Since we cannot say exactly where an electron is, …

1 MNW-L2 Since we cannot say exactly where an electron is, the Bohr picture of the atom, with electrons in neat orbits, cannot be correct. Quantum theory describes electron probability distributions:Quantum Mechanics and the hydrogen atomMNW-L2 Hydrogen Atom: Schr dinger Equation and Quantum NumbersPotential energy for the hydrogen atom:The time-independent Schr dinger equation in three dimensions is then:Equation 39-1 goes does the quantisation in QM come from ?The atomic problem is spherical so rewrite the equation in (r, , ) cossinrx= sinsinry= cosrz=Rewrite all derivatives in (r, , ), gives Schr dinger equation; = + + ErVmrrrm)(sin1sinsin122222222 hhThis is a partial differential equation, with 3 coordinates (derivatives);Use again the method of separation of variables:()()() ,,,YrRr= Bring r-dependence to left and angular dependence to right (divide by ):()()()() = = +,,21222 YYORrVEmrdrdRrdrdRQMhSeparation of variablesMNW-L2 where does the quantisation in QM come from ?

2 ()() = +RrVEmrdrdRrdrdR22221hRadial equationAngular equation()()()() = + = ,,sin1sinsin1,,222 YYYYOQMYYY 222sinsinsin+ = Once more separation of variables:()()() =,YDerive:2222sinsinsin11m= + = (again arbitrary constant)Simplest of the three: the azimuthal angle;()()0222= + mMNW-L2 where does the quantisation in QM come from ?A differential equationwith a boundary condition()()0222= + mand())(2 =+ Solutions:() ime= Boundary condition;()()() imimee= ==+ +2212=ime mis a positive or negative integerthis is a quantisation conditionGeneral: differential equation plus a boundary condition gives a quantisationMNW-L2 where does the quantisation in QM come from ?() ime= First coordinatewith integer m (positive and negative)Second coordinate0sinsinsin122= + mResults in()1+=lll withKl,2,1,0=andllKll,1,,1, + =mangularpartangularmomentumThird coordinate()()()121222+= +llhRrVEmrdrdRrdrdRDifferentialequationR esults in quantisation of energy2202222224henmenZRnZE = = with integer n (n>0)radialpartMNW-L2 angular wave functions + =2222sin1sinsin1 iLhwithKl,2,1,0=andllKll,1,,1, + =mOperators.

3 =iLzhThere is a class of functions that are simultaneous eigenfunctions),,(zyxLLLL=rAngular momentum()()() ,1,22lmlmYYLhll+=()() ,,lmlmzYmYLh=Spherical harmonics (Bolfuncties)() ,lmY 4100=Y ieYsin8311 = cos4310 =Y ieY =sin831,1 Vector space of solutions()1,2= dYlm ''''*mmllmllmdYY = () ()() () ,,,lmlmopP =+ = lParityMNW-L2 The radial part: finding the ground stateFind a solution for 0,0==mlERRrZeRrRm= + 0224'2"2 hPhysical intuition; no density for rtrial: ()arAerR/ =aReaARar = = /'2/2"aReaARar== ErZearam= 02224212 hmust hold for all values of r04022= ZemahPrefactor for 1/r:mZea22004h =Solution for thelength scale paramaterBohr radiusSolutions for the energy220222242hhemeZmaE = = Ground state in the Bohr model (n=1)()() = +RrVEmrdrdRrdrdR22221hMNW-L2 There are four different quantum numbers needed to specify the state of an electron in an The principal quantum number ngives the total The orbital quantum number gives the angular momentum; can take on integer values from 0 to Atom: Schr dinger Equation and Quantum Numbersll3.

4 The magnetic quantum number, m, gives the direction of the electron s angular momentum, and can take on integer values from to + .lllMNW-L2 Hydrogen Atom Wave FunctionsThe wave function of the ground state of hydrogen has the form:The probability of finding the electron in a volume dVaround a given point is then | | Probability DistributionsSpherical shell of thickness dr, inner radius rand outer radius r+ volume is dV=4 r2drDensity: | |2dV = | |24 r2drThe radial probablity distribution is then:Pr=4 r2| |2 Ground stateMNW-L2 This figure shows the three probability distributions for n= 2 and = 1(the distributions for m= +1and m= -1are the same), as well as the radial distribution for all n= Atom Wave FunctionslMNW-L2 Hydrogen Orbitals ( electron clouds)()2,trr Represents the probability to find a particleAt a location rat a time tThe probability densityThe probability distributionThe Nobel Prize in Physics 1954"for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction"Max BornMNW-L2 Atomic Hydrogen Radial partAnalysis of radial equation yields: =RnZEnlm22chemRe3048 = withWave functions:() () ( )hr/,,tniElmnlnlmerRtr = ()()() + =+ + nZLnZennnnaZunnZn 22!

5 12!121211/0lllllnaZr/2/= 220/4ea h=For numerical use: ()rruRnl=MNW-L2 Quantum analog of electromagnetic radiationClassical electric dipole radiationTransition dipole momentClassical oscillatorQuantum jump2reIrad&&r 22*1 dreIrad rThe atom does not radiate when it is in a stationary state !The atom has no dipole moment01*1== driir2026h fiifB=Intensity of spectral lines linkedto Einstein coefficient for absorption:MNW-L2 Selection rulesMathematical background: function of odd parity gives 0when integrated over spaceIn one dimension:dxxfdxxxifif = = )(*withifxxf =*)(()dxxfdxxfdxxfxdxfdxxfdxxfdxxf + =+ =+=000000)()()()()()()(0)(20 = dxxf)()(xfxf= if0=if)()(xfxf = i andf opposite parityi andf same parityElectric dipole radiation connects states of opposite parity !

6 MNW-L2 Selection rulesdepend on angularbehavior of the wave functionsrrrr Parity operator()() + =,,,,),,(),,(rrzyxzyxrrPrrAll quantum mechanical wave functionshave a definite parity()()rrrr = 0 ifrrIf and have opposite parityf i Rule about the mYlfunctions()),(),( mmYPYlll =MNW-L2 Allowed transitions between energy levels occur between states whose value of differ by one:Other, forbidden, transitions also occur but with much lower Atom: Schr dinger Equation and Quantum Numbersl selection rules, related to symmetry MNW-L2 Selection rules in Hydrogen atomIntensity of spectral lines given byififfire = = rr *1) Quantum numbernno restrictions2) Parity rule forlodd= l3) Laporte rule forl1rlrlr+=ifAngular momentum rule:so1 lFrom 2.

7 And = lLyman seriesBalmer series