learning University of San Francisco Modern Physics for Frommies IV The Universe - Sma to Large Lecture 3 Agenda Administrative Matters Atomic Physics Moecues Administrative Matters Wikis from previous course Note the ii in frommiis Modern Physics IV Lecture 3 1 Modern Physics IV Lecture 3 The current course wiki, now incudes a Gossary of Mathematica Symbos, Other gossaries can be found on Googe by searching mathematics symbos Quantum Mechanica View of Atoms Bohr mode discarded as an accurate description of nature Certain aspects have however been retained e.g. Eectrons in an atom exist ony in discrete states of definite energy, the stationary states Transitions between these states require the emission (or absorption of a photon. According to wave mechanics, eectrons do not trave in we defined circuar orbits aa Bohr. The eectron, because of its wave nature, is better thought of as spread out in space as a coud. The sie and shape of the eectron coud can be found by soving the Schrödinger equation for the atom and forming the probabiity distribution, ψ. Modern Physics IV Lecture 3 3 Modern Physics IV Lecture 3 4 1

2 This image cannot currenty be dispayed. Ground state of hydrogen Schrödinger s Equation in Spherica Coordinates x r y In Cartesian coordinates ħ m = where ( ) ψ ( x, y, ) Eψ x, y, x y = + + Separation of variabes: Assume a soution of the form ψ,, = ( x y ) X ( x) Y ( y) Z ( ) Modern Physics IV Lecture 3 5 Modern Physics IV Lecture 3 6 Spherica Time Independent Schrödinger Equation Want to do the same thing with spherica symmetry 1 ψ 1 ψ 1 ψ m r sin + θ + + ( E V ) ψ = 0 r r r r sinθ θ θ r sin θ ϕ ħ 1 e where for the hydrogen atom V = 4πε r Separation of variabes: Modern Physics IV Lecture 3 7 Modern Physics IV Lecture ( r ) = R( r) Θ( ) Φ ( ) Try soution of form ψ, θ, ϕ θ ϕ m ( ) ( ) ψ ( r, θ, ϕ) = R r Y θ, ϕ n ( θ ϕ ) = m ( θ ) m spherica harmonics: Y, P cos e where the P m associated LaGuerre functions: R imϕ are associated Legendre poynomias n ( r)

3 Quantum Numbers If we do QM in for a partice confined in a 1-D and 3-D potentia we or rigid box. (See Course II Lecture 4) The soutions are characteried by a singe quantum number (n) in the 1-D case and by three numbers (n x, n y and n ) in 3-D. These quantum numbers arise from the imposition of boundary conditions on the soutions. We might expect that in the 3-D probem of the hydrogen atom the soutions wi be characteried by numbers corresponding to Boundary conditions appied in 3-D. Restrictions on the vaues of these quantum numbers arise from the mathematics of the LaGuerre functions and the spherica harmonics. Actuay, we need a fourth number. There is an additiona degree of freedom which I wi treat in a few minutes. n = 1,,3, principe quantum number E0 m e 1 En = = n 4πε 0ħ n Bohr resut Resuts from boundary conditions on soution of the R part of the separated Schrödinger eqn. R part contains the potentia energy n aone determines the energy eves (actuay there is a sight deviation from this) Consequence of centra inverse square force. Modern Physics IV Lecture 3 9 Modern Physics IV Lecture 3 10 = 0,1,,3,, n 1 orbita anguar momentum quantum number ( ) Θ( θ ) associated with R r and parts of Sch. eqn. Cassicay L = r p or L = rmv tangentia h Quantu m boundary conditions L = ( + 1 ) π Note disageement with Bohr quantiation where h L = n π in particuar, the ground state has = 0 L = 0 The semicassica panetary mode with eectrons in orbits is not a good one Note: A these transitions have = ±1 Modern Physics IV Lecture February 01 Modern Physics IV Lecture 3 1 3

4 s p d f g h etc. = etc. Notation for states: n, e.g. 4d is n=4, = m =, + 1,,0,1,, 1, magnetic quantum number L is a vector quantity, conserved in a centra potentia ( ) The soution for Φ ϕ specifies that m is an integer reated to L's -component. h L = m π Aside on Anguar Momentum Modern Physics IV Lecture 3 13 Modern Physics IV Lecture 3 14 r θ v Partice of mass m moving with circuar speed v around an axis at radius r. L = r mv Magnitude: L = mvr sinθ Here, θ = 90 sinθ = 1 L = mvr Direction: to pane of r and v with sense determined by right hand rue. C = A B C = A B C = ABsinθ To the pane of AB Note that the vector product is not commutative Right Hand Rue A C A B = B A Again ook at Right Hand Rue Direction of advance of a right hand screw Modern Physics IV Lecture 3 15 Modern Physics IV Lecture 3 16 A C A B B A 4

5 ( ) L = + 1 ħ fixed L = mħ restricted Energy is dependent soey on n. Presence of mutipe s and ms for a given n states are degenerate This degeneracy is removed if directiona symmetry is broken by say a B or E fied. What about L x and L y? Space quantiation Note choice of axis is arbitrary. Modern Physics IV Lecture 3 17 Modern Physics IV Lecture 3 18 If L and L are known, knowedge of nd component 3rd is aso known. consequence of L = L + L + L Uncertainty principe: L ϕ =ħ x y If we know L exacty, we know nothing of ϕ we know nothing of L x and L y L Li L x y L y x Modern Physics IV Lecture 3 19 Modern Physics IV Lecture 3 0 5

6 Magnetic effects Norma Zeeman effect: Transition between 1s and p Spectra ines broaden and spit into 3 ines as B is appied and increased. 3 ines = norma Zeeman effect Appy externa magnetic fied Torque: N = µ B Potentia energy: U = µ B B Consider the eectron orbit to be a current oop with µ = IA ( e) π r q erv e µ = IA = A = = = L T π r v m e Vector: µ = L m eħ µ = m = µ Bm m eħ µ B = Bohr magneton m Modern Physics IV Lecture 3 1 Modern Physics IV Lecture 3 L quantied µ quantied Additiona potentia energy term: U = µ B = + µ m B B B Each degenerate energy eve,, is spit into +1 separate energy eves, m. The Stern-Gerach experiment: If B is inhomogeneous there wi be a net force as we as torque on the atom B has specified a direction in space ( axis) and the symmetry responsibe for the degeneracy has been broken. 1 February 01 Modern Physics IV Lecture 3 3 Modern Physics IV Lecture 3 4 6

7 Wofgang Paui: Eectron Spin Reativity besides n,, m need 4 th quantum number For 0 the states shoud separate according to m ines seen instead of the expected 3 (or +1 = odd) Haven t seen the whoe picture yet. G. Hoenbeck and S. Goudsmit: Propose intrinsic spin anguar momentum for the eectron s = ½ħ Another magnetic quantum number: m s = ± ½ 198, P. A. M. Dirac justifies this from reativity. iγ ψ = mψ Modern Physics IV Lecture 3 5 Modern Physics IV Lecture 3 6 Gives magnetic effects ike orbita anguar momentum. Intrinsic spin intrinsic magnetic dipoe moment New magnetic quantum numbers: m s = ±1/ Doubes number of states for a given n States are degenerate uness a spatia direction is specified, e.g. externa E or B fied Quantum state now specified by {n,, m, m s } Return to the Stern-Gerach experiment = 0 state wi give ines for m s = ± 1/ Fine structure: Even in the absence of externa fieds, very high resoution spectroscopy reveas spitting of spectra ines. Rest frame of eectron: nuceus orbits and appears as a current oop. Interacts with spin magnetic moment and breaks degeneracy Line separation is about 5 x 10-5 ev compared to the p 1s transition energy of 10. ev Hyperfine structure: Arises from the spin anguar momentum and consequent spin magnetism of the nuceon(s) Modern Physics IV Lecture 3 7 Modern Physics IV Lecture 3 8 7

8 s-wave states are sphericay symmetric, not so for 0 Quantum Statistics Consider a system of partices, say eectrons Wave function for the system is ψ, in voume eements at r1 and r P( r, r ) dv dv =ψ * ψ dv dv ( r r ) 1 we observe a probabiity of finding the partices It is easy to show that ( )( ) ψ * ψ = ψ * ψ i.e. no change in observabe for ψ ψ identica partices no observabe change if they are interchanged ψ *, ψ, ψ *, ψ, ( r r ) ( r r ) = ( r r ) ( r r ) Modern Physics IV Lecture 3 9 Modern Physics IV Lecture 3 30 So, under interchange possibiities ψ = ±ψ If identica partices interchange ψ = +ψ, they are said to obey Bose-Einstein statistics and are caed bosons. If identica partices interchange ψ = -ψ, they are said to obey Fermi-Dirac statistics and are caed fermions. Bosons have integra spin, e.g. photons, mesons, some atoms and nucei, Fermions have ½ integra spin, e.g. eptons, nuceons, some atoms and nucei,.. For fermions ψ ψ => ψ = 0 => Cannot have identica partices with the same set of quantum numbers. Paui Excusion Principe You can stick as many bosons into a quantum state as you want. Modern Physics IV Lecture 3 31 Modern Physics IV Lecture 3 3 8

9 Eectrons are fermions. Buid some eements. As eectrons are added the excusion principe wi have an effect. Hydrogen: 1 e in the 1s state 1s 1 He: e in the 1s state, m s =1 and -1, 1s No more e can be added to the 1s state without vioating the excusion principe! The K she is fied Li: 3 rd e has to go in s state s 1 Be: 4 th e in the s state, m s =1 and -1, s s state (subshe) is now fied B: 5 th e has to go in the p state p 1 p state has +1 = 3 vaues of m, each with vaues of m s, accommodating C, N, O, F and Ne as p p 6. p subshe is now fied, as is the L she Na: 11 th e has to go in 3s state 3s 1 etc. Modern Physics IV Lecture 3 33 Modern Physics IV Lecture s and 3p each with +1 m vaues each having vaues of m s. Weirdoes: 3d, 4d and 5d subshes fi up the transition metas foowed by the anthanides and the actinides Compicated inter eectron interactions mess things up If eectrons were bosons, they woud a sit in the ground state, 1s, and chemistry woud be very different. Modern Physics IV Lecture 3 35 Modern Physics IV Lecture

10 Ionic bonding: NaC Bonding in moecues Na Na has 11 e - 10 reside in inner cosed shes Last e - spends most of its time outside these shes. P for Na outer eectron The ast e - fees net attraction due to +1e, not a that strong C C has 17 e - 1 are in cosed shes 1s s p 6 3s Others are in non sphericay symmetric p states Modern Physics IV II Lecture Modern Physics IV II Lecture H m =0, unpaired 4 states m =±1, m s =±1 Excusion principe aows one more e - in m = 0 with spin oriented opposite to that of the ast C e - If an extra eectron happens to be in the vicinity it can be in this state and coud see an attraction due to C nuceus as much as +5e. Stronger than the +1e attraction between Na nuceus and its outer eectron charge distribution in side #36 and an ionic bond between Na and C. Covaent bonding: If H atoms are cose together, e H+H H - couds overap and e - orbit both nucei. Both H s in ground state. Eectron spins can be either parae (S = 1) or antiparae (S = 0) 1 st consider S = 1: Excusion principe e - with same quantum numbers must be in different paces, i.e. beong to different atoms. (+) nucei repe, no bond is formed. Modern Physics IV II Lecture Modern Physics IV II Lecture

11 S = 0: e - have different vaues for m s, spend a ot of time in the internucear region (+) nucei are attracted to the internucear e - and a bond is formed. In a wave picture, excusion principe destructive interference when S=1 and constructive when S=0. Energetics point of view: For S = 0, e - can occupy same space, space of atoms rather than 1 x is increased. H. U. P p can be ess energy is ess Moecue has ower energy than the separate atoms H is stabe Binding energy is 4.5 ev for H Modern Physics IV II Lecture Modern Physics IV II Lecture nm In the vicinity of r 0 we may approximate A B U + m n r r A, B constants for attractive, repusive parts of U m, n are sma integers Activation energy often need to break earier bonds H + O H O H and O must 1st be broken into H and O atoms spark U A = 0 for hypergoic materias, don t need spark Modern Physics IV II Lecture Modern Physics IV II Lecture

12 Energy storage in bioogica systems adenosine triphosphate ATP ADP + (phosphate group) + ENERGY Modern Physics IV II Lecture