Physical Chemistry 2021/8/21 Che mistry De partne nt of Fudan University 1
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 1 Physical Chemistry
Phusical chemiatry Chapter II Atomic Structure and Spectrum Hydrogen-like Atom: the model consists of a proton fixed at the origin and an electron that interacts with the proton through a coulombic potential 2 2 H 2 X The three spherical coordinates are associated spherical coordinate with the three spatial quantum numbers 2u r ar Or rasin 60o sin.o 方2r1a2,O 1 十 00 r2sin20 002 jy 十 e+ 2021/8/21 Che mistry De partne nt of Fudan University 2
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 2 Hydrogen-like Atom: the model consists of a proton fixed at the origin and an electron that interacts with the proton through a Coulombic potential. r Ze H V 2 2 2 2 2 2 2 = − + = − − ˆ spherical coordinate = + + + − r Ze E r r r r r r 2 2 2 2 2 2 2 2 2 1 1 1 2 sin sin sin
Phusical chemiatry Chapter II Atomic Structure and Spectrum Note that the angular and radial terms can be separated, we suggest that we can write the wavefunction as a product of radial and angular parts. (;0,)=R(r)Y(0,) Then the angular part is separated into two parts Y(O,d)=(O(0) and three parts are substituted into Schrodinger equation, we have 1 a OR ao sin 6-+ e+ r ar ar osin 6 ae a0 snap ao 2021/8/21 Che mistry De partne nt of Fudan University 3
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 3 Y(,)= ( )() + = − + + r Ze E r r R r R r 2 2 2 2 2 2 1 2 1 1 2 sin sin sin Note that the angular and radial terms can be separated, we suggest that we can write the wavefunction as a product of radial and angular parts. : (r,,)= R(r)Y(,) Then the angular part is separated into two parts: and three parts are substituted into Schrodinger equation, we have
Now the schrodinger equation can be written as three separate equations. 102OR2/2 radial equation E k r ar a 九2 sin e d de sin 6-+ksin 0 colatitude equation o de de 1dΦ azimuthal equation op do 2021/8/21 Che mistry De partne nt of Fudan University
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 4 k r Ze E r r R r R r = + 2 2 2 1 2 2 2 2 2 1 m d d − = 2 2 k m d d d d + = sin sin sin colatitude equation azimuthal equation radial equation Now the Schrodinger equation can be written as three separate equations
Phusical chemiatry Chapter II Atomic Structure and Spectrum We have seen that the azimuthal wave functions are e 2丌 This solution imposes the constraint the m be a quantum number and have values n=0,士1,±2,士3, sin e d de sin e +ksm2=m2k=1(+1) o de When this equation is solved it is found that k must equal l(l+1) with l=0,1,2,3.. and as above n=0,±1,土2,土3,, O(0)=P(cos0)=(1-cos 2 G(cos 0) G(e)=2a,cose 2021/8/21 Che mistry De partne nt of Fudan University 5
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 5 2 2 k m d d d d + = sin sin sin ( ) im e − = 2 1 ( ) P(cos ) ( cos ) G(cos) m 2 2 = = 1− − = = l m n n an G z 0 ( ) cos k = l(l +1) We have seen that the azimuthal wave functions are This solution imposes the constraint the m be a quantum number and have values m = 0, ±1, ±2, ±3, … When this equation is solved it is found that k must equal l(l+1) with l = 0, 1, 2, 3… and as above m = 0, ±1, ±2, ±3, …