4 Quantum Numbers for the Hydrogen Atom by kqm58610

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									                             The End                                                       4 Quantum Numbers for the
• All homework solutions are now available                                                      Hydrogen Atom
• Check CULearn for all scores (HW, Clickers, Exams)                       For hydrogenic atoms (one electron), energy levels only depend
                                                                           on n and we find the same formula as Bohr: En = −Z 2 ER / n2
   – Any requested changes are due today
                                                                           For multielectron atoms the energy also depends on ℓ.
• Final exam is Tuesday, December 15, 2009                                 There is a shorthand for giving the n and ℓ values.
  4:30 pm – 7:00 pm in G125 (this room)
                                                                                            2p                   Different letters correspond
                                                                                                                 to different values of ℓ
• Bring Calculator! Arrive on time, check seating rows.                                                          s     p     d      f     g       h…
                                                                             n=2                 ℓ=1
• Additional review help (in the Physics Help Room),                                                       l=   0      1     2      3     4       5
  today    Friday 1:30 – 4:00 pm
                                                                            The electrons also have another quantum number for
• I have to travel to Brookhaven National Lab next week
                                                                            their spin (ms = +/- ½).
  and so the exam will be proctored.




 Clicker question                                                                                 Hydrogen energy levels
  n = 1, 2, 3, … = Principal Quantum Number           En = −Z ER / n
                                                             2         2
                                                                                      ℓ=0        ℓ=1     ℓ=2
  ℓ = 0, 1, 2, … n-1 = angular momentum quantum number                                 (s)        (p)    (d)
    = s, p, d, f, …           L = l(l + 1)h                                n=3                                       E3 = − ER / 32 = −1.5 eV
                                                                                           3s     3p      3d
 m = 0, ±1, ±2, … ±ℓ     is the z-component of
                                                  Lz = mh
                         angular momentum                                  n=2                          E2 = − ER / 22 = −3.4 eV
                                                                                           2s     2p
 ms = ±1/2               from the spin contribution
A hydrogen atom electron is excited to an energy of −13.6/9 eV.
How many different quantum states could the electron be in?
                      E = −13.6/9 eV means n2 = 9 so n = 3
  A.   1
  B.   4              For n = 3, ℓ = 0,1,2 and thus
  C.   9              l = 0, m = 0                      x 2 for spin!
  D.   18             l = 1, m = -1, 0, +1                                  n=1            1s   E1 = − ER = −13.6 eV
  E.   more than 18   l = 2, m = -2, -1, 0, +1, +2




  Probability versus radius: P(r) = |Rnl(r)|2r 2                                 Higher n        average r bigger
                                                                                                 more spherical shells stacked within each other
 In spherical coordinates,                                                                       more nodes as function of r
 the volume element has
 an r2 term so probability                                                 n=1
 increases with r2.                                                        l=0

                                                                                                                      Probability finding
 Most probable radius for                                                                                             electron as function of r
 the n = 1 state is at the                                                 n=2
 Bohr radius aB                                                            l=0

 Most probable radius for all
 ℓ=n-1 states (those with only                                             n=3
 one peak) is at the radius                                                l=0
 predicted by Bohr (n2 aB).
                                                                                  0.05nm




                                                                                                        Radius (units of Bohr radius, a0)




                                                                                                                                                       1
Shapes of hydrogen wave functions:                                   Shapes of hydrogen wave functions:

ψ nlm (r ,θ , φ ) = Rnl (r ) f lm (θ ) g m (φ )                      ψ nlm (r ,θ , φ ) = Rnl (r ) f lm (θ ) g m (φ )
                                                                     l=1, called p-orbitals: angular dependence (n=2)
Look at s-orbitals (l=0): no angular dependence
                                                                            l=1, m=0: pz = dumbbell shaped.
                  n=1                           n=2
                                                                             l=1, m=-1: bagel shaped around z-axis (traveling wave)
                                                                             l=1, m=+1

                                                                                                       1     r − r / 2 a0 ⎛   3       ⎞
                                                                     n = 2, l = 1, m = 0 ⎯ ψ 211 =
                                                                                         ⎯→                     e         ⎜−    cos θ ⎟
                                                                                                          3               ⎜  4π       ⎟
                                                                                                     2 6 a0 a 0           ⎝           ⎠
                                                                                                         1 r − r / 2 a0 ⎛   3           ⎞   w/time dependence

                                                                     n = 2, l = 1, m = 1 ⎯ ψ 211 =
                                                                                         ⎯→                   e         ⎜−    sin θe iφ ⎟       eimφ+iΕt/h
                                                                                                             3          ⎜  8π           ⎟
                                                                                                     2 6a a 00          ⎝               ⎠

                                                                     Superposition applies:
                                                                      px=superposition (addition of m=-1 and m=+1)    Dumbbells
                                                                      py=superposition (subtraction of m=-1 and m=+1) (chemistry)




                                                                     Schrodinger finds quantization of energy and angular momentum:
           Physics vs Chemistry view of orbits:                         n=1, 2, 3 …       l=0, 1, 2, 3 (restricted to 0, 1, 2 … n-1)
      2p wave functions                   Dumbbell Orbits              En = − E1 / n 2                       | L |= l (l + 1) h
      (Physics view)                      (chemistry)
      (n=2, l=1)                                                     How does Schrodinger compare to what Bohr thought?
                                                                                                                    same
                                                                     I. The energy of the ground state solution is ________
                                                                     II. The angular momentum of the ground state solution is different
                                                                                                                              _______
                                           px                        III. The location of the electron is different
                                                                                                          _______
                                                      pz   py
   m=1                  m=-1
            m=0                     px=superposition                 a. same, same, same
                                    (addition of m=-1 and m=+1)      b. same, same, different                    Bohr got energy right,
                                    py=superposition                 c. same, different, different               but he said angular
                                    (subtraction of m=-1 and m=+1)   d. different, same, different               momentum L=nh, and
                                                                     e. different, different, different          thought the electron was
                                                                                                                 a point particle orbiting
                                                                                                                 around nucleus.




     Can Schrodinger make sense of the periodic table?
                                                                                          Schrödinger’s Cat
                                                                     A radioactive sample has a 50% chance of emitting an alpha particle.
                                                                     If it decays, a Geiger detector triggers the release of poison killing a
                                                                     cat in the box. Before opening the box, the cat is in a superposition
                                                                     of wave functions: ψ = 1ψ           + 1ψ
                                                                                                     2   dead      2   alive

                                                                      When does the wave function collapse to either dead or alive?
                                                                       No clear agreement. Interesting physics/philosophical question.




       Schrodinger’s solution for multi-electron atoms
Need to account for all the interactions among the electrons
Must solve for all electrons at once! (use matrices)
     V (for q1) = kqnucleus*q1/rn-1 + kq2q1/r2-1 + kq3q1/r3-1 + ….




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Good luck on
the final exam!


   Thank you for all your
attention and hard work this
         semester.




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