Chap 5: Absorption Spectrum of a one electron atom or ion

1. Absorption: A + h A* Ei+ h = Ef or h =Ef - Ei A Unexcited atom/molecule with energy Ei A* Excited Atom/Molecule with energy Ef after the absorption of a photon of energy h . Chap 5: Absorption Spectrum of a one electron atom or ion Quiz Next Week on Chap 5 and Section 4.6 & 4.7(2-D)

2. 4 photons hn hn hn hn Absorption When the Zero Magnetic Field (B=0) Mode n Energy 0 r E2 E1 V(r) attractive

3. 4 -1 photons hn hn hn hn Absorption When the Zero Magnetic Field (B=0) Mode n Energy A + h A* E1+ h = E2 or h =E2 – E1 0 r E2 E1 V(r) attractive

4. Magnetic field removes the degeneracy of the m-levels Since magnetic fields breaks the left/right hand symmetry of space Chap 5: M-states split in a B-field (removes degeneracy) m=+1 m=0 2s 2p m=-1 h 1s

5. No magnetic field m=+1 m=0 2s 2p Magnetic field m=-1 Energy Energy h m 1 0 -1 0 0 r r 1s E2 E2 E1 E1 V(r) attractive V(r) attractive

6. Chap. 5:H-atom and One electron Ions Eigen Functions and Values Ylm(. Spherical Harmonic is angular part of the eigen-function and is also the eigen function for the total angular momentum L2 L2 Ylm(=l(l+1)h2 Ylm( and Lz Ylm ( = mhYlm ( Where <|L|2>=L2 =l(l+1)h2 is the magnitude of the angular momentum squared and is a physical observable The magnitude of the angular momentum vector is L=√l(l+1)h2 In the Bohr limit! Magnitude of L~ l2h, l >>1 <|L|2>=L2 = l(l+1)h2 l2h2 for l>>1 Correspondence Principle and the its z-axis projection <Lz>=Lz =mh Sincel(l+1)h2 and mh are Eigen Values they can be measured Therefore l, m and En are physical Observables completely characterizes a Quantum State along with Electron spin

7. Chap 5: Angular Momentum Eigen Values Vector model of t bare angular momentum L: only the Magnitude L and one component, Lx, Ly or Lzcan be measured! <|L|2>=L2l(l+1)h2 z-axis projection <Lz>=Lz=mlh z Magnitude of L L= √l(l+1)h2 ml =+1 <Lz>=Lz= mlh L y ml =0 x L ml =-1

8. Chap 5: Classical Magnetic dipole moment   magnetic dipole moment <L> Attracted to high B-field ml < 0, = r2 m L~ -<L> Repelled from high B-field ml > 0

9. Chap 5: Stern-Gerlach Experiment for discovering electron spin Magnetic field(B) increases with z: U(z) =-mB potential energy in in the field mL~ <Lz> ~ ml Orbital Angular momentum mS~ <Sz> ~ ms <Sz> component of the spin angular momentum Fig. 5-11, p. 184

10. Vector model of the Spin angular momentum S=1/2 S =1/2 is the total spin quantum number B||z repelled from stronger B-field ms=+1/2 S-spin angular momentum <Sz>=Sz= msh x,y plane <|S|2>=S2= s(s+1)h2 z-axis projection <Sz>=Sz=msh ms=-1/2 attracted to stronger B-field

11. Chap 5: Classical Magnetic dipole moment   magnetic dipole moment <S> ~ <Sz> Attracted to high B-field ms=-1/2 =r2 m s~ -<S> Repelled from high B-field ms=+1/2

12. Chap. 5:H-atom and One electron Ions Eigen Functions and Values Ylm( Angular part of the eigen-function and is also the eigen function for the total angular momentum L2 Ylm(=l(l+1)h2 Ylm( and Lz Ylm( = mhYlm( <|L|2> = l(l+1)h2  l2h2 for l>>1 the Bohr limit! Magnitude of L~ (1) h, l >>1 and the its z-axis projection <Lz>=mh L2 Ylm(=l(l+1)h2 Ylm( and Lz Ylm( = mhYlm( Therefore n, l, m and En completely characterizes a Quantum State along with Electron spin angular momentum!

13. Chap 5: Angular Momentum Eigen Values Vector model of t bare angular momentum L: only the Magnitude L and one component, Lx, Ly or Lzcan be measured! <|L|2>=L2l(l+1)h2 z-axis projection <Lz>=Lz=mlh z Magnitude of L L= √l(l+1)h2 ml =+1 <Lz>=Lz= mlh L y ml =0 x L ml =-1

14. Chap 5: Angular Momentum Eigen functions Ypz=Y10( z  x <Lz>=0 L Angular eigen function for the State l=1; m=0 Ypz=Y10(=√(3/4π) cos() Angular momentum Eigen Function + Phase L=√2 h + Notice that only Eigen Values are knowable, i.e., measurable: E, L2, Lz - - Phase Ypz=Y10(= √(3/4π) cos() |Y10(|2 =(3/4π) cos2()

15. Chap 5: Angular Momentum Eigen functions Ypz= Ypz Ypy Ypz=Y10~cos() Linear Combinations of Eigen Functions Ypx=(1/√2){Y11 + Y1-1} ~ sin()cos() Ypy=(1/√2){Y11 - Y1-1} ~ sin()sin()

16. Chap. 5: 3-D Probability Density r|2dV =|R(r)Y()|2dV Probability per unit volume of finding the electron in the volume element: dV=dxdydz= {rd}{rsin(d}dr |R(r )|2 (r2dr) probability of finding the electron between r and r+dr |R(r )|2 r2 Prob per unit length |Y()|2sin(dd= |Y()|2d probability of finding the electron in the solid angle d=sin(dd |Y()|2 Prob per unit solid angle Fig. 5-1, p. 171

17. Chap. Solid Angle for a Sphere of Radius r The solid angleddA/r2 = sin(dddV= dr dA(surface area of dV), dA = r2sin(dd A=4πr2(surface area of the sphere)r2 = 4π solid angle portended(projected) by a sphere dA Differentail Area

18. Chap 5: Angular Momentum Eigen functions Ypz= Ypz Ypy Ypz=Y10~cos() Linear Combinations of Eigen Functions Ypx=(1/√2){Y11 + Y1-1} ~ sin()cos() Ypy=(1/√2){Y11 - Y1-1} ~ sin()sin()

19. z y x |Y00|2 |Y1±1|2 |Y10|2

20. Table 5-2, p. 175

21. Chap 5:1s ~R10( r ) Y00(, 2s~R20( r ) Y00(, 3s~R30( r ) Y00( Number of Nodes in Rnl (n-l-1): radial nodes l angular nodes a0 the Bohr is most probable radius Sphere Radius rs Prob < 0.05 of max prob for finding electron at r > rs

22. Chap 5:, 2pz~R20( r )Ypz(, 3pz R31( r )Ypz( Ypz ~ cos( |Ypz|2~ |cos( r  2pz~R20( r )Ypz( One electron orbital

23. Chap 5:, 2pz~R20( r )Ypz(, 3pz R31( r )Ypz( Ypz ~ cos( |Ypz|2~ |cos( r  L 2pz~R20( r )Ypz(

24. Chap 5:, 2pz~R20( r )Ypz(, 3pz R31( r )Ypz( 2pz~R20( r )Ypz( 2pz one electron orbital 2px~R21( r )Ypx( 2px one electron orbital 2py~R21( r )Ypy( 2py one electron orbital

25. Chap 5: Angular Momentum Eigen functions d-orbital L dzz ~ R32( r ) Y20( L r  r  L  dx2 -y2 ~ R32{Y22 +Y2-2}~cos(2) dxy ~ R32{Y22 - Y2-2}~ sin(2)

26. Chap 5: Aufbau Process; Atomic Ground State Electron Configuration P D P D P P P P P D

27. Chap 5: Periodic Table reflects the Electron Configuration; Atomic Properties Alkali Metals Rare Gases 2 8 Noble Metals Transition Metals 8 18 18 Halogens Alkaline earth Lanthanides Actinides

28. Chap 5: Periodic Table reflects the Electron Configuration: ionization Energies X X+ + e E=E(X+) - E(X)=IE1 and X+X2+ + e E=E(X2+) - E(X+)=IE2

29. Chap 5: Periodic Table reflects the Electron Configuration: ionization Energies

30. Chap 5: Periodic Table reflects the Electron Configuration: Electron Affinity X + e X-E = E(X-) - E(X)= - EA

31. Chap. 5:H-atom and One electron Ions Eigen Funct, and Values nlm(r,,) = Rnl( r )Ylm(: Eigen Function : Energy Eigen Values (same as Bohr) Principal quantum number Angular momentum quantum number Magnetic quantum number

32. Chap 5: Quantum Mechanical Energy levels The Energy Eigen Values are Independent of l and m! Therefore each n level has n, l levels (0,1, ..n-1), each with (2l+1) states and is therefore n2 degenerate. Consequently each (nl) level has (2l+1) m-states and is (2l+ 1) degenerate

33. Chap 5: Average QM distance of the electron from the Nucleus Chap 5: Average QM distance of the electron from the Nucleus The Average Quantum Mechanicaldistance between the Nucleus and the electron:<rnl> = (a0n2/Z){1+(1/2)[1- l(l+1)2/n2]}.For n>>1 and n~l reduces tothe Bohr Modelrn= (a0n2/Z) is also the most probable distance of the electron from the nucleusVisualizing atomic orbital probability density:http://www.phy.davidson.edu/stuhome/cabell_f/density.html The Average Quantum Mechanicaldistance between the Nucleus and the electron:<rnl> = (a0n2/Z){1+(1/2)[1- l(l+1)2/n2]}.For n>>1 and n~l reduces tothe Bohr Modelrn= (a0n2/Z) is also the most probable distance of the electron from the nucleusVisualizing atomic orbital probability density:http://www.phy.davidson.edu/stuhome/cabell_f/density.html